Introduction
Let q q q F q \mathbb{F}_q F q q q q S L ∞ ( F q ) = ⋃ n S L n ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q)=\bigcup_{n}\mathrm{SL}_n(\mathbb{F}_q) SL ∞ ( F q ) = ⋃ n SL n ( F q ) S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) A A A F q \mathbb{F}_q F q A i j = δ i j A_{ij}=\delta_{ij} A ij = δ ij i , j ∈ N i,j\in\mathbb{N} i , j ∈ N det A = 1 \det A=1 det A = 1
Our main goal is to classify the invariant random subgroups of the group S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) G G G Sub ( G ) \operatorname{Sub}(G) Sub ( G ) G G G Sub ( G ) \operatorname{Sub}(G) Sub ( G ) invariant random subgroup of G G G Sub ( G ) \operatorname{Sub}(G) Sub ( G ) δ N \delta_N δ N N ◃ G N\triangleleft G N ◃ G G G G G G G IRS ( G ) \operatorname{IRS}(G) IRS ( G ) G G G ergodic if it is an extreme point of the convex compact set IRS ( G ) \operatorname{IRS}(G) IRS ( G ) [1 ] . See e.g. [10 ] for a survey.
We show that the invariant random subgroups of S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) V = ∏ n ∈ N F q V=\prod_{n\in\mathbb{N}}\mathbb{F}_q V = ∏ n ∈ N F q F q \mathbb{F}_q F q S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) V V V A ⋅ v = A v A\cdot v=Av A ⋅ v = A v V ^ \widehat{V} V V V V S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) A ⋅ φ = ( A − 1 ) t φ A\cdot\varphi=(A^{-1})^{t}\varphi A ⋅ φ = ( A − 1 ) t φ
For each k , ℓ ∈ N 0 = N ∪ { 0 } k,\ell\in\mathbb{N}_0=\mathbb{N}\cup\{0\} k , ℓ ∈ N 0 = N ∪ { 0 } V k , ℓ = V k × V ^ ℓ V^{k,\ell}=V^k\times\widehat{V}^{\ell} V k , ℓ = V k × V ℓ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ Haar k , ℓ \operatorname{Haar}^{k,\ell} Haar k , ℓ Stab S L ∞ ( F q ) : V k , ℓ → Sub ( S L ∞ ( F q ) ) \operatorname{Stab}_{\mathrm{SL}_{\infty}(\mathbb{F}_q)}:V^{k,\ell}\to\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) Stab SL ∞ ( F q ) : V k , ℓ → Sub ( SL ∞ ( F q )) S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) μ k , ℓ \mu_{k,\ell} μ k , ℓ Haar k , ℓ \operatorname{Haar}^{k,\ell} Haar k , ℓ μ 0 , 0 = δ S L ∞ ( F q ) \mu_{0,0}=\delta_{\mathrm{SL}_{\infty}(\mathbb{F}_q)} μ 0 , 0 = δ SL ∞ ( F q ) μ ∞ = δ { 1 } \mu_{\infty}=\delta_{\{1\}} μ ∞ = δ { 1 }
Theorem 1.1. The ergodic invariant random subgroups of the group S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
IRS e r g o d i c ( S L ∞ ( F q ) ) = { μ k , ℓ : k , ℓ ∈ N 0 } ∪ { μ ∞ } . \operatorname{IRS}_{\mathrm{ergodic}}(\mathrm{SL}_{\infty}(\mathbb{F}_q))=\{\mu_{k,\ell}:k,\ell\in\mathbb{N}_0\}\cup\{\mu_{\infty}\}. IRS ergodic ( SL ∞ ( F q )) = { μ k , ℓ : k , ℓ ∈ N 0 } ∪ { μ ∞ } . These are pairwise distinct.
Theorem 1.1 has been conjectured by Simon Thomas [28 ] . One reading of Theorem 1.1 is as a far-reaching generalization of the fact that the group S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) IRS e r g o d i c ( S L ∞ ( F q ) ) \operatorname{IRS}_{\mathrm{ergodic}}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) IRS ergodic ( SL ∞ ( F q )) μ ∞ \mu_{\infty} μ ∞ μ k , ℓ \mu_{k,\ell} μ k , ℓ IRS ( S L ∞ ( F q ) ) \operatorname{IRS}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) IRS ( SL ∞ ( F q )) [15 ] .
There is an alternative interpretation of each invariant random subgroup μ k , ℓ \mu_{k,\ell} μ k , ℓ Gr k ( V ) × Gr ℓ ( V ^ ) \operatorname{Gr}_k(V) \times\operatorname{Gr}_{\ell}(\widehat{V}) Gr k ( V ) × Gr ℓ ( V ) § 7.5.2 for details).
It is interesting to note a difference between the topological and measurable dynamics of the S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) [31 ] has classified the minimal closed orbits (i.e. uniformly recurrent subgroups ) for this the action. The only such orbits are the trivial ones, namely the singleton { N } \{N\} { N } N = { I } N=\{I\} N = { I } N = S L ∞ ( F q ) N=\mathrm{SL}_{\infty}(\mathbb{F}_q) N = SL ∞ ( F q )
Example. It is a consequence of Theorem 1.1 that the only meaningful information that can be “extracted” given a random tuple x = ( v 1 , … , v k , φ 1 , … , φ ℓ ) ∈ V k , ℓ x=(v^1,\ldots,v^k,\varphi^1,\ldots,\varphi^\ell)\in V^{k,\ell} x = ( v 1 , … , v k , φ 1 , … , φ ℓ ) ∈ V k , ℓ k k k ℓ \ell ℓ x x x v ∈ V 1 , 0 v\in V^{1,0} v ∈ V 1 , 0 Haar 1 , 0 \operatorname{Haar}^{1,0} Haar 1 , 0
{ A ∈ S L ∞ ( F q ) : A v = v } = { A ∈ S L ∞ ( F q ) : A v = λ v for some λ ∈ F q } . \{A\in\mathrm{SL}_{\infty}(\mathbb{F}_q):Av=v\}=\{A\in\mathrm{SL}_{\infty}(\mathbb{F}_q):Av=\lambda v\text{ for some }\lambda\in\mathbb{F}_q\}. { A ∈ SL ∞ ( F q ) : A v = v } = { A ∈ SL ∞ ( F q ) : A v = λ v for some λ ∈ F q } . We also consider the finitary general linear group G L ∞ ( F q ) = ⋃ n G L n ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q)=\bigcup_n\mathrm{GL}_n(\mathbb{F}_q) GL ∞ ( F q ) = ⋃ n GL n ( F q ) S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) q − 1 q-1 q − 1 K ≤ F q × K\leq\mathbb{F}_q^\times K ≤ F q × k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 x ∈ V k , ℓ x\in V^{k,\ell} x ∈ V k , ℓ
Stab G L ∞ ( F q ) K ( x ) = { g ∈ G L ∞ ( F q ) : g x = x and det g ∈ K } . \operatorname{Stab}^{K}_{\mathrm{GL}_{\infty}(\mathbb{F}_q)}(x)=\{g\in\mathrm{GL}_{\infty}(\mathbb{F}_q):gx=x\text{ and }\det g\in K\}. Stab GL ∞ ( F q ) K ( x ) = { g ∈ GL ∞ ( F q ) : g x = x and det g ∈ K } . Pushing forward the measure Haar k , ℓ \operatorname{Haar}^{k,\ell} Haar k , ℓ Stab G L ∞ ( F q ) K \operatorname{Stab}^{K}_{\mathrm{GL}_{\infty}(\mathbb{F}_q)} Stab GL ∞ ( F q ) K Sub ( G L ∞ ( F q ) ) \operatorname{Sub}(\mathrm{GL}_{\infty}(\mathbb{F}_q)) Sub ( GL ∞ ( F q )) μ k , ℓ K \mu^K_{k,\ell} μ k , ℓ K μ ∞ = δ { I } \mu_{\infty}=\delta_{\{I\}} μ ∞ = δ { I }
Theorem 1.2. The ergodic invariant random subgroups of the group G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q )
{ μ k , ℓ K : k , ℓ ∈ N 0 and K ≤ F q × is a subgroup } ∪ { μ ∞ } . \{\mu^K_{k,\ell}:k,\ell\in\mathbb{N}_0\text{ and }K\leq\mathbb{F}_q^\times\text{ is a subgroup}\}\cup\{\mu_{\infty}\}. { μ k , ℓ K : k , ℓ ∈ N 0 and K ≤ F q × is a subgroup } ∪ { μ ∞ } . These are pairwise distinct.
Classification of invariant measures on V k , ℓ V^{k,\ell} V k , ℓ
As part of the proof of Theorem 1.1 we are led to classify the S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ
Theorem 1.3. Let k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 μ \mu μ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ μ \mu μ x = ( v 1 , … , v k , φ 1 , … , φ ℓ ) ∈ V k , ℓ x=(v^1,\ldots,v^k,\varphi^1,\ldots,\varphi^\ell)\in V^{k,\ell} x = ( v 1 , … , v k , φ 1 , … , φ ℓ ) ∈ V k , ℓ { v 1 , … , v k } \{v^1,\ldots,v^k\} { v 1 , … , v k } { φ 1 , … , φ ℓ } \{\varphi^1,\ldots,\varphi^\ell\} { φ 1 , … , φ ℓ } μ = Haar k , ℓ \mu=\operatorname{Haar}^{k,\ell} μ = Haar k , ℓ
The proof of the above Theorem 1.3 depends on the pointwise ergodic theorem for actions of amenable groups and on various linear-algebraic counting problems over the field F q \mathbb{F}_q F q Theorem 7.11 for the full classification of all S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ Gr k ( V ) × Gr ℓ ( V ^ ) \operatorname{Gr}_k(V)\times\operatorname{Gr}_{\ell}(\widehat{V}) Gr k ( V ) × Gr ℓ ( V )
Remark 1.4. In the case where q q q k = 0 k=0 k = 0 ℓ = 0 \ell=0 ℓ = 0 [2 Theorem 4.1] with a different proof. While it is likely that the proof method of [2 ] can be extended also when q q q k k k ℓ \ell ℓ
Thoma characters
Traces and characters in the sense of Thoma play an important role in our work. A trace on a given discrete group G G G φ : G → C \varphi:G\to\mathbb{C} φ : G → C φ ( e ) = 1 \varphi(e)=1 φ ( e ) = 1 π \pi π G G G tr π dim π \frac{\operatorname{tr}\pi}{\dim\pi} d i m π tr π π \pi π Tr ( G ) \operatorname{Tr}(G) Tr ( G ) G G G [4 ] for more details.
The extreme points of the convex set Tr ( G ) \operatorname{Tr}(G) Tr ( G ) Thoma characters of G G G Ch ( G ) \operatorname{Ch}(G) Ch ( G ) SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) r ∈ N 0 r\in\mathbb{N}_0 r ∈ N 0 χ r : SL ∞ ( F q ) → [ 0 , 1 ] \chi_r:\operatorname{SL}_{\infty}(\mathbb{F}_q)\to[0,1] χ r : SL ∞ ( F q ) → [ 0 , 1 ]
χ k ( g ) = q − r ⋅ rk ( g − I ) ∀ g ∈ SL ∞ ( F q ) . \chi_k(g)=q^{-r\cdot\operatorname{rk}(g-I)}\qquad\forall g\in\operatorname{SL}_{\infty}(\mathbb{F}_q). χ k ( g ) = q − r ⋅ rk ( g − I ) ∀ g ∈ SL ∞ ( F q ) . Note that the rank of the matrix g − I g-I g − I g ∈ GL ∞ ( F q ) g\in\operatorname{GL}_{\infty}(\mathbb{F}_q) g ∈ GL ∞ ( F q ) χ ∞ = δ I \chi_{\infty}=\delta_I χ ∞ = δ I
Theorem 1.5 (Skudlarek [27 ] ). Ch ( SL ∞ ( F q ) ) = { χ r : r ∈ N 0 } ∪ { χ ∞ } \operatorname{Ch}(\operatorname{SL}_{\infty}(\mathbb{F}_q))=\{\chi_r:r\in\mathbb{N}_0\}\cup\{\chi_{\infty}\} Ch ( SL ∞ ( F q )) = { χ r : r ∈ N 0 } ∪ { χ ∞ }
We point out that our proof of Theorem 1.1 relies crucially and in several ways on the a priori knowledge of Skudlarek’s character classification for SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
To an invariant random subgroup μ \mu μ G G G Vershik trace χ μ \chi_\mu χ μ
χ μ ( g ) = μ ( { H ∈ Sub ( G ) : g ∈ H } ) ∀ g ∈ G . \chi_\mu(g)=\mu(\{H\in\operatorname{Sub}(G):g\in H\})\qquad\forall g\in G. χ μ ( g ) = μ ({ H ∈ Sub ( G ) : g ∈ H }) ∀ g ∈ G . The Vershik trace χ μ \chi_\mu χ μ μ k , ℓ \mu_{k,\ell} μ k , ℓ SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) χ r \chi_r χ r r = k + ℓ r=k+\ell r = k + ℓ
The Vershik trace map IRS ( G ) → Tr ( G ) \operatorname{IRS}(G)\to\operatorname{Tr}(G) IRS ( G ) → Tr ( G ) G G G
The finitary symmetric and alternating groups S ∞ = Sym f i n ( N ) S_{\infty}=\operatorname{Sym}_{\mathrm{fin}}(\mathbb{N}) S ∞ = Sym fin ( N ) A ∞ = Alt f i n ( N ) A_{\infty}=\operatorname{Alt}_{\mathrm{fin}}(\mathbb{N}) A ∞ = Alt fin ( N ) § 4.3 for a detailed discussion). Even so, the Vershik trace map for these two groups is “close enough” to be a bijection and carries a great deal of information. We will make essential use of this behavior in the proof of Theorem 1.1, as the alternating group A ∞ A_{\infty} A ∞ SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
In the case of the group SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
Other groups and future directions
The two groups A ∞ A_{\infty} A ∞ S ∞ S_{\infty} S ∞ q = 1 q=1 q = 1 SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) [29 ] . The invariant random subgroups were classified by Vershik in [35 ] . The calculation of the Vershik traces and the description of these as a weighted average over the set of Thoma characters are done in [32 ] . These results will be described precisely in § 4.3 .
For the permutation groups S ∞ S_{\infty} S ∞ A ∞ A_{\infty} A ∞ [32 ] ). This turns out to be very far from the situation for the group S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
The fact that S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) [4 Chapter 11] for details, though this point of view will not be pursued in the sequel). This has led Gorin, Kerov, and Vershik to consider the group G L B ( F q ) \mathrm{GLB}(\mathbb{F}_q) GLB ( F q ) F q \mathbb{F}_q F q [11 ] they show that G L B ( F q ) \mathrm{GLB}(\mathbb{F}_q) GLB ( F q ) ∞ \infty ∞ G L B ( F q ) \mathrm{GLB}(\mathbb{F}_q) GLB ( F q )
The characters of several other families of infinite-dimensional groups are known, opening the door to a classification of their invariant random subgroups. These include the infinite-dimensional real Lie groups U ∞ ( R ) U_{\infty}(\mathbb{R}) U ∞ ( R ) [37 , 6 , 36 ] , S O ∞ ( R ) SO_{\infty}(\mathbb{R}) S O ∞ ( R ) S p ∞ ( R ) Sp_{\infty}(\mathbb{R}) S p ∞ ( R ) [7 ] . Note that S ∞ S_{\infty} S ∞ G L ∞ ( R ) \mathrm{GL}_{\infty}(\mathbb{R}) GL ∞ ( R ) [17 ] and it may be of interest to classify their invariant random subgroups.
Lastly, we mention the infinite-dimensional Lie groups over a finite field, i.e. U ∞ ( F q ) U_{\infty}(\mathbb{F}_q) U ∞ ( F q ) S O ∞ ( F q ) SO_{\infty}(\mathbb{F}_q) S O ∞ ( F q ) S p ∞ ( F q ) Sp_{\infty}(\mathbb{F}_q) S p ∞ ( F q ) q q q [24 ] . Thomas Thomas [31 ] has shown that S p ∞ ( F 2 n ) Sp_{\infty}(\mathbb{F}_{2^n}) S p ∞ ( F 2 n )
Permutation stability
A discrete group G G G permutation stable if every almost-action of G G G
The connection between invariant random subgroups and permutation stability for amenable groups was established by Becker, Lubotzky and Thom [5 ] . The precise statement is the following. An invariant random subgroup of G G G co-sofic if it is a limit in the weak-* topology of invariant random subgroups supported on finite index subgroups.
Theorem 1.6 ([5 ] ). A finitely generated amenable group G G G G G G
The work [5 ] is one of the motivations for the study of invariant random subgroups of amenable groups. For example, in [23 ] , the authors use Theorem 1.6 to show that the uncountable family of the B. H. Neumann groups are permutation stable. The analysis of the invariant random subgroups of these groups crucially depends on the above-mentioned classification of the invariant random subgroups of S ∞ S_{\infty} S ∞ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) Theorem 1.1 can be used to establish the permutation stability of an analogous family of groups.
Theorem 1.7. Let P = ( n i ) \mathcal{P}=(n_i) P = ( n i ) n 1 ≥ 3 n_1\geq3 n 1 ≥ 3 ⨂ i P S L n i ( F q ) \bigotimes_i \mathrm{PSL}_{n_i}(\mathbb{F}_q) ⨂ i PSL n i ( F q )
The abstract definition of the diagonal product operation is involved, and we do not reproduce or invoke it here. See e.g. [9 ] where the same group is defined and shown to be Hilbert–Schmidt stable. We only consider here a concrete and hands-on construction of the group ⨂ i PSL n i ( F q ) \bigotimes_i \operatorname{PSL}_{n_i}(\mathbb{F}_q) ⨂ i PSL n i ( F q ) ∏ i PSL n i ( F q ) \prod_i \operatorname{PSL}_{n_i}(\mathbb{F}_q) ∏ i PSL n i ( F q ) ∏ i A n i \prod_i A_{n_i} ∏ i A n i § 9 for details on all this.
Proof outline
Our strategy to prove Theorem 1.1 is as follows. Fix a non-trivial ergodic invariant random subgroup μ \mu μ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 fixed-point map
FP : Sub ( S L ∞ ( F q ) ) ⟶ V k , ℓ \operatorname{FP}:\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q))\longrightarrow V^{k,\ell} FP : Sub ( SL ∞ ( F q )) ⟶ V k , ℓ which associates to a μ \mu μ H H H FP ( H ) ∈ V k , ℓ \operatorname{FP}(H)\in V^{k,\ell} FP ( H ) ∈ V k , ℓ H = Stab S L ∞ ( F q ) ( FP ( H ) ) H=\operatorname{Stab}_{\mathrm{SL}_{\infty}(\mathbb{F}_q)}(\operatorname{FP}(H)) H = Stab SL ∞ ( F q ) ( FP ( H )) H H H μ \mu μ FP ( H ) \operatorname{FP}(H) FP ( H ) Haar k , ℓ \operatorname{Haar}^{k,\ell} Haar k , ℓ
μ = ( Stab S L ∞ ( F q ) ) ∗ FP ∗ μ = ( Stab S L ∞ ( F q ) ) ∗ Haar k , ℓ = μ k , ℓ \mu=\bigl(\operatorname{Stab}_{\mathrm{SL}_{\infty}(\mathbb{F}_q)}\bigr)_*\operatorname{FP}_*\mu=\bigl(\operatorname{Stab}_{\mathrm{SL}_{\infty}(\mathbb{F}_q)}\bigr)_*\operatorname{Haar}^{k,\ell}=\mu_{k,\ell} μ = ( Stab SL ∞ ( F q ) ) ∗ FP ∗ μ = ( Stab SL ∞ ( F q ) ) ∗ Haar k , ℓ = μ k , ℓ as required. To construct the map FP \operatorname{FP} FP A ∞ A_\infty A ∞
Let us provide a bit more detail. A finitary basis is a basis for the vector space ⨁ n ∈ N F q \bigoplus_{n\in\mathbb{N}}\mathbb{F}_q ⨁ n ∈ N F q f f f i f : A ∞ ↪ S L ∞ ( F q ) i_f:A_\infty\hookrightarrow\mathrm{SL}_{\infty}(\mathbb{F}_q) i f : A ∞ ↪ SL ∞ ( F q ) f f f A ∞ ( f ) A_\infty(f) A ∞ ( f ) Sub ( S L ∞ ( F q ) ) → Sub ( A ∞ ( f ) ) \operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q))\to\operatorname{Sub}(A_\infty(f)) Sub ( SL ∞ ( F q )) → Sub ( A ∞ ( f )) ν f \nu_f ν f A ∞ ( f ) A_\infty(f) A ∞ ( f ) μ \mu μ
Identify the invariant random subgroup ν f \nu_f ν f f f f
Use the joint distribution of the measures ν f \nu_f ν f f f f f f f k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 FP : Sub ( S L ∞ ( F q ) ) → V k , ℓ \operatorname{FP}:\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q))\to V^{k,\ell} FP : Sub ( SL ∞ ( F q )) → V k , ℓ
Show that the map FP \operatorname{FP} FP μ = ( Stab S L ∞ ( F q ) ) ∗ FP ∗ μ \mu=\bigl(\operatorname{Stab}_{\mathrm{SL}_{\infty}(\mathbb{F}_q)}\bigr)_*\operatorname{FP}_*\mu μ = ( Stab SL ∞ ( F q ) ) ∗ FP ∗ μ FP ∗ μ = Haar k , ℓ \operatorname{FP}_*\mu=\operatorname{Haar}^{k,\ell} FP ∗ μ = Haar k , ℓ
Before elaborating further, we need to sketch the invariant random subgroup structure of the group A ∞ A_\infty A ∞ α = ( α i ) i ∈ N \alpha=(\alpha_i)_{i\in\mathbb{N}} α = ( α i ) i ∈ N N \mathbb{N} N α \alpha α N \mathbb{N} N i i i α i \alpha_i α i H H H A ∞ A_\infty A ∞ H ∈ Sub ( A ∞ ) H\in\operatorname{Sub}(A_\infty) H ∈ Sub ( A ∞ ) A ∞ A_\infty A ∞ A ∞ A_\infty A ∞ § 4.3 ).
By the above classification, the invariant random subgroup ν f \nu_f ν f μ \mu μ f f f N \mathbb{N} N σ ∈ A ∞ ( f ) \sigma\in A_\infty(f) σ ∈ A ∞ ( f ) μ \mu μ H H H σ \sigma σ
For the invariant random subgroup μ k , ℓ \mu_{k,\ell} μ k , ℓ k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 H ∈ Sub ( S L ∞ ( F q ) ) H\in\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) H ∈ Sub ( SL ∞ ( F q )) x = ( v , φ ) = ( v 1 , … , v k , φ 1 , … , φ ℓ ) ∈ V k , ℓ x=(v,\varphi)=(v^1,\ldots,v^k,\varphi^1,\ldots,\varphi^\ell)\in V^{k,\ell} x = ( v , φ ) = ( v 1 , … , v k , φ 1 , … , φ ℓ ) ∈ V k , ℓ f f f F q r \mathbb{F}_q^r F q r r = k + ℓ r=k+\ell r = k + ℓ n ∈ N n\in\mathbb{N} n ∈ N
x n = ( v n 1 , … , v n k , φ n 1 , … , φ n ℓ ) ∈ F q r . (1) x_n=\left(v_n^1,\ldots,v_n^k,\varphi_n^1,\ldots,\varphi_n^\ell\right)\in\mathbb{F}_q^r. \tag*{(1)} x n = ( v n 1 , … , v n k , φ n 1 , … , φ n ℓ ) ∈ F q r . (1) Here the subscripts v n v_n v n φ n \varphi_n φ n n n n v v v φ \varphi φ f f f σ \sigma σ x x x σ \sigma σ H H H ν f \nu_f ν f A ∞ A_\infty A ∞ N \mathbb{N} N q r q^r q r q − r q^{-r} q − r
Let us return to the analysis of a general ergodic invariant random subgroup μ \mu μ μ \mu μ μ k , ℓ \mu_{k,\ell} μ k , ℓ f f f ν f \nu_f ν f A ∞ A_\infty A ∞ q r q^r q r r ∈ N 0 r \in\mathbb{N}_0 r ∈ N 0 μ \mu μ H ∈ Sub ( SL ∞ ( F q ) ) H \in\operatorname{Sub}(\operatorname{SL}_\infty(\mathbb{F}_q)) H ∈ Sub ( SL ∞ ( F q )) k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 x ∈ V k , ℓ x \in V^{k,\ell} x ∈ V k , ℓ k + ℓ = r k+\ell=r k + ℓ = r (1) for all finitary bases f f f H ↦ x H \mapsto x H ↦ x
Organization
In § 2 we define the finitary special and general linear groups and consider their properties and subgroup structure. This section contains all the algebraic information to be used throughout the work. In Section 3 we prove abstractly that any action of GL ∞ ( F q ) \operatorname{GL}_\infty(\mathbb{F}_q) GL ∞ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ
The discussion of the main ideas of the work begins in earnest in § 4 , where we recall the definitions of invariant random subgroups and traces, generally speaking and for the finitary alternating group A ∞ A_\infty A ∞ § 5 we use Skudlarek’s classification of the characters of the group SL ∞ ( F q ) \operatorname{SL}_\infty(\mathbb{F}_q) SL ∞ ( F q ) § 5.2 ) and implement stage (1) of the strategy outlined in § 1.5 (§ 5.3 ). In § 6 we introduce the theory of the “coloring equivalence relation”, a random coloring given by an invariant random subgroup μ \mu μ GL ∞ ( F q ) \operatorname{GL}_\infty(\mathbb{F}_q) GL ∞ ( F q )
§ 7 is standalone and involves ergodic theory. In that section we prove the measure classification result Theorem 1.3 . As a corollary, we classify the SL ∞ ( F q ) \operatorname{SL}_\infty(\mathbb{F}_q) SL ∞ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ § 7.5 ). In § 8 we define the map FP and finish the proof of our main results Theorems 1.1 and Theorem 1.2. This section contains a discussion of topological structure of the space of ergodic invariant random subgroups of SL ∞ ( F q ) \operatorname{SL}_\infty(\mathbb{F}_q) SL ∞ ( F q ) GL ∞ ( F q ) \operatorname{GL}_\infty(\mathbb{F}_q) GL ∞ ( F q ) § 9 we recall the notion of permutation stability and use the classification of IRS ( SL ∞ ( F q ) ) \operatorname{IRS}(\operatorname{SL}_\infty(\mathbb{F}_q)) IRS ( SL ∞ ( F q )) Theorem 1.7 , that is, show that the diagonal products ⨂ PSL n i ( F q ) \bigotimes\operatorname{PSL}_{n_i}(\mathbb{F}_q) ⨂ PSL n i ( F q )
Standing assumptions and notations
All our probability measures are Borel.
We let N 0 = N ∪ { 0 } \mathbb{N}_0=\mathbb{N}\cup\{0\} N 0 = N ∪ { 0 } N > n \mathbb{N}_{>n} N > n n n n F q \mathbb{F}_q F q q q q 0 n , k 0_{n,k} 0 n , k M n , k ( F q ) M_{n,k}(\mathbb{F}_q) M n , k ( F q )
We use the following two standard asymptotic notations. The notation o ( 1 ) o(1) o ( 1 ) n → ∞ n\to\infty n → ∞ O ( n ) O(n) O ( n ) f f f ∣ f ( n ) ∣ ≤ c n |f(n)|\le cn ∣ f ( n ) ∣ ≤ c n c > 0 c>0 c > 0 n n n F q \mathbb{F}_q F q
Grant support
AL is supported by the ISF grants 1788/22 and 3553/26 and the BSF grant 2022105.
AI statement
Most of the research was done prior to the year 2026. No AI was used towards the research. AI was used to look up relevant references, including properties of Singer cycles.
The finitary special linear group
Let q q q F q \mathbb{F}_q F q S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q )
The groups S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q )
Consider the F q \mathbb{F}_q F q ⨁ n ∈ N F q \bigoplus_{n\in\mathbb{N}}\mathbb{F}_q ⨁ n ∈ N F q e = ( e n ) n = 1 ∞ e=(e^n)_{n=1}^{\infty} e = ( e n ) n = 1 ∞ finitary basis for the F q \mathbb{F}_q F q ⨁ n ∈ N F q \bigoplus_{n\in\mathbb{N}}\mathbb{F}_q ⨁ n ∈ N F q f = ( f n ) n = 1 ∞ f=(f^n)_{n=1}^{\infty} f = ( f n ) n = 1 ∞ f n = e n f^n=e^n f n = e n n ∈ N n\in\mathbb{N} n ∈ N
The finitary general linear group G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) F q \mathbb{F}_q F q ⨁ n ∈ N F q \bigoplus_{n\in\mathbb{N}}\mathbb{F}_q ⨁ n ∈ N F q G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) det : G L ∞ ( F q ) → F q × \det:\mathrm{GL}_{\infty}(\mathbb{F}_q)\to\mathbb{F}_q^{\times} det : GL ∞ ( F q ) → F q × S L ∞ ( F q ) = ker det \mathrm{SL}_{\infty}(\mathbb{F}_q)=\ker\det SL ∞ ( F q ) = ker det S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) F q × \mathbb{F}_q^{\times} F q ×
An element of the group G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) A A A F q \mathbb{F}_q F q N \mathbb{N} N A i j = δ i j A_{ij}=\delta_{ij} A ij = δ ij ( i , j ) ∈ N × N (i,j)\in\mathbb{N}\times\mathbb{N} ( i , j ) ∈ N × N S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
Let f f f I ⊂ N I\subset\mathbb{N} I ⊂ N S L I ( f ) \mathrm{SL}_{I}(f) SL I ( f ) g ∈ S L ∞ ( F q ) g\in\mathrm{SL}_{\infty}(\mathbb{F}_q) g ∈ SL ∞ ( F q ) span F q { f i : i ∈ I } \operatorname{span}_{\mathbb{F}_q}\{f^i:i\in I\} span F q { f i : i ∈ I } span F q { f i : i ∈ N ∖ I } \operatorname{span}_{\mathbb{F}_q}\{f^i:i\in\mathbb{N}\setminus I\} span F q { f i : i ∈ N ∖ I } S L I ( f ) \mathrm{SL}_{I}(f) SL I ( f ) supported on the index set I I I f f f . The group S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
S L ∞ ( F q ) = lim I ⊂ N , ∣ I ∣ < ∞ S L I ( f ) . \mathrm{SL}_{\infty}(\mathbb{F}_q)=\lim_{\substack{I\subset\mathbb{N},\,|I|<\infty}}\mathrm{SL}_{I}(f). SL ∞ ( F q ) = I ⊂ N , ∣ I ∣ < ∞ lim SL I ( f ) . We will sometimes write S L n ( f ) \mathrm{SL}_{n}(f) SL n ( f ) S L { 1 , … , n } ( f ) \mathrm{SL}_{\{1,\ldots,n\}}(f) SL { 1 , … , n } ( f ) I ⊂ N I\subset\mathbb{N} I ⊂ N ∣ I ∣ = n |I|=n ∣ I ∣ = n
S L n ( F q ) → S L I ( f ) ⊆ S L ∞ ( F q ) \mathrm{SL}_{n}(\mathbb{F}_q)\to\mathrm{SL}_{I}(f)\subseteq\mathrm{SL}_{\infty}(\mathbb{F}_q) SL n ( F q ) → SL I ( f ) ⊆ SL ∞ ( F q ) which is an isomorphism onto its image. The image of a matrix A ∈ S L n ( F q ) A\in\mathrm{SL}_{n}(\mathbb{F}_q) A ∈ SL n ( F q ) e e e [ A ] I [A]_{I} [ A ] I I ⊂ N I\subset\mathbb{N} I ⊂ N S L I ( F q ) \mathrm{SL}_{I}(\mathbb{F}_q) SL I ( F q ) S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
All the notations and remarks in the previous paragraph apply mutatis mutandis to the general linear group G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q )
For each finitary basis f f f i f i_f i f S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) f f f A ∞ ( f ) = i f ( A ∞ ) A_{\infty}(f)=i_f(A_{\infty}) A ∞ ( f ) = i f ( A ∞ ) I ⊂ N I\subset\mathbb{N} I ⊂ N A I ( f ) A_{I}(f) A I ( f ) A ∞ ( f ) A_{\infty}(f) A ∞ ( f ) { f n : n ∈ I } \{f^n:n\in I\} { f n : n ∈ I }
For each element g ∈ G L ∞ ( F q ) g\in\mathrm{GL}_{\infty}(\mathbb{F}_q) g ∈ GL ∞ ( F q ) g − I g-I g − I ⨁ n ∈ N F q \bigoplus_{n\in\mathbb{N}}\mathbb{F}_q ⨁ n ∈ N F q
rk ~ ( g ) = rk ( g − I ) ∀ g ∈ G L ∞ ( F q ) . \widetilde{\operatorname{rk}}(g)=\operatorname{rk}(g-I)\quad\forall\,g\in\mathrm{GL}_{\infty}(\mathbb{F}_q). rk ( g ) = rk ( g − I ) ∀ g ∈ GL ∞ ( F q ) . Note that rk ~ ( g ) \widetilde{\operatorname{rk}}(g) rk ( g ) g g g g g g rk ~ ( g ) \widetilde{\operatorname{rk}}(g) rk ( g ) I I I g g g
Lastly, we set some common notation for specific matrices. Let n ≥ 2 n\geq2 n ≥ 2 1 ≤ i , j ≤ n 1\leq i,j\leq n 1 ≤ i , j ≤ n σ i j ∈ G L n ( F q ) \sigma_{ij}\in\mathrm{GL}_{n}(\mathbb{F}_q) σ ij ∈ GL n ( F q ) i i i j j j E i j ∈ G L n ( F q ) E_{ij}\in\mathrm{GL}_{n}(\mathbb{F}_q) E ij ∈ GL n ( F q ) 1 1 1 ( i , j ) (i,j) ( i , j ) α ∈ F q × \alpha\in\mathbb{F}_q^{\times} α ∈ F q × D i ( α ) = diag ( 1 , … , α , … , 1 ) ∈ G L n ( F q ) D_i(\alpha)=\operatorname{diag}(1,\ldots,\alpha,\ldots,1)\in\mathrm{GL}_{n}(\mathbb{F}_q) D i ( α ) = diag ( 1 , … , α , … , 1 ) ∈ GL n ( F q ) α \alpha α i i i
The standard and dual actions
We let V V V V ^ \widehat{V} V ∏ n ∈ N F q \prod_{n\in\mathbb{N}} \mathbb{F}_q ∏ n ∈ N F q G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) A ⋅ v = A v A\cdot v=Av A ⋅ v = A v A ⋅ φ = ( A − 1 ) T φ A\cdot\varphi=(A^{-1})^{\mathsf{T}}\varphi A ⋅ φ = ( A − 1 ) T φ A ∈ G L ∞ ( F q ) A\in\mathrm{GL}_{\infty}(\mathbb{F}_q) A ∈ GL ∞ ( F q ) v ∈ V v\in V v ∈ V φ ∈ V ^ \varphi\in\widehat{V} φ ∈ V S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) V V V V ^ \widehat{V} V
For each fixed k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 V k , ℓ V^{k,\ell} V k , ℓ V k × V ^ ℓ V^k\times\widehat{V}^{\ell} V k × V ℓ G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) M ∞ × k ( F q ) × M ℓ × ∞ ( F q ) M_{\infty\times k}(\mathbb{F}_q)\times M_{\ell\times\infty}(\mathbb{F}_q) M ∞ × k ( F q ) × M ℓ × ∞ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ V ℓ , k V^{\ell,k} V ℓ , k G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q )
We set up some additional notation. For each n ∈ N n\in\mathbb{N} n ∈ N V n = F q n V_n=\mathbb{F}_q^n V n = F q n V > n = ∏ i = n + 1 ∞ F q V_{>n}=\prod_{i=n+1}^{\infty}\mathbb{F}_q V > n = ∏ i = n + 1 ∞ F q V V V V = V n × V > n V=V_n\times V_{>n} V = V n × V > n v ∈ V v\in V v ∈ V v ≤ n ∈ V n v_{\leq n}\in V_n v ≤ n ∈ V n v > n ∈ V > n v_{>n}\in V_{>n} v > n ∈ V > n V ^ \widehat{V} V V ^ = V ^ n × V ^ > n \widehat{V}=\widehat{V}_n\times\widehat{V}_{>n} V = V n × V > n V ^ n \widehat{V}_n V n V n V_n V n v ≤ n v_{\leq n} v ≤ n v > n v_{>n} v > n v v v F q m \mathbb{F}_q^m F q m m > n m>n m > n
It is natural to interpret elements in V ^ \widehat{V} V V V V v v v φ \varphi φ V V V V ^ \widehat{V} V V × V ^ → F q V\times\widehat{V}\to\mathbb{F}_q V × V → F q
⟨ ⋅ , ⋅ ⟩ n : V × V ^ → F q , ⟨ v , φ ⟩ n = ⟨ v ≤ n , φ ≤ n ⟩ \langle\cdot,\cdot\rangle_n:V\times\widehat{V}\to\mathbb{F}_q,\qquad\langle v,\varphi\rangle_n=\langle v_{\leq n},\varphi_{\leq n}\rangle ⟨ ⋅ , ⋅ ⟩ n : V × V → F q , ⟨ v , φ ⟩ n = ⟨ v ≤ n , φ ≤ n ⟩ for each n ∈ N n\in\mathbb{N} n ∈ N G L m ( F q ) \mathrm{GL}_m(\mathbb{F}_q) GL m ( F q ) ⟨ ⋅ , ⋅ ⟩ n \langle\cdot,\cdot\rangle_n ⟨ ⋅ , ⋅ ⟩ n n ≥ m n\geq m n ≥ m ⟨ g v , g φ ⟩ n = ⟨ v , φ ⟩ n \langle gv,g\varphi\rangle_n=\langle v,\varphi\rangle_n ⟨ g v , g φ ⟩ n = ⟨ v , φ ⟩ n g ∈ G L m ( F q ) g\in\mathrm{GL}_m(\mathbb{F}_q) g ∈ GL m ( F q ) x = ( v , φ ) ∈ V k , ℓ x=(v,\varphi)\in V^{k,\ell} x = ( v , φ ) ∈ V k , ℓ n ∈ N n\in\mathbb{N} n ∈ N M n ( v , φ ) ∈ M ℓ × k ( F q ) M_n(v,\varphi)\in M_{\ell\times k}(\mathbb{F}_q) M n ( v , φ ) ∈ M ℓ × k ( F q )
( M n ( v , φ ) ) i j = ⟨ v j , φ i ⟩ n . \left(M_n(v,\varphi)\right)_{ij}=\langle v^j,\varphi^i\rangle_n. ( M n ( v , φ ) ) ij = ⟨ v j , φ i ⟩ n . Extension by permutations
Let H ≤ G L ∞ ( F q ) H\leq\mathrm{GL}_{\infty}(\mathbb{F}_q) H ≤ GL ∞ ( F q ) g ∈ G L ∞ ( F q ) g\in\mathrm{GL}_{\infty}(\mathbb{F}_q) g ∈ GL ∞ ( F q ) extendable by permutations to H H H I ⊂ N I\subset\mathbb{N} I ⊂ N ∣ I ∣ = n |I|=n ∣ I ∣ = n A ∈ G L n ( F q ) A\in\mathrm{GL}_n(\mathbb{F}_q) A ∈ GL n ( F q ) g = [ A ] I g=[A]_I g = [ A ] I m ∈ N m\in\mathbb{N} m ∈ N J ⊂ N J\subset\mathbb{N} J ⊂ N ∣ J ∣ = k |J|=k ∣ J ∣ = k max I < m < min J \max I<m<\min J max I < m < min J P ∈ G L k ( F q ) P\in\mathrm{GL}_k(\mathbb{F}_q) P ∈ GL k ( F q )
[ ( A 0 0 P ) ] I ∪ J ∈ H . \left[\begin{pmatrix} A & 0 \\ 0 & P \end{pmatrix}\right]_{I\cup J}\in H. [ ( A 0 0 P ) ] I ∪ J ∈ H . The subset of all matrices g ∈ G L ∞ ( F q ) g\in\mathrm{GL}_{\infty}(\mathbb{F}_q) g ∈ GL ∞ ( F q ) H H H G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) H ∗ H^* H ∗ H ⊆ H ∗ H\subseteq H^* H ⊆ H ∗ H H H H ∗ H^* H ∗ G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q )
Singer cycles
We discuss Singer cycles, a special type of group element. These are well-known in the theory of finite linear groups. See [16 ] for a general overview.
Definition 2.1. Let q q q G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q ) q n − 1 q^n-1 q n − 1 S L n ( F q ) \mathrm{SL}_n(\mathbb{F}_q) SL n ( F q ) q n − 1 q − 1 \frac{q^n-1}{q-1} q − 1 q n − 1
We list basic properties of Singer cycles. A reference for these can be found in [16 ] . Singer cycles have a maximal order in their corresponding group. In the G L n \mathrm{GL}_n GL n F q n \mathbb{F}_{q^n} F q n s ∈ G L n ( F q ) s\in\mathrm{GL}_n(\mathbb{F}_q) s ∈ GL n ( F q ) M n ( F q ) M_n(\mathbb{F}_q) M n ( F q ) F q n \mathbb{F}_{q^n} F q n s s s M n ( F q ) M_n(\mathbb{F}_q) M n ( F q ) s s s G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q )
∣ C l G L n ( F q ) ( s ) ∣ = ∣ G L n ( F q ) ∣ q n − 1 = q n 2 + O ( n ) . (2) \left|\mathrm{Cl}_{\mathrm{GL}_n(\mathbb{F}_q)}(s)\right|=\frac{|\mathrm{GL}_n(\mathbb{F}_q)|}{q^n-1}=q^{n^2+O(n)}. \tag*{(2)} Cl GL n ( F q ) ( s ) = q n − 1 ∣ GL n ( F q ) ∣ = q n 2 + O ( n ) . (2) For S L n \mathrm{SL}_n SL n F q n × → F q × \mathbb{F}_{q^n}^{\times}\to\mathbb{F}_q^{\times} F q n × → F q × S L n \mathrm{SL}_n SL n
∣ C l S L n ( F q ) ( s ) ∣ = ∣ S L n ( F q ) ∣ ( q n − 1 ) / ( q − 1 ) = q n 2 + O ( n ) . (3) \left|\mathrm{Cl}_{\mathrm{SL}_n(\mathbb{F}_q)}(s)\right|=\frac{|\mathrm{SL}_n(\mathbb{F}_q)|}{(q^n-1)/(q-1)}=q^{n^2+O(n)}. \tag*{(3)} Cl SL n ( F q ) ( s ) = ( q n − 1 ) / ( q − 1 ) ∣ SL n ( F q ) ∣ = q n 2 + O ( n ) . (3) Singer cycles have been useful in the study of finite linear groups. In particular, the types of maximal subgroups of G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q ) S L n ( F q ) \mathrm{SL}_n(\mathbb{F}_q) SL n ( F q ) [18 , 14 , 13 ] for this theory and for subgroups containing related types of elements. We will need the following consequence of this classification (we encourage the reader interested in the fastest route towards our main result Theorem 1.1 to skip the proof and move onward to Sections 3 and § 4 upon first reading).
Proposition 2.2. Let M M M S L n ( F q ) \mathrm{SL}_n(\mathbb{F}_q) SL n ( F q )
∣ M ∣ ≤ q 1 2 n 2 + O ( n ) . |M|\leq q^{\frac{1}{2}n^2+O(n)}. ∣ M ∣ ≤ q 2 1 n 2 + O ( n ) . Proof. By Lemma 2.3 in [13 ] , when n > 20 n>20 n > 20 S L n ( F q ) \mathrm{SL}_n(\mathbb{F}_q) SL n ( F q )
The normalizer of the special unitary group S U n ( F q 1 / 2 ) \mathrm{SU}_n(\mathbb{F}_{q^{1/2}}) SU n ( F q 1/2 ) n n n q q q
The normalizer of the derived subgroup of the special orthogonal group Ω n ( F q ) \Omega_n(\mathbb{F}_q) Ω n ( F q ) n n n q q q
The normalizer of the symplectic group S p n ( F q ) \mathrm{Sp}_n(\mathbb{F}_q) Sp n ( F q ) n n n
The normalizer of G L n / f ( F q f ) ∩ S L n ( F q ) \mathrm{GL}_{n/f}(\mathbb{F}_{q^f})\cap\mathrm{SL}_n(\mathbb{F}_q) GL n / f ( F q f ) ∩ SL n ( F q ) f f f n n n G L n / f ( F q f ) \mathrm{GL}_{n/f}(\mathbb{F}_{q^f}) GL n / f ( F q f ) G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q )
The definitions of these groups are given in [20 ] . We bound the sizes of these normalizers using known orders of linear groups. Let K K K S U n ( F q 1 / 2 ) \mathrm{SU}_n(\mathbb{F}_{q^{1/2}}) SU n ( F q 1/2 ) Ω n ( F q ) \Omega_n(\mathbb{F}_q) Ω n ( F q ) S p n ( F q ) \mathrm{Sp}_n(\mathbb{F}_q) Sp n ( F q ) [20 ] we have N G L n ( F q ) ( K ) ≤ Δ N_{\mathrm{GL}_n(\mathbb{F}_q)}(K)\leq\Delta N GL n ( F q ) ( K ) ≤ Δ Δ ≤ G L n ( F q ) \Delta\leq\mathrm{GL}_n(\mathbb{F}_q) Δ ≤ GL n ( F q ) Δ \Delta Δ q 1 2 n 2 + O ( n ) q^{\frac{1}{2}n^2+O(n)} q 2 1 n 2 + O ( n )
∣ N S L n ( F q ) ( K ) ∣ ≤ ∣ N G L n ( F q ) ( K ) ∣ ≤ ∣ Δ ∣ = q 1 2 n 2 + O ( n ) . |N_{\mathrm{SL}_n(\mathbb{F}_q)}(K)|\leq|N_{\mathrm{GL}_n(\mathbb{F}_q)}(K)|\leq|\Delta|=q^{\frac{1}{2}n^2+O(n)}. ∣ N SL n ( F q ) ( K ) ∣ ≤ ∣ N GL n ( F q ) ( K ) ∣ ≤ ∣Δ∣ = q 2 1 n 2 + O ( n ) . Now let K = G L n / f ( F q f ) ∩ S L n ( F q ) K=\mathrm{GL}_{n/f}(\mathbb{F}_{q^f})\cap\mathrm{SL}_n(\mathbb{F}_q) K = GL n / f ( F q f ) ∩ SL n ( F q ) N G L n ( F q ) ( K ) ≅ G L n / f ( F q f ) ⋊ Gal ( F q f / F q ) N_{\mathrm{GL}_n(\mathbb{F}_q)}(K)\cong\mathrm{GL}_{n/f}(\mathbb{F}_{q^f})\rtimes\operatorname{Gal}(\mathbb{F}_{q^f}/\mathbb{F}_q) N GL n ( F q ) ( K ) ≅ GL n / f ( F q f ) ⋊ Gal ( F q f / F q )
∣ N S L n ( F q ) ( K ) ∣ ≤ ∣ Gal ( F q f / F q ) ∣ ⋅ ∣ G L n / f ( F q f ) ∣ = f ( q f ) ( n f ) 2 + O ( n / f ) = q n 2 f + O ( n ) ≤ q 1 2 n 2 + O ( n ) , |N_{\mathrm{SL}_n(\mathbb{F}_q)}(K)|\leq|\operatorname{Gal}(\mathbb{F}_{q^f}/\mathbb{F}_q)|\cdot|\mathrm{GL}_{n/f}(\mathbb{F}_{q^f})|=f(q^f)^{\left(\frac{n}{f}\right)^2+O(n/f)}=q^{\frac{n^2}{f}+O(n)}\leq q^{\frac{1}{2}n^2+O(n)}, ∣ N SL n ( F q ) ( K ) ∣ ≤ ∣ Gal ( F q f / F q ) ∣ ⋅ ∣ GL n / f ( F q f ) ∣ = f ( q f ) ( f n ) 2 + O ( n / f ) = q f n 2 + O ( n ) ≤ q 2 1 n 2 + O ( n ) , as required. □ \square □
On maximal subgroups of G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q )
We state some results in the theory of finite general linear groups and their subgroups. These are to be used in Section 3 below.
Proposition 2.3. Suppose n ≥ 3 n\geq3 n ≥ 3 S S S G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q )
G L n − 1 ( F q ) = ( 1 0 1 , n − 1 0 n − 1 , 1 G L n − 1 ( F q ) ) \mathrm{GL}_{n-1}(\mathbb{F}_q)=\left(\begin{array}{cc}1&0_{1,n-1}\\0_{n-1,1}&\mathrm{GL}_{n-1}(\mathbb{F}_q)\end{array}\right) GL n − 1 ( F q ) = ( 1 0 n − 1 , 1 0 1 , n − 1 GL n − 1 ( F q ) ) Then there is a subgroup K ≤ F q × K\leq\mathbb{F}_q^{\times} K ≤ F q × S S S
Q ( K ) = ( K 0 1 , n − 1 0 n − 1 , 1 G L n − 1 ( F q ) ) (4) Q(K)=\begin{pmatrix} K & 0_{1,n-1} \\ 0_{n-1,1} & \mathrm{GL}_{n-1}(\mathbb{F}_q) \end{pmatrix} \tag*{(4)} Q ( K ) = ( K 0 n − 1 , 1 0 1 , n − 1 GL n − 1 ( F q ) ) (4) P V ( K ) = ( K ( F q n − 1 ) t 0 n − 1 , 1 G L n − 1 ( F q ) ) (5) P_V(K)=\begin{pmatrix} K & (\mathbb{F}_q^{n-1})^{\mathsf{t}} \\ 0_{n-1,1} & \mathrm{GL}_{n-1}(\mathbb{F}_q) \end{pmatrix} \tag*{(5)} P V ( K ) = ( K 0 n − 1 , 1 ( F q n − 1 ) t GL n − 1 ( F q ) ) (5) P H ( K ) = ( K 0 1 , n − 1 F q n − 1 G L n − 1 ( F q ) ) = P V ( K ) t . (6) P_H(K)=\begin{pmatrix} K & 0_{1,n-1} \\ \mathbb{F}_q^{n-1} & \mathrm{GL}_{n-1}(\mathbb{F}_q) \end{pmatrix}=P_V(K)^{\mathsf{t}}. \tag*{(6)} P H ( K ) = ( K F q n − 1 0 1 , n − 1 GL n − 1 ( F q ) ) = P V ( K ) t . (6) Proof. The case n ≥ 4 n \ge4 n ≥ 4 [8 Theorem 2] . The sharper case n ≥ 3 n \ge3 n ≥ 3 [25 ] . □
Corollary 2.4. Suppose n ≥ 3 n \ge3 n ≥ 3 S S S G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q ) G L n − 1 ( F q ) \mathrm{GL}_{n-1}(\mathbb{F}_q) GL n − 1 ( F q ) Proposition 2.3 . If q n − 1 q^n-1 q n − 1 [ G L n ( F q ) : S ] [\mathrm{GL}_n(\mathbb{F}_q):S] [ GL n ( F q ) : S ]
(1) S = P V = P V ( { 1 } ) S=P_V=P_V(\{1\}) S = P V = P V ({ 1 }) S = ( 1 ( F q n − 1 ) t 0 n − 1 , 1 G L n − 1 ( F q ) ) S=\begin{pmatrix} 1 & (\mathbb{F}_q^{n-1})^{\mathsf{t}} \\ 0_{n-1,1} & \mathrm{GL}_{n-1}(\mathbb{F}_q) \end{pmatrix} S = ( 1 0 n − 1 , 1 ( F q n − 1 ) t GL n − 1 ( F q ) )
(2) S = P H = P H ( { 1 } ) = P V t S=P_H=P_H(\{1\})=P_V^{\mathsf{t}} S = P H = P H ({ 1 }) = P V t
(3) S = P V ∩ P H = Q ( { 1 } ) = ( 1 0 1 , n − 1 0 n − 1 , 1 G L n − 1 ( F q ) ) S=P_V\cap P_H=Q(\{1\})=\begin{pmatrix} 1 & 0_{1,n-1} \\ 0_{n-1,1} & \mathrm{GL}_{n-1}(\mathbb{F}_q) \end{pmatrix} S = P V ∩ P H = Q ({ 1 }) = ( 1 0 n − 1 , 1 0 1 , n − 1 GL n − 1 ( F q ) ) Proof. Let us look at the indices of the proper subgroups of G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q ) G L n − 1 ( F q ) \mathrm{GL}_{n-1}(\mathbb{F}_q) GL n − 1 ( F q ) K ≤ F q × K\le\mathbb{F}_q^{\times} K ≤ F q ×
[ G L n ( F q ) : Q ( K ) ] = q n − 1 ( q n − 1 ) / ∣ K ∣ [\mathrm{GL}_n(\mathbb{F}_q):Q(K)]=q^{n-1}(q^n-1)/|K| [ GL n ( F q ) : Q ( K )] = q n − 1 ( q n − 1 ) /∣ K ∣ and
[ G L n ( F q ) : P V ( K ) ] = [ G L n ( F q ) : P H ( K ) ] = ( q n − 1 ) / ∣ K ∣ . [\mathrm{GL}_n(\mathbb{F}_q):P_V(K)]=[\mathrm{GL}_n(\mathbb{F}_q):P_H(K)]=(q^n-1)/|K|. [ GL n ( F q ) : P V ( K )] = [ GL n ( F q ) : P H ( K )] = ( q n − 1 ) /∣ K ∣. Since ∣ K ∣ |K| ∣ K ∣ q q q K K K K = { 1 } K=\{1\} K = { 1 }
Permutational actions of G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q )
Fix a prime power q q q G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ C C C γ ∈ C N \gamma\in C^{\mathbb{N}} γ ∈ C N I = { i 1 < ⋯ < i n } ⊂ N I=\{i_1<\cdots<i_n\}\subset\mathbb{N} I = { i 1 < ⋯ < i n } ⊂ N γ I = ( γ i k ) k = 1 n ∈ C n \gamma_I=(\gamma_{i_k})_{k=1}^n\in C^n γ I = ( γ i k ) k = 1 n ∈ C n
Definition 3.1. Let C C C G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) C N C^{\mathbb{N}} C N local if for every n ∈ N n\in\mathbb{N} n ∈ N I ⊂ N I\subset\mathbb{N} I ⊂ N ∣ I ∣ = n |I|=n ∣ I ∣ = n A ∈ G L n ( F q ) A\in\mathrm{GL}_n(\mathbb{F}_q) A ∈ GL n ( F q )
γ I = δ I ⇒ ( [ A ] I γ ) I = ( [ A ] I δ ) I ∀ γ , δ ∈ C N \gamma_I=\delta_I\Rightarrow([A]_I\gamma)_I=([A]_I\delta)_I\quad\forall\gamma,\delta\in C^{\mathbb{N}} γ I = δ I ⇒ ([ A ] I γ ) I = ([ A ] I δ ) I ∀ γ , δ ∈ C N and
( [ A ] I γ ) N ∖ I = γ N ∖ I ∀ γ ∈ C N . ([A]_I\gamma)_{\mathbb{N}\setminus I}=\gamma_{\mathbb{N}\setminus I}\quad\forall\gamma\in C^{\mathbb{N}}. ([ A ] I γ ) N ∖ I = γ N ∖ I ∀ γ ∈ C N . The action is called permutational if permutation matrices act in the standard manner. Namely, if P σ ∈ G L ∞ ( F q ) P_\sigma\in\mathrm{GL}_{\infty}(\mathbb{F}_q) P σ ∈ GL ∞ ( F q ) σ ∈ S ∞ \sigma\in S_{\infty} σ ∈ S ∞
( P σ ⋅ γ ) i = γ σ ( i ) (P_\sigma\cdot\gamma)_i=\gamma_{\sigma(i)} ( P σ ⋅ γ ) i = γ σ ( i ) for every γ ∈ C N \gamma\in C^{\mathbb{N}} γ ∈ C N
We will refer to elements of C C C colors , for reasons that will become clearer in the next sections. A local action as in Definition 3.1 induces for every n ∈ N n\in\mathbb{N} n ∈ N G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q ) C n C^n C n n n n G L n ( F q ) = G L { 1 , … , n } ( F q ) ≤ G L ∞ ( F q ) \mathrm{GL}_n(\mathbb{F}_q)=\mathrm{GL}_{\{1,\ldots,n\}}(\mathbb{F}_q)\le\mathrm{GL}_{\infty}(\mathbb{F}_q) GL n ( F q ) = GL { 1 , … , n } ( F q ) ≤ GL ∞ ( F q )
Theorem 3.2. Let C C C ∣ C ∣ = q r |C|=q^{r} ∣ C ∣ = q r r ∈ N r\in\mathbb{N} r ∈ N r e d ∈ C \mathrm{red}\in C red ∈ C G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_{q}) GL ∞ ( F q ) C N C^{\mathbb{N}} C N
(1) the constant vector ρ = ( r e d ) n ∈ N ∈ C N \rho=(\mathrm{red})_{n\in\mathbb{N}}\in C^{\mathbb{N}} ρ = ( red ) n ∈ N ∈ C N
(2) if A ∈ G L n ( F q ) A\in\mathrm{GL}_{n}(\mathbb{F}_{q}) A ∈ GL n ( F q ) 1 1 1 A A A C n C^{n} C n ρ n = ( r e d ) i = 1 n \rho_{n}=(\mathrm{red})_{i=1}^{n} ρ n = ( red ) i = 1 n
There exist k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_{0} k , ℓ ∈ N 0 k + ℓ = r k+\ell=r k + ℓ = r π : C → F q k × F q ℓ \pi:C\to\mathbb{F}_{q}^{k}\times\mathbb{F}_{q}^{\ell} π : C → F q k × F q ℓ π N : C N → V k , ℓ \pi^{\mathbb{N}}:C^{\mathbb{N}}\to V^{k,\ell} π N : C N → V k , ℓ G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_{q}) GL ∞ ( F q ) π \pi π π ( r e d ) = ( 0 , 0 ) ∈ F q k × F q ℓ \pi(\mathrm{red})=(0,0)\in\mathbb{F}_{q}^{k}\times\mathbb{F}_{q}^{\ell} π ( red ) = ( 0 , 0 ) ∈ F q k × F q ℓ
For the rest of this section, we will assume that the objects which appear in Theorem 3.2 are fixed. In particular, we are given a local permutational action on color vectors satisfying the above two additional assumptions.
Remark 3.3. The material of the current Section 3 3 3 Theorem 3.2 . A reader interested in the fastest route to understand Theorems 1.1 and 1.2 is encouraged to skip the rest of Section 3 3 3
We will be making the following identification implicitly: for any pair n , m ∈ N ∪ { ∞ } n,m\in\mathbb{N}\cup\{\infty\} n , m ∈ N ∪ { ∞ } n ≤ m n\le m n ≤ m C n C^{n} C n { γ ∈ C m : γ i = r e d for all i > n } \{\gamma\in C^{m}:\gamma_{i}=\mathrm{red}\text{ for all }i>n\} { γ ∈ C m : γ i = red for all i > n } C m C^{m} C m G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_{q}) GL ∞ ( F q ) G L n ( F q ) \mathrm{GL}_{n}(\mathbb{F}_{q}) GL n ( F q ) G L m ( F q ) \mathrm{GL}_{m}(\mathbb{F}_{q}) GL m ( F q ) C m C^{m} C m
Remark 3.4. The largest matrix which plays a role in the proof of Theorem 3.2 has size 4 × 4 4\times4 4 × 4 G L n ( F q ) \mathrm{GL}_{n}(\mathbb{F}_{q}) GL n ( F q ) C n C^{n} C n n ≥ 4 n\ge4 n ≥ 4 n ≥ 4 n\ge4 n ≥ 4 G L 3 ( F q ) \mathrm{GL}_{3}(\mathbb{F}_{q}) GL 3 ( F q ) G L 2 ( F q ) \mathrm{GL}_{2}(\mathbb{F}_{q}) GL 2 ( F q )
Color types
The first step in the proof of Theorem 3.2 is to exhibit a product structure on the set of colors C C C C = C V × C H C=C_{V}\times C_{H} C = C V × C H k + ℓ = r k+\ell=r k + ℓ = r C V , C H C_{V},C_{H} C V , C H ∣ C V ∣ = q k |C_{V}|=q^{k} ∣ C V ∣ = q k ∣ C H ∣ = q ℓ |C_{H}|=q^{\ell} ∣ C H ∣ = q ℓ
We will use the distinguished element r e d ∈ C \mathrm{red}\in C red ∈ C C V × { r e d } C_{V}\times\{\mathrm{red}\} C V × { red } { r e d } × C H \{\mathrm{red}\}\times C_{H} { red } × C H
Let c ∈ C c\in C c ∈ C r e d \mathrm{red} red n ≥ 3 n\ge3 n ≥ 3 G L n ( F q ) \mathrm{GL}_{n}(\mathbb{F}_{q}) GL n ( F q )
v c = ( c , r e d , … , r e d ) ∈ C n . v_{c}=(c,\mathrm{red},\ldots,\mathrm{red})\in C^{n}. v c = ( c , red , … , red ) ∈ C n . Let S c = Stab G L n ( F q ) ( v c ) S_{c}=\operatorname{Stab}_{\mathrm{GL}_{n}(\mathbb{F}_{q})}(v_{c}) S c = Stab GL n ( F q ) ( v c ) S c S_{c} S c G L n − 1 ( F q ) \mathrm{GL}_{n-1}(\mathbb{F}_{q}) GL n − 1 ( F q ) n − 1 n-1 n − 1 s ∈ G L n ( F q ) s\in\mathrm{GL}_{n}(\mathbb{F}_{q}) s ∈ GL n ( F q ) § 2.4 ). Under Assumption (2) of Theorem 3.2 , the cyclic group ⟨ s ⟩ \langle s\rangle ⟨ s ⟩ G L n ( F q ) \mathrm{GL}_{n}(\mathbb{F}_{q}) GL n ( F q ) v c v_{c} v c ∣ ⟨ s ⟩ ∣ = q n − 1 |\langle s\rangle|=q^{n}-1 ∣ ⟨ s ⟩ ∣ = q n − 1 ∣ G L n ( F q ) : S c ∣ |\mathrm{GL}_{n}(\mathbb{F}_{q}):S_{c}| ∣ GL n ( F q ) : S c ∣ S c S_{c} S c G L n ( F q ) \mathrm{GL}_{n}(\mathbb{F}_{q}) GL n ( F q ) Corollary 2.4 that S c S_{c} S c P V , P H P_{V},P_{H} P V , P H P V ∩ P H P_{V}\cap P_{H} P V ∩ P H
Definition 3.5. A color c ∈ C c\in C c ∈ C r e d \mathrm{red} red vertical (resp. horizontal ) if S c = Stab G L n ( F q ) ( v c ) = P V S_{c}=\operatorname{Stab}_{\mathrm{GL}_{n}(\mathbb{F}_{q})}(v_{c})=P_{V} S c = Stab GL n ( F q ) ( v c ) = P V S c = P H S_{c}=P_{H} S c = P H C V C_{V} C V C H C_{H} C H pure if it is either vertical or horizontal.
It is not hard to check that the above definition does not depend on the choice of n n n C ≅ C V × C H C\cong C_{V}\times C_{H} C ≅ C V × C H ℓ \ell ℓ k + ℓ = r k+\ell=r k + ℓ = r C V N ≅ V k , 0 C_V^N \cong V^{k,0} C V N ≅ V k , 0 C H N ≅ V 0 , ℓ C_H^N \cong V^{0,\ell} C H N ≅ V 0 , ℓ
Lemma 3.6. Let c , d ∈ C c,d\in C c , d ∈ C
E 21 ⋅ ( c , d ) = ( c ′ , d ′ ) ∈ C 2 and E 12 − 1 ⋅ ( c , d ) = ( c ′ ′ , d ′ ′ ) ∈ C 2 . E_{21}\cdot(c,d)=(c',d')\in C^2 \qquad\text{and}\qquad E_{12}^{-1}\cdot(c,d)=(c'',d'')\in C^2. E 21 ⋅ ( c , d ) = ( c ′ , d ′ ) ∈ C 2 and E 12 − 1 ⋅ ( c , d ) = ( c ′′ , d ′′ ) ∈ C 2 . We have
If d ∈ C V d\in C_V d ∈ C V d ′ ∈ C V d'\in C_V d ′ ∈ C V c ∈ C V c\in C_V c ∈ C V c ′ ∈ C V c'\in C_V c ′ ∈ C V
If d ∈ C H d\in C_H d ∈ C H d ′ ′ ∈ C H d''\in C_H d ′′ ∈ C H c ∈ C H c\in C_H c ∈ C H c ′ ′ ∈ C H c''\in C_H c ′′ ∈ C H
Proof. We only show item (1), as the proof of the second item (2) is analogous. Assume that d ∈ C V d\in C_V d ∈ C V d ′ ∈ C V d'\in C_V d ′ ∈ C V B ∈ P V B\in P_V B ∈ P V v d ′ = ( d ′ , red , red ) ∈ C 3 v_{d'}=(d',\operatorname{red},\operatorname{red})\in C^3 v d ′ = ( d ′ , red , red ) ∈ C 3 ( 1 0 0 B ) \left(\begin{smallmatrix}1&0\\0&B\end{smallmatrix}\right) ( 1 0 0 B ) ( c ′ , d ′ , red , red ) (c',d',\operatorname{red},\operatorname{red}) ( c ′ , d ′ , red , red ) B = ( 1 y 0 2 , 1 A ) B=\left(\begin{smallmatrix}1&y\\0_{2,1}&A\end{smallmatrix}\right) B = ( 1 0 2 , 1 y A ) A ∈ GL 2 ( F q ) A\in\operatorname{GL}_2(\mathbb{F}_q) A ∈ GL 2 ( F q ) y ∈ M 1 × 2 ( F q ) y\in M_{1\times2}(\mathbb{F}_q) y ∈ M 1 × 2 ( F q )
( 1 y 0 2 , 1 A ) ⋅ ( c ′ , d ′ , red , red ) = ( 1 y 0 2 , 1 A ) ( E 21 0 2 0 2 I 2 ) ⋅ ( c , d , red , red ) = ( E 21 0 2 0 2 I 2 ) ( 1 y 0 2 , 1 A ) ⋅ ( c , d , red , red ) = ( E 21 0 2 0 2 I 2 ) ⋅ ( c , d , red , red ) = ( c ′ , d ′ , red , red ) . \begin{aligned}
\begin{pmatrix}1&y\\0_{2,1}&A\end{pmatrix}\cdot(c',d',\operatorname{red},\operatorname{red})
=
\begin{pmatrix}1&y\\0_{2,1}&A\end{pmatrix}
\begin{pmatrix}E_{21}&0_2\\0_2&I_2\end{pmatrix}
\cdot(c,d,\operatorname{red},\operatorname{red}) \\
&=
\begin{pmatrix}E_{21}&0_2\\0_2&I_2\end{pmatrix}
\begin{pmatrix}1&y\\0_{2,1}&A\end{pmatrix}
\cdot(c,d,\operatorname{red},\operatorname{red}) \\
&=
\begin{pmatrix}E_{21}&0_2\\0_2&I_2\end{pmatrix}\cdot(c,d,\operatorname{red},\operatorname{red})
=(c',d',\operatorname{red},\operatorname{red}).
\end{aligned} ( 1 0 2 , 1 y A ) ⋅ ( c ′ , d ′ , red , red ) = ( 1 0 2 , 1 y A ) ( E 21 0 2 0 2 I 2 ) ⋅ ( c , d , red , red ) = ( E 21 0 2 0 2 I 2 ) ( 1 0 2 , 1 y A ) ⋅ ( c , d , red , red ) = ( E 21 0 2 0 2 I 2 ) ⋅ ( c , d , red , red ) = ( c ′ , d ′ , red , red ) . We get d ′ ∈ C V d'\in C_V d ′ ∈ C V c ∈ C V c\in C_V c ∈ C V
( 1 y 0 2 , 1 A ) ⋅ ( c ′ , d ′ , red , red ) = ( 1 y 0 2 , 1 A ) ( E 21 0 2 0 2 I 2 ) ⋅ ( c , d , red , red ) = ( E 21 0 2 0 2 I 2 ) ( 1 y 0 2 , 1 A ) ( 1 0 0 1 , 2 0 1 1 0 2 , 1 0 2 , 1 I 2 ) ⋅ ( c , d , red , red ) = ( c ′ , d ′ , red , red ) . \begin{aligned}
\begin{pmatrix}1&y\\0_{2,1}&A\end{pmatrix}\cdot(c',d',\operatorname{red},\operatorname{red})
=
\begin{pmatrix}1&y\\0_{2,1}&A\end{pmatrix}
\begin{pmatrix}E_{21}&0_2\\0_2&I_2\end{pmatrix}
\cdot(c,d,\operatorname{red},\operatorname{red}) \\
&=
\begin{pmatrix}E_{21}&0_2\\0_2&I_2\end{pmatrix}
\begin{pmatrix}1&y\\0_{2,1}&A\end{pmatrix}
\begin{pmatrix}1&0&0_{1,2}\\0&1&1\\0_{2,1}&0_{2,1}&I_2\end{pmatrix}
\cdot(c,d,\operatorname{red},\operatorname{red}) \\
&=(c',d',\operatorname{red},\operatorname{red}).
\end{aligned} ( 1 0 2 , 1 y A ) ⋅ ( c ′ , d ′ , red , red ) = ( 1 0 2 , 1 y A ) ( E 21 0 2 0 2 I 2 ) ⋅ ( c , d , red , red ) = ( E 21 0 2 0 2 I 2 ) ( 1 0 2 , 1 y A ) 1 0 0 2 , 1 0 1 0 2 , 1 0 1 , 2 1 I 2 ⋅ ( c , d , red , red ) = ( c ′ , d ′ , red , red ) . In the last equation, the second and third matrices act trivially because c , d ∈ C V c,d\in C_V c , d ∈ C V v c ′ = ( c ′ , red , red ) v_{c'}=(c',\operatorname{red},\operatorname{red}) v c ′ = ( c ′ , red , red ) P V P_V P V c ′ ∈ C V c'\in C_V c ′ ∈ C V □ \square □
Orbits of pure colors
The goal of this section is classify the restriction of the GL n ( F q ) \operatorname{GL}_n(\mathbb{F}_q) GL n ( F q ) v c = ( c , red , … , red ) ∈ C n v_c=(c,\operatorname{red},\ldots,\operatorname{red})\in C^n v c = ( c , red , … , red ) ∈ C n c ∈ C c\in C c ∈ C
The action of the group GL ∞ ( F q ) \operatorname{GL}_\infty(\mathbb{F}_q) GL ∞ ( F q ) C N C^{\mathbb{N}} C N GL 1 ( F q ) ≅ F q × \operatorname{GL}_1(\mathbb{F}_q)\cong\mathbb{F}_q^\times GL 1 ( F q ) ≅ F q × C C C α ⋅ c \alpha\cdot c α ⋅ c α ∈ F q × \alpha\in\mathbb{F}_q^\times α ∈ F q × c ∈ C c\in C c ∈ C Theorem 3.2 and by fact the action is local, we have α ⋅ red = red \alpha\cdot\operatorname{red}=\operatorname{red} α ⋅ red = red α ∈ F q × \alpha\in\mathbb{F}_q^\times α ∈ F q × 0 ⋅ c = red 0\cdot c=\operatorname{red} 0 ⋅ c = red c ∈ C c\in C c ∈ C
We now claim that the GL 1 ( F q ) \operatorname{GL}_1(\mathbb{F}_q) GL 1 ( F q ) C C C α I = D 1 ( α ) ⋯ D n ( α ) ∈ GL n ( F q ) \alpha I=D_1(\alpha)\cdots D_n(\alpha)\in\operatorname{GL}_n(\mathbb{F}_q) α I = D 1 ( α ) ⋯ D n ( α ) ∈ GL n ( F q ) α ∈ F q × \alpha\in\mathbb{F}_q^\times α ∈ F q × α I ⋅ v c = v α ⋅ c \alpha I\cdot v_c=v_{\alpha\cdot c} α I ⋅ v c = v α ⋅ c c ∈ C c\in C c ∈ C D i ( α ) D_i(\alpha) D i ( α ) i i i GL 1 ( F q ) \operatorname{GL}_1(\mathbb{F}_q) GL 1 ( F q ) α I \alpha I α I P V P_V P V P H P_H P H GL n − 1 ( F q ) \operatorname{GL}_{n-1}(\mathbb{F}_q) GL n − 1 ( F q ) C ∖ { r e d } C \setminus\{\mathrm{red}\} C ∖ { red } Theorem 3.2 . We later extend it to an F q \mathbb{F}_q F q C V C_V C V C H C_H C H
Proposition 3.7. Let n ≥ 3 n \ge3 n ≥ 3 c ∈ C c \in C c ∈ C
O c = { ρ } ∪ GL n ( F q ) ⋅ v c ⊆ C n O_c = \{\rho\} \cup\operatorname{GL}_n(\mathbb{F}_q) \cdot v_c \subseteq C^n O c = { ρ } ∪ GL n ( F q ) ⋅ v c ⊆ C n be the orbit of v c v_c v c C n C^n C n ρ = ( r e d , … , r e d ) \rho= (\mathrm{red},\ldots,\mathrm{red}) ρ = ( red , … , red )
T c = F q ⋅ c ⊆ C T_c = \mathbb{F}_q \cdot c \subseteq C T c = F q ⋅ c ⊆ C be the set of multiples of c c c θ c : T c → F q \theta_c : T_c \to\mathbb{F}_q θ c : T c → F q α ⋅ c \alpha\cdot c α ⋅ c α \alpha α c c c T c ⊆ C V T_c \subseteq C_V T c ⊆ C V T c ⊆ C H T_c \subseteq C_H T c ⊆ C H O c O_c O c T c n T_c^n T c n C n C^n C n θ c n : O c = T c n → F q n \theta_c^n : O_c = T_c^n \to\mathbb{F}_q^n θ c n : O c = T c n → F q n GL n ( F q ) \operatorname{GL}_n(\mathbb{F}_q) GL n ( F q ) F q n \mathbb{F}_q^n F q n
Proof. Let c ∈ C V c \in C_V c ∈ C V GL n ( F q ) ⋅ v c \operatorname{GL}_n(\mathbb{F}_q) \cdot v_c GL n ( F q ) ⋅ v c GL n ( F q ) \operatorname{GL}_n(\mathbb{F}_q) GL n ( F q ) GL n ( F q ) / P V ≅ F q n ∖ { 0 } \operatorname{GL}_n(\mathbb{F}_q)/P_V \cong\mathbb{F}_q^n \setminus\{0\} GL n ( F q ) / P V ≅ F q n ∖ { 0 } GL n ( F q ) \operatorname{GL}_n(\mathbb{F}_q) GL n ( F q ) v c v_c v c e 1 e^1 e 1 ρ ∈ O c \rho\in O_c ρ ∈ O c 0 ∈ F q n 0 \in\mathbb{F}_q^n 0 ∈ F q n GL n ( F q ) \operatorname{GL}_n(\mathbb{F}_q) GL n ( F q ) GL n ( F q ) \operatorname{GL}_n(\mathbb{F}_q) GL n ( F q ) φ c : O c → F q n \varphi_c : O_c \to\mathbb{F}_q^n φ c : O c → F q n φ c ( ρ ) = 0 \varphi_c(\rho) = 0 φ c ( ρ ) = 0 φ c ( v c ) = e 1 \varphi_c(v_c) = e^1 φ c ( v c ) = e 1
The fact that T c ⊆ C V T_c \subseteq C_V T c ⊆ C V T c ⊆ C H T_c \subseteq C_H T c ⊆ C H O c = T c n O_c = T_c^n O c = T c n φ c \varphi_c φ c θ c n \theta_c^n θ c n
Claim. Let 1 ≤ i , j ≤ n 1 \le i,j \le n 1 ≤ i , j ≤ n γ ∈ O c ⊆ C n \gamma\in O_c \subseteq C^n γ ∈ O c ⊆ C n γ i = γ j \gamma_i = \gamma_j γ i = γ j φ c ( γ ) i = φ c ( γ ) j \varphi_c(\gamma)_i = \varphi_c(\gamma)_j φ c ( γ ) i = φ c ( γ ) j
Proof. The GL ∞ ( F q ) \operatorname{GL}_\infty(\mathbb{F}_q) GL ∞ ( F q ) γ i = γ j \gamma_i = \gamma_j γ i = γ j σ i j ⋅ γ = γ \sigma_{ij} \cdot\gamma= \gamma σ ij ⋅ γ = γ φ c \varphi_c φ c σ i j φ c ( γ ) = φ c ( γ ) \sigma_{ij}\varphi_c(\gamma) = \varphi_c(\gamma) σ ij φ c ( γ ) = φ c ( γ ) φ c ( γ ) i = φ c ( γ ) j \varphi_c(\gamma)_i = \varphi_c(\gamma)_j φ c ( γ ) i = φ c ( γ ) j
Claim. Any pure color d ∈ C V d \in C_V d ∈ C V E 21 ⋅ ( d , r e d ) = ( d , d ) E_{21} \cdot(d,\mathrm{red}) = (d,d) E 21 ⋅ ( d , red ) = ( d , d )
Proof. Let d ∈ C V d \in C_V d ∈ C V d = r e d d = \mathrm{red} d = red E 21 ⋅ ( r e d , r e d ) = ( r e d , r e d ) E_{21} \cdot(\mathrm{red},\mathrm{red}) = (\mathrm{red},\mathrm{red}) E 21 ⋅ ( red , red ) = ( red , red ) Theorem 3.2 , in which case the claim holds. We assume from now on d ≠ r e d d \ne\mathrm{red} d = red
Recall that we identify GL 2 ( F q ) \operatorname{GL}_2(\mathbb{F}_q) GL 2 ( F q ) C 2 C^2 C 2 C n C^n C n 3 , … , n 3,\ldots,n 3 , … , n ( d , r e d ) (d,\mathrm{red}) ( d , red ) v d v_d v d φ d : O d → F q n \varphi_d : O_d \to\mathbb{F}_q^n φ d : O d → F q n d d d φ d ( ( d , r e d ) ) = φ d ( v d ) = e 1 \varphi_d((d,\mathrm{red})) = \varphi_d(v_d) = e^1 φ d (( d , red )) = φ d ( v d ) = e 1 E 21 e 1 = e 1 + e 2 E_{21}e^1 = e^1 + e^2 E 21 e 1 = e 1 + e 2
E 21 ⋅ ( d , r e d ) = ( d ′ , d ′ ) E_{21} \cdot(d,\mathrm{red}) = (d',d') E 21 ⋅ ( d , red ) = ( d ′ , d ′ ) for some color d ′ ∈ C d' \in C d ′ ∈ C d ′ ∈ C V d' \in C_V d ′ ∈ C V d ′ ′ ∈ C V d'' \in C_V d ′′ ∈ C V
E 21 ⋅ ( d ′ , r e d ) = ( d ′ ′ , d ′ ′ ) . E_{21} \cdot(d',\mathrm{red}) = (d'',d''). E 21 ⋅ ( d ′ , red ) = ( d ′′ , d ′′ ) . Altogether, we get
E 21 E 31 ⋅ ( d , r e d , r e d ) = E 21 ⋅ ( d ′ , r e d , d ′ ) = ( d ′ ′ , d ′ ′ , d ′ ) . (7) E_{21}E_{31} \cdot(d,\mathrm{red},\mathrm{red}) = E_{21} \cdot(d',\mathrm{red},d') = (d'',d'',d'). \tag*{(7)} E 21 E 31 ⋅ ( d , red , red ) = E 21 ⋅ ( d ′ , red , d ′ ) = ( d ′′ , d ′′ , d ′ ) . (7) On the other hand, applying φ d \varphi_d φ d
φ d ( E 21 E 31 ⋅ ( d , r e d , r e d ) ) = E 21 E 31 e 1 = ( 1 , 1 , 1 ) t . \varphi_d\bigl(E_{21}E_{31} \cdot(d,\mathrm{red},\mathrm{red})\bigr) = E_{21}E_{31}e^1 = (1,1,1)^t. φ d ( E 21 E 31 ⋅ ( d , red , red ) ) = E 21 E 31 e 1 = ( 1 , 1 , 1 ) t . Thus the colors appearing on the right-hand side of eq. (7) are equal to each other, namely d ′ = d ′ ′ d' = d'' d ′ = d ′′
E 21 ⋅ ( d , r e d ) = E 21 ⋅ ( d ′ , r e d ) . E_{21} \cdot(d,\mathrm{red}) = E_{21} \cdot(d',\mathrm{red}). E 21 ⋅ ( d , red ) = E 21 ⋅ ( d ′ , red ) . We obtain d = d ′ d = d' d = d ′
To ease notations, and having the color c c c O = O c O=O_c O = O c T = T c T=T_c T = T c φ = φ c \varphi=\varphi_c φ = φ c θ = θ c \theta=\theta_c θ = θ c O = T n O=T^n O = T n φ = θ n \varphi=\theta^n φ = θ n T n ⊆ O T^n\subseteq O T n ⊆ O
Consider a point γ ∈ T n \gamma\in T^n γ ∈ T n γ ∈ O \gamma\in O γ ∈ O φ ( γ ) = ( θ ( γ i ) ) i \varphi(\gamma)=(\theta(\gamma_i))_i φ ( γ ) = ( θ ( γ i ) ) i 1 ≤ i ≤ n 1\leq i\leq n 1 ≤ i ≤ n γ \gamma γ γ = ρ \gamma=\rho γ = ρ γ \gamma γ d ∈ T c d\in T_c d ∈ T c 1 ≤ i ≤ n 1\leq i\leq n 1 ≤ i ≤ n
γ = σ 1 i ⋅ v d = σ 1 i D 1 ( θ ( d ) ) ⋅ v c ∈ O , \gamma=\sigma_{1i}\cdot v_d=\sigma_{1i}D_1(\theta(d))\cdot v_c\in O, γ = σ 1 i ⋅ v d = σ 1 i D 1 ( θ ( d )) ⋅ v c ∈ O , and
φ ( γ ) = φ ( σ 1 i D 1 ( θ ( d ) ) ⋅ v c ) = σ 1 i D 1 ( θ ( d ) ) e 1 = ( 0 , … , θ ( d ) , … , 0 ) t \varphi(\gamma)=\varphi(\sigma_{1i}D_1(\theta(d))\cdot v_c)=\sigma_{1i}D_1(\theta(d))e_1=(0,\ldots,\theta(d),\ldots,0)^t φ ( γ ) = φ ( σ 1 i D 1 ( θ ( d )) ⋅ v c ) = σ 1 i D 1 ( θ ( d )) e 1 = ( 0 , … , θ ( d ) , … , 0 ) t as required.
For the induction step, suppose two entries γ i \gamma_i γ i γ j \gamma_j γ j 1 ≤ i ≠ j ≤ n 1\leq i\neq j\leq n 1 ≤ i = j ≤ n α ∈ F q \alpha\in\mathbb{F}_q α ∈ F q α ⋅ γ j = γ i \alpha\cdot\gamma_j=\gamma_i α ⋅ γ j = γ i D j ( α ) ⋅ γ D_j(\alpha)\cdot\gamma D j ( α ) ⋅ γ i i i j j j E i j − 1 D j ( α ) ⋅ γ E_{ij}^{-1}D_j(\alpha)\cdot\gamma E ij − 1 D j ( α ) ⋅ γ γ \gamma γ E i j − 1 D j ( α ) ⋅ γ ∈ O E_{ij}^{-1}D_j(\alpha)\cdot\gamma\in O E ij − 1 D j ( α ) ⋅ γ ∈ O γ ∈ O \gamma\in O γ ∈ O
φ ( γ ) = D j ( α − 1 ) E i j φ ( E i j − 1 D j ( α ) ⋅ γ ) = D j ( α − 1 ) E i j ( θ ( γ 1 ) , … , 0 , … , θ ( γ i ) , … , θ ( γ n ) ) t = ( θ ( γ k ) ) k , \begin{aligned}
\varphi(\gamma)=D_j(\alpha^{-1})E_{ij}\varphi(E_{ij}^{-1}D_j(\alpha)\cdot\gamma) \\
&=D_j(\alpha^{-1})E_{ij}(\theta(\gamma_1),\ldots,0,\ldots,\theta(\gamma_i),\ldots,\theta(\gamma_n))^t=(\theta(\gamma_k))_k,
\end{aligned} φ ( γ ) = D j ( α − 1 ) E ij φ ( E ij − 1 D j ( α ) ⋅ γ ) = D j ( α − 1 ) E ij ( θ ( γ 1 ) , … , 0 , … , θ ( γ i ) , … , θ ( γ n ) ) t = ( θ ( γ k ) ) k , as required.
Lastly, the case where c ∈ C H c\in C_H c ∈ C H GL n ( F q ) / P H ≅ F q n ∖ { 0 } \operatorname{GL}_n(\mathbb{F}_q)/P_H\cong\mathbb{F}_q^n\setminus\{0\} GL n ( F q ) / P H ≅ F q n ∖ { 0 } GL n ( F q ) \operatorname{GL}_n(\mathbb{F}_q) GL n ( F q )
Vector space structure on pure colors
We define an addition operator on the set C V C_V C V C H C_H C H c , d ∈ C V c,d\in C_V c , d ∈ C V c + d = d ′ c+d=d' c + d = d ′
E 21 ⋅ ( c , d ) = ( c ′ , d ′ ) ∈ C 2 . E_{21}\cdot(c,d)=(c',d')\in C^2. E 21 ⋅ ( c , d ) = ( c ′ , d ′ ) ∈ C 2 . For a pair of horizontal colors c , d ∈ C H c,d\in C_H c , d ∈ C H c + d = d ′ c+d=d' c + d = d ′
E 12 − 1 ⋅ ( c , d ) = ( c ′ , d ′ ) ∈ C 2 . E_{12}^{-1}\cdot(c,d)=(c',d')\in C^2. E 12 − 1 ⋅ ( c , d ) = ( c ′ , d ′ ) ∈ C 2 . Notice that by Theorem 3.6, in both cases the result c + d c+d c + d c c c d d d
Proposition 3.8. The operations + : C V × C V → C V +:C_V\times C_V\to C_V + : C V × C V → C V F q × C V → C V \mathbb{F}_q\times C_V\to C_V F q × C V → C V C V C_V C V F q \mathbb{F}_q F q GL n ( F q ) \operatorname{GL}_n(\mathbb{F}_q) GL n ( F q ) C V n C_V^n C V n C H C_H C H GL n ( F q ) \operatorname{GL}_n(\mathbb{F}_q) GL n ( F q ) C H n C_H^n C H n
Proof. We verify the vector space axioms over F q \mathbb{F}_q F q C V C_V C V c , d ∈ C V c,d\in C_V c , d ∈ C V
E 31 E 32 ⋅ ( c , d , r e d ) = E 31 ⋅ ( c , d , d ) = ( c ′ , d , c + d ) E_{31}E_{32}\cdot(c,d,\mathrm{red})=E_{31}\cdot(c,d,d)=(c',d,c+d) E 31 E 32 ⋅ ( c , d , red ) = E 31 ⋅ ( c , d , d ) = ( c ′ , d , c + d ) for some c ′ ∈ C V c'\in C_V c ′ ∈ C V c 1 ′ , c 2 ′ ∈ C V c'_1,c'_2\in C_V c 1 ′ , c 2 ′ ∈ C V E 32 E_{32} E 32 E 31 E_{31} E 31 E 32 E 31 E_{32}E_{31} E 32 E 31 d + c d+c d + c
We check associativity of addition. Let c 1 , c 2 , c 3 ∈ C V c_1,c_2,c_3\in C_V c 1 , c 2 , c 3 ∈ C V
E 41 E 42 E 43 ⋅ ( c 1 , c 2 , c 3 , r e d ) = ( c 1 ′ , c 2 ′ , c 3 , c 1 + ( c 2 + c 3 ) ) . E_{41}E_{42}E_{43}\cdot(c_1,c_2,c_3,\mathrm{red})=(c'_1,c'_2,c_3,c_1+(c_2+c_3)). E 41 E 42 E 43 ⋅ ( c 1 , c 2 , c 3 , red ) = ( c 1 ′ , c 2 ′ , c 3 , c 1 + ( c 2 + c 3 )) . Since all these elementary matrices commute, the application of E 43 E 41 E 42 E_{43}E_{41}E_{42} E 43 E 41 E 42 ( c 1 , c 2 , c 3 , r e d ) (c_1,c_2,c_3,\mathrm{red}) ( c 1 , c 2 , c 3 , red )
c 1 + ( c 2 + c 3 ) = c 3 + ( c 1 + c 2 ) = ( c 1 + c 2 ) + c 3 c_1+(c_2+c_3)=c_3+(c_1+c_2)=(c_1+c_2)+c_3 c 1 + ( c 2 + c 3 ) = c 3 + ( c 1 + c 2 ) = ( c 1 + c 2 ) + c 3 as required. Further, by Theorem 3.7, the distinguished color r e d ∈ C V \mathrm{red}\in C_V red ∈ C V ( − 1 ) ⋅ c (-1)\cdot c ( − 1 ) ⋅ c ( C V , + ) (C_V,+) ( C V , + )
We consider the axioms of the scalar multiplication. Let α , β ∈ F q \alpha,\beta\in\mathbb{F}_q α , β ∈ F q α ⋅ ( β ⋅ c ) = ( α β ) ⋅ c \alpha\cdot(\beta\cdot c)=(\alpha\beta)\cdot c α ⋅ ( β ⋅ c ) = ( α β ) ⋅ c 1 ⋅ c = c 1\cdot c=c 1 ⋅ c = c
( α α + β ) = ( α 0 α + β β ) e 1 = θ c ( E 21 D 1 ( α ) D 2 ( β ) E 21 − 1 ⋅ v c ) = θ c ( ( c ′ , α ⋅ c + β ⋅ c ) ) . \begin{pmatrix}\alpha\\ \alpha+\beta\end{pmatrix}
=
\begin{pmatrix}\alpha&0\\ \alpha+\beta&\beta\end{pmatrix}e_1
=
\theta_c(E_{21}D_1(\alpha)D_2(\beta)E_{21}^{-1}\cdot v_c)
=
\theta_c((c',\alpha\cdot c+\beta\cdot c)). ( α α + β ) = ( α α + β 0 β ) e 1 = θ c ( E 21 D 1 ( α ) D 2 ( β ) E 21 − 1 ⋅ v c ) = θ c (( c ′ , α ⋅ c + β ⋅ c )) . This implies the axiom ( α + β ) ⋅ c = α ⋅ c + β ⋅ c (\alpha+\beta)\cdot c=\alpha\cdot c+\beta\cdot c ( α + β ) ⋅ c = α ⋅ c + β ⋅ c α I \alpha I α I E 21 E_{21} E 21
( α ⋅ c ′ ′ , α ⋅ ( c + d ) ) = ( α I 2 ) E 21 ⋅ ( c , d ) = E 21 ( α I 2 ) ⋅ ( c , d ) = E 21 ⋅ ( α ⋅ c , α ⋅ d ) = ( c ′ , α ⋅ c + α ⋅ d ) . (\alpha\cdot c'',\alpha\cdot(c+d))=(\alpha I_2)E_{21}\cdot(c,d)=E_{21}(\alpha I_2)\cdot(c,d)=E_{21}\cdot(\alpha\cdot c,\alpha\cdot d)=(c',\alpha\cdot c+\alpha\cdot d). ( α ⋅ c ′′ , α ⋅ ( c + d )) = ( α I 2 ) E 21 ⋅ ( c , d ) = E 21 ( α I 2 ) ⋅ ( c , d ) = E 21 ⋅ ( α ⋅ c , α ⋅ d ) = ( c ′ , α ⋅ c + α ⋅ d ) . The distributivity axiom α ⋅ ( c + d ) = α ⋅ c + α ⋅ d \alpha\cdot(c+d)=\alpha\cdot c+\alpha\cdot d α ⋅ ( c + d ) = α ⋅ c + α ⋅ d C V C_V C V F q \mathbb{F}_q F q
It remains to show that the subset C V n C_V^n C V n G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q ) G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q ) C V n C_V^n C V n F q n \mathbb{F}_q^n F q n D i ( α ) D_i(\alpha) D i ( α ) C V n C_V^n C V n G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q ) E i j E_{ij} E ij D i ( α ) D_i(\alpha) D i ( α ) 1 ≤ i , j ≤ n 1\leq i,j\leq n 1 ≤ i , j ≤ n α ∈ F q × \alpha\in\mathbb{F}_q^\times α ∈ F q × C V n C_V^n C V n G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q )
It remains to show that the action of the elementary matrices E i j E_{ij} E ij C V n C_V^n C V n E 21 E_{21} E 21 c , d ∈ C V c,d\in C_V c , d ∈ C V c ′ ∈ C c'\in C c ′ ∈ C
E 21 ⋅ ( c , d ) = ( c ′ , c + d ) . E_{21}\cdot(c,d)=(c',c+d). E 21 ⋅ ( c , d ) = ( c ′ , c + d ) . We need to show that c ′ = c c'=c c ′ = c c ′ ∈ C V c'\in C_V c ′ ∈ C V
E 21 − 1 E 31 E 21 ⋅ ( c , d , r e d ) = E 21 − 1 E 31 ⋅ ( c ′ , c + d , r e d ) = E 21 − 1 ⋅ ( c ′ , c + d , c ′ ) = ( c , d , c ′ ) . E_{21}^{-1}E_{31}E_{21}\cdot(c,d,\mathrm{red})=E_{21}^{-1}E_{31}\cdot(c',c+d,\mathrm{red})=E_{21}^{-1}\cdot(c',c+d,c')=(c,d,c'). E 21 − 1 E 31 E 21 ⋅ ( c , d , red ) = E 21 − 1 E 31 ⋅ ( c ′ , c + d , red ) = E 21 − 1 ⋅ ( c ′ , c + d , c ′ ) = ( c , d , c ′ ) . The two matrices E 21 E_{21} E 21 E 31 E_{31} E 31
E 31 ⋅ ( c , d , r e d ) = ( c , d , c ) . E_{31}\cdot(c,d,\mathrm{red})=(c,d,c). E 31 ⋅ ( c , d , red ) = ( c , d , c ) . We conclude that c = c ′ c=c' c = c ′ C H C_H C H C H n C_H^n C H n G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q )
Colors as products of pure colors
In this section we prove Theorem 3.2 . By Theorem 3.8 the sets C V C_V C V C H C_H C H F q \mathbb{F}_q F q G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q ) C V n C_V^n C V n C H n C_H^n C H n C C C C V × C H C_V\times C_H C V × C H
Lemma 3.9. Let c ∈ C c\in C c ∈ C
E 21 ⋅ ( c , r e d ) = ( c , d ) E_{21}\cdot(c,\mathrm{red})=(c,d) E 21 ⋅ ( c , red ) = ( c , d ) and
E 12 − 1 ⋅ ( c , r e d ) = ( c , d ′ ) E_{12}^{-1}\cdot(c,\mathrm{red})=(c,d') E 12 − 1 ⋅ ( c , red ) = ( c , d ′ ) for some pure colors d ∈ C V d\in C_V d ∈ C V d ′ ∈ C H d'\in C_H d ′ ∈ C H
Proof. We prove only the first assertion for E 21 E_{21} E 21 C V C_V C V E 12 − 1 E_{12}^{-1} E 12 − 1 C H C_H C H E 21 − 1 ⋅ ( c , r e d ) = ( c 1 , d ) E_{21}^{-1} \cdot(c,\mathrm{red}) = (c_1,d) E 21 − 1 ⋅ ( c , red ) = ( c 1 , d ) c 1 ∈ C c_1 \in C c 1 ∈ C d ∈ C V d \in C_V d ∈ C V c = c 1 c=c_1 c = c 1 c 2 , c 3 , d 2 , d 3 ∈ C c_2,c_3,d_2,d_3 \in C c 2 , c 3 , d 2 , d 3 ∈ C
E 21 − 1 E 31 E 21 ⋅ ( c , r e d , r e d ) = E 21 − 1 E 31 ⋅ ( c 1 , d , r e d ) = E 21 − 1 ⋅ ( c 2 , d , d 2 ) = ( c 3 , d 3 , d 2 ) . (8) E_{21}^{-1}E_{31}E_{21}\cdot(c,\mathrm{red},\mathrm{red})=E_{21}^{-1}E_{31}\cdot(c_1,d,\mathrm{red})=E_{21}^{-1}\cdot(c_2,d,d_2)=(c_3,d_3,d_2). \tag*{(8)} E 21 − 1 E 31 E 21 ⋅ ( c , red , red ) = E 21 − 1 E 31 ⋅ ( c 1 , d , red ) = E 21 − 1 ⋅ ( c 2 , d , d 2 ) = ( c 3 , d 3 , d 2 ) . (8) The matrices E 21 E_{21} E 21 E 31 E_{31} E 31 E 21 − 1 E 31 E 21 = E 31 E_{21}^{-1}E_{31}E_{21}=E_{31} E 21 − 1 E 31 E 21 = E 31
( c 3 , d 3 , d 2 ) = ( c 1 , r e d , d ) . (c_3,d_3,d_2)=(c_1,\mathrm{red},d). ( c 3 , d 3 , d 2 ) = ( c 1 , red , d ) . Hence, the right-most equality in eq. (8) implies that
E 21 ⋅ ( c 1 , r e d ) = ( c 2 , d ) . E_{21}\cdot(c_1,\mathrm{red})=(c_2,d). E 21 ⋅ ( c 1 , red ) = ( c 2 , d ) . Next, we have the following, where the right-most equality uses the second claim from the proof of Theorem 3.7 applied for d ∈ C V d \in C_V d ∈ C V
E 32 − 1 E 31 E 21 ⋅ ( c , r e d , r e d ) = E 32 − 1 E 31 ⋅ ( c 1 , d , r e d ) = E 32 − 1 ⋅ ( c 2 , d , d ) = ( c 2 , d , r e d ) . E_{32}^{-1}E_{31}E_{21}\cdot(c,\mathrm{red},\mathrm{red})=E_{32}^{-1}E_{31}\cdot(c_1,d,\mathrm{red})=E_{32}^{-1}\cdot(c_2,d,d)=(c_2,d,\mathrm{red}). E 32 − 1 E 31 E 21 ⋅ ( c , red , red ) = E 32 − 1 E 31 ⋅ ( c 1 , d , red ) = E 32 − 1 ⋅ ( c 2 , d , d ) = ( c 2 , d , red ) . The standard commutator identity [ E 32 , E 21 ] = E 31 [E_{32},E_{21}]=E_{31} [ E 32 , E 21 ] = E 31
E 32 − 1 E 31 E 21 = E 21 E 32 − 1 . E_{32}^{-1}E_{31}E_{21}=E_{21}E_{32}^{-1}. E 32 − 1 E 31 E 21 = E 21 E 32 − 1 . Acting with the matrix on right-hand side of the above on the vector ( c , r e d , r e d ) (c,\mathrm{red},\mathrm{red}) ( c , red , red ) ( c 1 , d , r e d ) (c_1,d,\mathrm{red}) ( c 1 , d , red ) c 1 = c 2 c_1=c_2 c 1 = c 2
E 21 ⋅ ( c , r e d ) = ( c 1 , d ) = ( c 2 , d ) = E 21 ⋅ ( c 1 , r e d ) E_{21}\cdot(c,\mathrm{red})=(c_1,d)=(c_2,d)=E_{21}\cdot(c_1,\mathrm{red}) E 21 ⋅ ( c , red ) = ( c 1 , d ) = ( c 2 , d ) = E 21 ⋅ ( c 1 , red ) so that c = c 1 c=c_1 c = c 1
Note that Theorem 3.9 shows in particular that pure colors do exist.
Proof of Theorem 3.2 . We define a map
π : C → C V × C H \pi:C\to C_V\times C_H π : C → C V × C H as follows. For each color c ∈ C c\in C c ∈ C
E 21 ⋅ ( c , r e d ) = ( c , π V ( c ) ) and E 12 − 1 ⋅ ( c , r e d ) = ( c , π H ( c ) ) . E_{21}\cdot(c,\mathrm{red})=(c,\pi_V(c))\quad\text{and}\quad E_{12}^{-1}\cdot(c,\mathrm{red})=(c,\pi_H(c)). E 21 ⋅ ( c , red ) = ( c , π V ( c )) and E 12 − 1 ⋅ ( c , red ) = ( c , π H ( c )) . Note that the first entries in the above pairs are equal to c c c π \pi π
π ( c ) = ( π V ( c ) , π H ( c ) ) \pi(c)=(\pi_V(c),\pi_H(c)) π ( c ) = ( π V ( c ) , π H ( c )) for all c ∈ C c\in C c ∈ C π \pi π π n : C n → C V n × C H n \pi^n:C^n\to C_V^n\times C_H^n π n : C n → C V n × C H n GL n ( F q ) \operatorname{GL}_n(\mathbb{F}_q) GL n ( F q ) GL n ( F q ) \operatorname{GL}_n(\mathbb{F}_q) GL n ( F q ) C V n C_V^n C V n C H n C_H^n C H n π N : C N → C V N × C H N ≅ V k , ℓ \pi^N:C^N\to C_V^N\times C_H^N\cong V^{k,\ell} π N : C N → C V N × C H N ≅ V k , ℓ GL ∞ ( F q ) \operatorname{GL}_\infty(\mathbb{F}_q) GL ∞ ( F q )
Let us first show that the product map π n \pi^n π n GL n ( F q ) \operatorname{GL}_n(\mathbb{F}_q) GL n ( F q )
Π V n = ( I n 0 n I n I n ) ∈ GL 2 n ( F q ) . \Pi_V^n=\begin{pmatrix}I_n&0_n\\I_n&I_n\end{pmatrix}\in\operatorname{GL}_{2n}(\mathbb{F}_q). Π V n = ( I n I n 0 n I n ) ∈ GL 2 n ( F q ) . Denote as usual ρ = ( r e d , … , r e d ) ∈ C n \rho=(\mathrm{red},\ldots,\mathrm{red})\in C^n ρ = ( red , … , red ) ∈ C n ρ \rho ρ γ ∈ C n \gamma\in C^n γ ∈ C n
Π V n ⋅ ( γ , ρ ) = ( γ , π V n ( γ ) ) . \Pi_V^n\cdot(\gamma,\rho)=(\gamma,\pi^n_V(\gamma)). Π V n ⋅ ( γ , ρ ) = ( γ , π V n ( γ )) . Let A ∈ GL n ( F q ) A\in\operatorname{GL}_n(\mathbb{F}_q) A ∈ GL n ( F q )
( γ , A π V n ( γ ) ) = ( I n 0 n 0 n A ) Π V n ⋅ ( γ , ρ ) , (\gamma,A\pi^n_V(\gamma))=
\begin{pmatrix}I_n&0_n\\0_n&A\end{pmatrix}
\Pi_V^n\cdot(\gamma,\rho), ( γ , A π V n ( γ )) = ( I n 0 n 0 n A ) Π V n ⋅ ( γ , ρ ) , while on the other hand
( γ , π V n ( A ⋅ γ ) ) = ( A − 1 0 n 0 n I n ) Π V n ( A 0 n 0 n I n ) ⋅ ( γ , ρ ) . (\gamma,\pi^n_V(A\cdot\gamma))=
\begin{pmatrix}A^{-1}&0_n\\0_n&I_n\end{pmatrix}
\Pi_V^n
\begin{pmatrix}A&0_n\\0_n&I_n\end{pmatrix}
\cdot(\gamma,\rho). ( γ , π V n ( A ⋅ γ )) = ( A − 1 0 n 0 n I n ) Π V n ( A 0 n 0 n I n ) ⋅ ( γ , ρ ) . Denote by M 1 M_1 M 1 M 2 M_2 M 2 ( γ , ρ ) (\gamma,\rho) ( γ , ρ ) M 1 M_1 M 1 M 2 M_2 M 2 ( γ , ρ ) (\gamma,\rho) ( γ , ρ ) M 2 − 1 M 1 M_2^{-1}M_1 M 2 − 1 M 1 M 2 − 1 M 1 = ( I n 0 n 0 n A ) M_2^{-1}M_1=\begin{pmatrix}I_n&0_n\\0_n&A\end{pmatrix} M 2 − 1 M 1 = ( I n 0 n 0 n A ) ( γ , ρ ) (\gamma,\rho) ( γ , ρ ) (4) . We conclude that A π V n ( γ ) = π V n ( A ⋅ γ ) A\pi_V^n(\gamma)=\pi_V^n(A\cdot\gamma) A π V n ( γ ) = π V n ( A ⋅ γ ) π H n \pi_H^n π H n
Next, we wish to define a map ϕ : C V × C H → C \phi:C_V\times C_H\to C ϕ : C V × C H → C π \pi π
Φ 1 = E 23 − 1 E 31 and Φ 2 = E 31 E 23 − 1 . \Phi_1=E_{23}^{-1}E_{31}\qquad\text{and}\qquad\Phi_2=E_{31}E_{23}^{-1}. Φ 1 = E 23 − 1 E 31 and Φ 2 = E 31 E 23 − 1 . Given a vertical c V ∈ C V c_V\in C_V c V ∈ C V c H ∈ C H c_H\in C_H c H ∈ C H
Φ 1 ⋅ ( c V , c H , red ) = ( c V , c H ′ , ϕ 1 ( c V , c H ) ) , Φ 2 ⋅ ( c V , c H , red ) = ( c V ′ , c H , ϕ 2 ( c V , c H ) ) , \Phi_1\cdot(c_V,c_H,\operatorname{red})=(c_V,c_H',\phi_1(c_V,c_H)),\qquad\Phi_2\cdot(c_V,c_H,\operatorname{red})=(c_V',c_H,\phi_2(c_V,c_H)), Φ 1 ⋅ ( c V , c H , red ) = ( c V , c H ′ , ϕ 1 ( c V , c H )) , Φ 2 ⋅ ( c V , c H , red ) = ( c V ′ , c H , ϕ 2 ( c V , c H )) , for some pair of maps ϕ 1 , ϕ 2 : C V × C H → C \phi_1,\phi_2:C_V\times C_H\to C ϕ 1 , ϕ 2 : C V × C H → C c V ′ , c H ′ ∈ C c_V',c_H'\in C c V ′ , c H ′ ∈ C Φ 1 \Phi_1 Φ 1 Φ 2 \Phi_2 Φ 2 E 31 E_{31} E 31 E 23 − 1 E_{23}^{-1} E 23 − 1
We claim that ϕ 1 = ϕ 2 \phi_1=\phi_2 ϕ 1 = ϕ 2
( c V , c H ′ , ϕ 1 ( c V , c H ) ) = Φ 1 ⋅ ( c V , c H , red ) = Φ 1 Φ 2 − 1 Φ 2 ⋅ ( c V , c H , red ) = E 21 − 1 ⋅ ( c V ′ , c H , ϕ 2 ( c V , c H ) ) (c_V,c_H',\phi_1(c_V,c_H))=\Phi_1\cdot(c_V,c_H,\operatorname{red})=\Phi_1\Phi_2^{-1}\Phi_2\cdot(c_V,c_H,\operatorname{red})=E_{21}^{-1}\cdot(c_V',c_H,\phi_2(c_V,c_H)) ( c V , c H ′ , ϕ 1 ( c V , c H )) = Φ 1 ⋅ ( c V , c H , red ) = Φ 1 Φ 2 − 1 Φ 2 ⋅ ( c V , c H , red ) = E 21 − 1 ⋅ ( c V ′ , c H , ϕ 2 ( c V , c H )) and the support of E 21 − 1 E_{21}^{-1} E 21 − 1 ϕ \phi ϕ
Now we show that ϕ = π − 1 \phi=\pi^{-1} ϕ = π − 1 c V ∈ C V c_V\in C_V c V ∈ C V c H ∈ C H c_H\in C_H c H ∈ C H c = ϕ ( c V , c H ) ∈ C c=\phi(c_V,c_H)\in C c = ϕ ( c V , c H ) ∈ C π V ( c ) = c V \pi_V(c)=c_V π V ( c ) = c V π H ( c ) = c H \pi_H(c)=c_H π H ( c ) = c H E 43 E_{43} E 43 Φ 1 \Phi_1 Φ 1 ( c V , c H , red , red ) (c_V,c_H,\operatorname{red},\operatorname{red}) ( c V , c H , red , red )
E 43 Φ 1 ⋅ ( c V , c H , red , red ) = ( c V , c H ′ , c , π V ( c ) ) . E_{43}\Phi_1\cdot(c_V,c_H,\operatorname{red},\operatorname{red})=(c_V,c_H',c,\pi_V(c)). E 43 Φ 1 ⋅ ( c V , c H , red , red ) = ( c V , c H ′ , c , π V ( c )) . We argue that c V = π V ( c ) c_V=\pi_V(c) c V = π V ( c ) σ 14 \sigma_{14} σ 14
( E 43 Φ 1 ) − 1 σ 14 E 43 Φ 1 = ( 1 0 1 1 0 1 0 0 0 0 0 − 1 0 0 − 1 0 ) . (E_{43}\Phi_1)^{-1}\sigma_{14}E_{43}\Phi_1=
\begin{pmatrix}
1&0&1&1\\
0&1&0&0\\
0&0&0&-1\\
0&0&-1&0
\end{pmatrix}. ( E 43 Φ 1 ) − 1 σ 14 E 43 Φ 1 = 1 0 0 0 0 1 0 0 1 0 0 − 1 1 0 − 1 0 . This matrix fixes the second coordinate and acts on the remaining ones, which all lie in C V C_V C V ( c V , c H , red , red ) (c_V,c_H,\operatorname{red},\operatorname{red}) ( c V , c H , red , red ) π H ( c ) = c H \pi_H(c)=c_H π H ( c ) = c H π ∘ ϕ = id \pi\circ\phi=\operatorname{id} π ∘ ϕ = id
To show ϕ ∘ π = id \phi\circ\pi=\operatorname{id} ϕ ∘ π = id c ∈ C c\in C c ∈ C
Π = E 13 − 1 E 21 = ( 1 0 − 1 1 1 0 0 0 1 ) \Pi=E_{13}^{-1}E_{21}=
\begin{pmatrix}
1&0&-1\\
1&1&0\\
0&0&1
\end{pmatrix} Π = E 13 − 1 E 21 = 1 1 0 0 1 0 − 1 0 1 has the effect of applying both maps π V \pi_V π V π H \pi_H π H
Π ⋅ ( c , red , red ) = ( c , π V ( c ) , π H ( c ) ) . \Pi\cdot(c,\operatorname{red},\operatorname{red})=(c,\pi_V(c),\pi_H(c)). Π ⋅ ( c , red , red ) = ( c , π V ( c ) , π H ( c )) . Take the vector v c = ( c , red , red , red ) v_c=(c,\operatorname{red},\operatorname{red},\operatorname{red}) v c = ( c , red , red , red ) Π \Pi Π Φ 1 \Phi_1 Φ 1 c c c ϕ ( π ( c ) ) \phi(\pi(c)) ϕ ( π ( c )) σ 14 \sigma_{14} σ 14
( ( 1 0 3 0 3 Φ 1 ) ( Π 0 3 0 3 1 ) ) − 1 σ 14 ( ( 1 0 3 0 3 Φ 1 ) ( Π 0 3 0 3 1 ) ) = ( 1 0 0 0 0 1 0 0 0 − 1 0 − 1 0 − 1 − 1 0 ) \left(\begin{pmatrix}1&0_3\\0_3&\Phi_1\end{pmatrix}\begin{pmatrix}\Pi&0_3\\0_3&1\end{pmatrix}\right)^{-1}
\sigma_{14}
\left(\begin{pmatrix}1&0_3\\0_3&\Phi_1\end{pmatrix}\begin{pmatrix}\Pi&0_3\\0_3&1\end{pmatrix}\right)
=
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
0&-1&0&-1\\
0&-1&-1&0
\end{pmatrix} ( ( 1 0 3 0 3 Φ 1 ) ( Π 0 3 0 3 1 ) ) − 1 σ 14 ( ( 1 0 3 0 3 Φ 1 ) ( Π 0 3 0 3 1 ) ) = 1 0 0 0 0 1 − 1 − 1 0 0 0 − 1 0 0 − 1 0 stabilizes v c v_c v c ( 1 0 3 0 3 B ) \begin{pmatrix}1&0_3\\0_3&B\end{pmatrix} ( 1 0 3 0 3 B ) B B B ϕ ∘ π = id \phi\circ\pi=\operatorname{id} ϕ ∘ π = id π \pi π
By Theorem 3.8 the sets C V C_V C V C H C_H C H F q \mathbb{F}_q F q ∣ C V ∣ = q k |C_V| = q^k ∣ C V ∣ = q k ∣ C H ∣ = q ℓ |C_H| = q^\ell ∣ C H ∣ = q ℓ k , ℓ ≥ 0 k,\ell\ge0 k , ℓ ≥ 0 π \pi π q k + ℓ = ∣ C ∣ = q r q^{k+\ell} = |C| = q^r q k + ℓ = ∣ C ∣ = q r k + ℓ = r k+\ell= r k + ℓ = r π \pi π ρ \rho ρ ( r e d , r e d ) ∈ C V × C H (\mathrm{red},\mathrm{red}) \in C_V \times C_H ( red , red ) ∈ C V × C H □ \square □
Invariant random subgroups and traces
In this section we recall the notions of invariant random subgroups and Thoma traces. In addition, we discuss the classification of these objects for the finitary alternating group A ∞ A_\infty A ∞
Invariant random subgroups
Invariant random subgroups. Let G G G Chabauty space of the group G G G
Sub ( G ) = { H : H ≤ G } . \operatorname{Sub}(G) = \{H : H \le G\}. Sub ( G ) = { H : H ≤ G } . The space Sub ( G ) \operatorname{Sub}(G) Sub ( G ) { 0 , 1 } G \{0,1\}^{G} { 0 , 1 } G Sub ( G ) \operatorname{Sub}(G) Sub ( G ) { 0 , 1 } G \{0,1\}^{G} { 0 , 1 } G Chabauty topology . The group G G G Sub ( G ) \operatorname{Sub}(G) Sub ( G )
g ⋅ H = g H g − 1 ∀ H ∈ Sub ( G ) , g ∈ G . g \cdot H = gHg^{-1} \quad\forall H \in\operatorname{Sub}(G),\ g \in G. g ⋅ H = g H g − 1 ∀ H ∈ Sub ( G ) , g ∈ G . Let M ( Sub ( G ) ) \mathcal{M}(\operatorname{Sub}(G)) M ( Sub ( G )) Sub ( G ) \operatorname{Sub}(G) Sub ( G ) ∗ * ∗ G G G M ( Sub ( G ) ) \mathcal{M}(\operatorname{Sub}(G)) M ( Sub ( G )) invariant random subgroup is a G G G M ( Sub ( G ) ) \mathcal{M}(\operatorname{Sub}(G)) M ( Sub ( G )) IRS ( G ) \operatorname{IRS}(G) IRS ( G )
The set IRS ( G ) \operatorname{IRS}(G) IRS ( G ) ∗ * ∗ M ( Sub ( G ) ) \mathcal{M}(\operatorname{Sub}(G)) M ( Sub ( G )) ergodic if it is an extreme point of the space IRS ( G ) \operatorname{IRS}(G) IRS ( G ) G G G Sub ( G ) \operatorname{Sub}(G) Sub ( G ) δ { I } ∈ IRS ( G ) \delta_{\{I\}} \in\operatorname{IRS}(G) δ { I } ∈ IRS ( G ) trivial .
Assume that the countable group G G G ( X , ν ) (X,\nu) ( X , ν ) Stab G : X → Sub ( G ) \operatorname{Stab}_G : X \to\operatorname{Sub}(G) Stab G : X → Sub ( G ) ν \nu ν G G G G G G [1 ] .
Definition 4.1. Let H H H G G G
Sub ( G ) → Sub ( H ) , K ↦ K ∩ H ∀ K ∈ Sub ( G ) . \operatorname{Sub}(G) \to\operatorname{Sub}(H), \quad K \mapsto K \cap H \quad\forall K \in\operatorname{Sub}(G). Sub ( G ) → Sub ( H ) , K ↦ K ∩ H ∀ K ∈ Sub ( G ) . The intersection map is H H H IRS ( G ) → IRS ( H ) \operatorname{IRS}(G) \to\operatorname{IRS}(H) IRS ( G ) → IRS ( H )
See e.g. [1 , 10 , 22 ] for more details about invariant random subgroups.
Thoma and Vershik traces
We recall the notion of a trace in the sense of Thoma [Th06a, Th06b]. These are generalizations of the normalized trace functions of finite-dimensional unitary representations of a group. A very good reference for theory of Thoma traces is [4 ] .
Definition 4.2. Let G G G χ : G → C \chi: G \to\mathbb{C} χ : G → C trace of G G G
(1) χ \chi χ χ ( e ) = 1 \chi(e) = 1 χ ( e ) = 1
(2) χ \chi χ χ ( g − 1 h g ) = χ ( h ) \chi(g^{-1}hg) = \chi(h) χ ( g − 1 h g ) = χ ( h ) g , h ∈ G g,h \in G g , h ∈ G
(3) χ \chi χ
∑ i , j = 1 n a j ‾ a i χ ( g j − 1 g i ) ≥ 0 (9) \sum_{i,j=1}^{n} \overline{a_j}a_i\chi(g_j^{-1}g_i) \ge0 \tag*{(9)} i , j = 1 ∑ n a j a i χ ( g j − 1 g i ) ≥ 0 (9) for all n ∈ N n \in\mathbb{N} n ∈ N a 1 , … , a n ∈ C a_1,\ldots,a_n \in\mathbb{C} a 1 , … , a n ∈ C g 1 , … , g n ∈ G g_1,\ldots,g_n \in G g 1 , … , g n ∈ G Tr ( G ) \operatorname{Tr}(G) Tr ( G ) G G G ℓ ∞ ( G ) \ell^\infty(G) ℓ ∞ ( G ) Thoma dual of G G G Ch ( G ) \operatorname{Ch}(G) Ch ( G ) Ch ( G ) \operatorname{Ch}(G) Ch ( G ) characters of G G G
We mention that the terminology in Definition 4.2 is not completely standardized, and varies across the literature. There is a natural map IRS ( G ) → Tr ( G ) \operatorname{IRS}(G) \to\operatorname{Tr}(G) IRS ( G ) → Tr ( G )
Definition 4.3. Let G G G μ \mu μ G G G
χ μ : G → [ 0 , 1 ] , χ μ ( g ) = μ ( { H ∈ Sub ( G ) : g ∈ H } ) \chi_\mu:G \to[0,1],\qquad\chi_\mu(g)=\mu\bigl(\{H \in\operatorname{Sub}(G):g \in H\}\bigr) χ μ : G → [ 0 , 1 ] , χ μ ( g ) = μ ( { H ∈ Sub ( G ) : g ∈ H } ) is a trace of the group G G G Vershik trace associated to μ \mu μ μ ↦ χ μ \mu\mapsto\chi_\mu μ ↦ χ μ Vershik trace map .
To see that the Vershik trace χ μ \chi_\mu χ μ μ \mu μ χ μ = ∫ Sub ( G ) 1 H d μ ( H ) \chi_\mu=\int_{\operatorname{Sub}(G)}1_H\,d\mu(H) χ μ = ∫ Sub ( G ) 1 H d μ ( H ) 1 H 1_H 1 H
We recall that a metrizable compact convex subset X X X Choquet simplex if for every point x 0 ∈ X x_0 \in X x 0 ∈ X λ \lambda λ X X X x 0 = ∫ X x d λ ( x ) x_0=\int_X x\,d\lambda(x) x 0 = ∫ X x d λ ( x ) Tr ( G ) \operatorname{Tr}(G) Tr ( G ) IRS ( G ) \operatorname{IRS}(G) IRS ( G ) [30 ] and [26 Chapter 12] respectively. Given a trace or an invariant random subgroup of G G G ergodic decomposition .
Thoma characters and invariant random subgroups of the finitary alternating group
Let A ∞ A_\infty A ∞
As mentioned, the Thoma characters of A ∞ A_\infty A ∞ [29 ] . The classification of the invariant random subgroups of the group S ∞ S_\infty S ∞ [35 ] with a slight inaccuracy. In [32 ] Thomas presents a corrected argument for this classification. To get the argument for A ∞ A_\infty A ∞ IRS ( S ∞ ) \operatorname{IRS}(S_\infty) IRS ( S ∞ ) [23 ] presents a derivation of the invariant random subgroups of A ∞ A_\infty A ∞ S ∞ S_\infty S ∞ [32 ] for the group S ∞ S_\infty S ∞ A ∞ A_\infty A ∞
Thoma characters of A ∞ A_\infty A ∞
Let ( β i ) i ∈ N (\beta_i)_{i\in\mathbb{N}} ( β i ) i ∈ N ( γ i ) i ∈ N (\gamma_i)_{i\in\mathbb{N}} ( γ i ) i ∈ N
∑ i = 1 ∞ β i + ∑ i = 1 ∞ γ i ≤ 1. \sum_{i=1}^{\infty}\beta_i+\sum_{i=1}^{\infty}\gamma_i\le1. i = 1 ∑ ∞ β i + i = 1 ∑ ∞ γ i ≤ 1. Define the function χ ( β ; γ ) : A ∞ → C \chi_{(\beta;\gamma)}:A_\infty\to\mathbb{C} χ ( β ; γ ) : A ∞ → C
χ ( β ; γ ) ( σ ) = ∏ k = 2 ∞ ( ∑ i = 1 ∞ β i k + ( − 1 ) k + 1 ∑ i = 1 ∞ γ i k ) c k ( σ ) ∀ σ ∈ A ∞ \chi_{(\beta;\gamma)}(\sigma)=\prod_{k=2}^{\infty}\left(\sum_{i=1}^{\infty}\beta_i^k+(-1)^{k+1}\sum_{i=1}^{\infty}\gamma_i^k\right)^{c_k(\sigma)}
\qquad\forall\sigma\in A_\infty χ ( β ; γ ) ( σ ) = k = 2 ∏ ∞ ( i = 1 ∑ ∞ β i k + ( − 1 ) k + 1 i = 1 ∑ ∞ γ i k ) c k ( σ ) ∀ σ ∈ A ∞ where for each k ≥ 2 k \ge2 k ≥ 2 c k ( σ ) c_k(\sigma) c k ( σ ) k k k σ ∈ A ∞ \sigma\in A_\infty σ ∈ A ∞
Theorem 4.4 ([29 Satz 6] ). The functions χ ( β ; γ ) \chi_{(\beta;\gamma)} χ ( β ; γ ) A ∞ A_\infty A ∞ χ ( β ; γ ) = χ ( β ′ ; γ ′ ) \chi_{(\beta;\gamma)} = \chi_{(\beta';\gamma')} χ ( β ; γ ) = χ ( β ′ ; γ ′ ) ( β , γ ) = ( β ′ , γ ′ ) (\beta,\gamma) = (\beta',\gamma') ( β , γ ) = ( β ′ , γ ′ ) ( β , γ ) = ( γ ′ , β ′ ) (\beta,\gamma) = (\gamma',\beta') ( β , γ ) = ( γ ′ , β ′ )
Invariant random subgroups of A ∞ A_\infty A ∞
We describe the ergodic invariant random subgroups of the group A ∞ A_\infty A ∞ D ⊂ [ 0 , 1 ] N 0 D \subset[0,1]^{\mathbb{N}_0} D ⊂ [ 0 , 1 ] N 0 α = ( α i ) i ∈ N 0 \alpha= (\alpha_i)_{i\in\mathbb{N}_0} α = ( α i ) i ∈ N 0
The sequence ( α i ) i = 1 ∞ (\alpha_i)_{i=1}^{\infty} ( α i ) i = 1 ∞
∑ i = 0 ∞ α i = 1 \sum_{i=0}^{\infty} \alpha_i = 1 ∑ i = 0 ∞ α i = 1
Fix a sequence α ∈ D \alpha\in D α ∈ D I α = { i ∈ N : α i > 0 } I_\alpha= \{i \in\mathbb{N} : \alpha_i > 0\} I α = { i ∈ N : α i > 0 } S α = ⨁ i ∈ I α { ± 1 } S_\alpha= \bigoplus_{i\in I_\alpha} \{\pm1\} S α = ⨁ i ∈ I α { ± 1 } s α s_\alpha s α
s α : S α → { ± 1 } , s α ( t ) = ∏ i ∈ I α t i ∀ t ∈ S α . s_\alpha: S_\alpha\to\{\pm1\}, \qquad s_\alpha(t) = \prod_{i\in I_\alpha} t_i \quad\forall t \in S_\alpha. s α : S α → { ± 1 } , s α ( t ) = i ∈ I α ∏ t i ∀ t ∈ S α . Denote E α = ker s α ≤ S α E_\alpha= \ker s_\alpha\leq S_\alpha E α = ker s α ≤ S α T T T E α E_\alpha E α ν α T \nu_\alpha^T ν α T A ∞ A_\infty A ∞ α ∈ D \alpha\in D α ∈ D T ∈ Sub ( E α ) T \in\operatorname{Sub}(E_\alpha) T ∈ Sub ( E α )
The fixed sequence α ∈ D \alpha\in D α ∈ D p α p_\alpha p α N 0 \mathbb{N}_0 N 0 p α ( i ) = α i p_\alpha(i) = \alpha_i p α ( i ) = α i μ α \mu_\alpha μ α ( N 0 ) N (\mathbb{N}_0)^{\mathbb{N}} ( N 0 ) N c ∈ ( N 0 ) N c \in(\mathbb{N}_0)^{\mathbb{N}} c ∈ ( N 0 ) N N \mathbb{N} N n ∈ N n \in\mathbb{N} n ∈ N i ∈ N 0 i \in\mathbb{N}_0 i ∈ N 0 α i \alpha_i α i n ∈ N n \in\mathbb{N} n ∈ N c ∈ ( N 0 ) N c \in(\mathbb{N}_0)^{\mathbb{N}} c ∈ ( N 0 ) N i ∈ N 0 i \in\mathbb{N}_0 i ∈ N 0
B i ( c ) = { n ∈ N : c n = i } . B_i(c) = \{n \in\mathbb{N} : c_n = i\}. B i ( c ) = { n ∈ N : c n = i } . Let H α , T ( c ) ∈ Sub ( A ∞ ) H_{\alpha,T}(c) \in\operatorname{Sub}(A_\infty) H α , T ( c ) ∈ Sub ( A ∞ ) σ ∈ A ∞ \sigma\in A_\infty σ ∈ A ∞
σ ( B i ( c ) ) ⊆ B i ( c ) \sigma(B_i(c)) \subseteq B_i(c) σ ( B i ( c )) ⊆ B i ( c ) i ∈ N i \in\mathbb{N} i ∈ N
σ ∣ B 0 ( c ) = id \sigma|_{B_0(c)} = \operatorname{id} σ ∣ B 0 ( c ) = id
( sgn ( σ ∣ B i ( c ) ) ) i ∈ I α ∈ T (\operatorname{sgn}(\sigma|_{B_i(c)}))_{i\in I_\alpha} \in T ( sgn ( σ ∣ B i ( c ) ) ) i ∈ I α ∈ T
In other words, the subgroup H α , T ( c ) H_{\alpha,T}(c) H α , T ( c ) 0 ∈ N 0 0 \in\mathbb{N}_0 0 ∈ N 0 T T T μ α \mu_\alpha μ α A ∞ A_\infty A ∞ H α , T : ( N 0 ) N → Sub ( A ∞ ) H_{\alpha,T} : (\mathbb{N}_0)^{\mathbb{N}} \to\operatorname{Sub}(A_\infty) H α , T : ( N 0 ) N → Sub ( A ∞ ) ν α T \nu_\alpha^T ν α T A ∞ A_\infty A ∞
Theorem 4.5 (Vershik, [33 Theorem 9.1] ). For any ergodic invariant random subgroup ν \nu ν A ∞ A_\infty A ∞ α ∈ D \alpha\in D α ∈ D T ∈ Sub ( E α ) T \in\operatorname{Sub}(E_\alpha) T ∈ Sub ( E α ) ν = ν α T \nu= \nu_\alpha^T ν = ν α T
There are pairs ( α , T ) (\alpha,T) ( α , T ) ( α , T ′ ) (\alpha,T') ( α , T ′ ) ν α T = ν α T ′ \nu_\alpha^T = \nu_\alpha^{T'} ν α T = ν α T ′ T ≠ T ′ T \ne T' T = T ′ α \alpha α [32 Remark 1.2] .
The Vershik trace map for A ∞ A_\infty A ∞
We describe the Vershik traces associated to the ergodic invariant random subgroups of A ∞ A_\infty A ∞ χ ν α T \chi_{\nu_\alpha^T} χ ν α T T = E α T = E_\alpha T = E α
Fix a sequence α ∈ D \alpha\in D α ∈ D S α S_\alpha S α S α → { ± 1 } S_\alpha\to\{\pm1\} S α → { ± 1 } S α ^ \widehat{S_\alpha} S α
S α ^ ≅ ∏ i ∈ I α { 0 , 1 } \widehat{S_\alpha} \cong\prod_{i\in I_\alpha} \{0,1\} S α ≅ i ∈ I α ∏ { 0 , 1 } by means of the pairing
⨁ i ∈ I α { ± 1 } × ∏ i ∈ I α { 0 , 1 } → { ± 1 } , ( t , u ) ↦ ∏ i ∈ I α t i u i ∀ t ∈ S α , u ∈ S α ^ . \bigoplus_{i\in I_\alpha}\{\pm1\}\times\prod_{i\in I_\alpha}\{0,1\}\to\{\pm1\},\qquad(t,u)\mapsto\prod_{i\in I_\alpha}t_i^{u_i}\qquad\forall t\in S_\alpha,u\in\widehat{S_\alpha}. i ∈ I α ⨁ { ± 1 } × i ∈ I α ∏ { 0 , 1 } → { ± 1 } , ( t , u ) ↦ i ∈ I α ∏ t i u i ∀ t ∈ S α , u ∈ S α . Under this identification, the multiplication homomorphism s α s_\alpha s α s α = ( 1 ) i ∈ I α ∈ S α ^ s_\alpha=(1)_{i\in I_\alpha}\in\widehat{S_\alpha} s α = ( 1 ) i ∈ I α ∈ S α E α E_\alpha E α E α ^ ≅ S α ^ / ⟨ s α ⟩ \widehat{E_\alpha}\cong\widehat{S_\alpha}/\langle s_\alpha\rangle E α ≅ S α / ⟨ s α ⟩
Given the fixed sequence α ∈ D \alpha\in\mathcal{D} α ∈ D u ∈ E α ^ u\in\widehat{E_\alpha} u ∈ E α χ α u \chi_\alpha^u χ α u A ∞ A_\infty A ∞
χ α u ( σ ) = ∏ k = 2 ∞ ( ∑ i ∈ I α ( − 1 ) ( k + 1 ) u i α i k ) c k ( σ ) ∀ σ ∈ A ∞ . \chi_\alpha^u(\sigma)=\prod_{k=2}^{\infty}\left(\sum_{i\in I_\alpha}(-1)^{(k+1)u_i}\alpha_i^k\right)^{c_k(\sigma)}\qquad\forall\sigma\in A_\infty. χ α u ( σ ) = k = 2 ∏ ∞ ( i ∈ I α ∑ ( − 1 ) ( k + 1 ) u i α i k ) c k ( σ ) ∀ σ ∈ A ∞ . The character χ α u \chi_\alpha^u χ α u χ ( β ; γ ) \chi_{(\beta;\gamma)} χ ( β ; γ )
β i = { α i , if u i = 0 , 0 , if u i = 1 and γ i = { 0 , if u i = 0 , α i , if u i = 1. \beta_i=
\begin{cases}
\alpha_i,&\text{if }u_i=0,\\
0,&\text{if }u_i=1
\end{cases}
\qquad\text{and}\qquad
\gamma_i=
\begin{cases}
0,&\text{if }u_i=0,\\
\alpha_i,&\text{if }u_i=1.
\end{cases} β i = { α i , 0 , if u i = 0 , if u i = 1 and γ i = { 0 , α i , if u i = 0 , if u i = 1. We note that this formula is independent of the choice of representative for u u u S α ^ \widehat{S_\alpha} S α χ α u = χ ( β ; γ ) \chi_\alpha^u=\chi_{(\beta;\gamma)} χ α u = χ ( β ; γ ) χ α u + s = χ ( γ ; β ) \chi_\alpha^{u+s}=\chi_{(\gamma;\beta)} χ α u + s = χ ( γ ; β ) ν α T \nu_\alpha^T ν α T
Theorem 4.6 ([32 Theorem 1.7] ). Let α ∈ D \alpha\in\mathcal{D} α ∈ D T ≤ E α T\leq E_\alpha T ≤ E α E α / T ^ \widehat{E_\alpha/T} E α / T E α E_\alpha E α T T T ν α T \nu_\alpha^T ν α T
χ ν α T = ∫ E α / T ^ χ α u d u \chi_{\nu_\alpha^T}=\int_{\widehat{E_\alpha/T}}\chi_\alpha^u\,du χ ν α T = ∫ E α / T χ α u d u where the measure d u du d u E α / T ^ \widehat{E_\alpha/T} E α / T
The proof in [[32 ] ] is for invariant random subgroups of S ∞ S_\infty S ∞ S α S_\alpha S α E α E_\alpha E α
The orbit equivalence map
We take this opportunity to introduce several objects that involve the finitary alternating group and that will be useful in § 6.1 below:
Let E ( N ) ⊆ 2 N × N \mathcal{E}(\mathbb{N})\subseteq2^{\mathbb{N}\times\mathbb{N}} E ( N ) ⊆ 2 N × N N \mathbb{N} N A ∞ A_\infty A ∞ N \mathbb{N} N A ∞ ↷ 2 N × N A_\infty\curvearrowright2^{\mathbb{N}\times\mathbb{N}} A ∞ ↷ 2 N × N A ∞ ↷ E ( N ) A_\infty\curvearrowright\mathcal{E}(\mathbb{N}) A ∞ ↷ E ( N )
Definition 4.7. Let O : Sub ( A ∞ ) → E ( N ) \mathcal{O}:\operatorname{Sub}(A_\infty)\to\mathcal{E}(\mathbb{N}) O : Sub ( A ∞ ) → E ( N )
O ( H ) = { ( i , σ ⋅ i ) : i ∈ N , σ ∈ H } ∀ H ∈ Sub ( A ∞ ) . \mathcal{O}(H)=\{(i,\sigma\cdot i):i\in\mathbb{N},\ \sigma\in H\}\qquad\forall H\in\operatorname{Sub}(A_\infty). O ( H ) = {( i , σ ⋅ i ) : i ∈ N , σ ∈ H } ∀ H ∈ Sub ( A ∞ ) . Let N : E ( N ) → N ∪ { ∞ } \mathcal{N}:\mathcal{E}(\mathbb{N})\to\mathbb{N}\cup\{\infty\} N : E ( N ) → N ∪ { ∞ }
We define a map ϱ : ( N 0 ) N → E ( N ) \varrho:(\mathbb{N}_0)^{\mathbb{N}}\to\mathcal{E}(\mathbb{N}) ϱ : ( N 0 ) N → E ( N ) ϱ ( c ) \varrho(c) ϱ ( c ) c ∈ ( N 0 ) N c\in(\mathbb{N}_0)^{\mathbb{N}} c ∈ ( N 0 ) N P c P_c P c N \mathbb{N} N
P c = { B k ( c ) : k ∈ N , B k ( c ) ≠ ∅ } ∪ { { i } : i ∈ B 0 ( c ) } . P_c=\{B_k(c):k\in\mathbb{N},\ B_k(c)\ne\varnothing\}\cup\{\{i\}:i\in B_0(c)\}. P c = { B k ( c ) : k ∈ N , B k ( c ) = ∅ } ∪ {{ i } : i ∈ B 0 ( c )} . Let α ∈ D \alpha\in\mathcal{D} α ∈ D μ α \mu_\alpha μ α ϱ \varrho ϱ λ α \lambda_\alpha λ α D \mathcal{D} D μ α \mu_\alpha μ α O \mathcal{O} O N \mathcal{N} N
Proposition 4.8. The maps O \mathcal{O} O N \mathcal{N} N O \mathcal{O} O A ∞ A_{\infty} A ∞ N \mathcal{N} N A ∞ A_{\infty} A ∞ α ∈ D \alpha\in\mathcal{D} α ∈ D T ∈ Sub ( E α ) T \in\operatorname{Sub}(E_{\alpha}) T ∈ Sub ( E α ) ν α T \nu_{\alpha}^{T} ν α T O ∗ ν α T = λ α \mathcal{O}_{*}\nu_{\alpha}^{T}=\lambda_{\alpha} O ∗ ν α T = λ α N \mathcal{N} N λ α \lambda_{\alpha} λ α ∣ I α ∣ \lvert I_{\alpha}\rvert ∣ I α ∣ α 0 = 0 \alpha_{0}=0 α 0 = 0 ∞ \infty ∞ α 0 > 0 \alpha_{0}>0 α 0 > 0
We remark that Kingman [19 ] has shown that the λ α \lambda_{\alpha} λ α S ∞ S_{\infty} S ∞ E ( N ) \mathcal{E}(\mathbb{N}) E ( N )
Pointed partitions
We digress further to overview the notion of pointed partitions of N \mathbb{N} N A ∞ A_{\infty} A ∞ [35 ] .
Definition 4.9. A pointed partition of the set N \mathbb{N} N P P P B P ∈ P B_{P}\in P B P ∈ P N \mathbb{N} N E ∗ ( N ) \mathcal{E}^{*}(\mathbb{N}) E ∗ ( N )
The set E ∗ ( N ) \mathcal{E}^{*}(\mathbb{N}) E ∗ ( N ) S ∞ S_{\infty} S ∞ N \mathbb{N} N S ∞ S_{\infty} S ∞ E ∗ ( N ) \mathcal{E}^{*}(\mathbb{N}) E ∗ ( N ) E ( N ) \mathcal{E}(\mathbb{N}) E ( N )
Here is one way to construct pointed partitions. Given a point c ∈ ( Z ≥ − 1 ) N c\in(\mathbb{Z}_{\geq-1})^{\mathbb{N}} c ∈ ( Z ≥ − 1 ) N B k ( c ) = { n ∈ N : c n = k } B_{k}(c)=\{n\in\mathbb{N}:c_{n}=k\} B k ( c ) = { n ∈ N : c n = k } k ∈ Z ≥ − 1 k\in\mathbb{Z}_{\geq-1} k ∈ Z ≥ − 1 P c ∈ E ∗ ( N ) P_{c}\in\mathcal{E}^{*}(\mathbb{N}) P c ∈ E ∗ ( N )
P c = { B k ( c ) : k ∈ N ∪ { − 1 } } ∪ { { i } : i ∈ B 0 ( c ) } , P_{c}=\{B_{k}(c):k\in\mathbb{N}\cup\{-1\}\}\cup\{\{i\}:i\in B_{0}(c)\}, P c = { B k ( c ) : k ∈ N ∪ { − 1 }} ∪ {{ i } : i ∈ B 0 ( c )} , whose distinguished block is B − 1 ( c ) B_{-1}(c) B − 1 ( c )
Let D ∗ ⊆ [ 0 , 1 ] Z ≥ − 1 \mathcal{D}^{*}\subseteq[0,1]^{\mathbb{Z}_{\geq-1}} D ∗ ⊆ [ 0 , 1 ] Z ≥ − 1 ( α i ) i ∈ Z ≥ − 1 (\alpha_{i})_{i\in\mathbb{Z}_{\geq-1}} ( α i ) i ∈ Z ≥ − 1
The subsequence ( α i ) i = 1 ∞ (\alpha_{i})_{i=1}^{\infty} ( α i ) i = 1 ∞
∑ i = − 1 ∞ α i = 1 \sum_{i=-1}^{\infty}\alpha_{i}=1 ∑ i = − 1 ∞ α i = 1
Given each sequence α ∈ D ∗ \alpha\in\mathcal{D}^{*} α ∈ D ∗ p α p_{\alpha} p α Z ≥ − 1 \mathbb{Z}_{\geq-1} Z ≥ − 1 p α ( i ) = α i p_{\alpha}(i)=\alpha_{i} p α ( i ) = α i i ∈ Z ≥ − 1 i\in\mathbb{Z}_{\geq-1} i ∈ Z ≥ − 1 μ α \mu_{\alpha} μ α ( Z ≥ − 1 ) N (\mathbb{Z}_{\geq-1})^{\mathbb{N}} ( Z ≥ − 1 ) N μ α \mu_{\alpha} μ α c ↦ P c c\mapsto P_{c} c ↦ P c λ α ∗ \lambda_{\alpha}^{*} λ α ∗
Proposition 4.10. Any ergodic A ∞ A_{\infty} A ∞ E ∗ ( N ) \mathcal{E}^{*}(\mathbb{N}) E ∗ ( N ) λ α ∗ \lambda_{\alpha}^{*} λ α ∗ α ∈ D ∗ \alpha\in\mathcal{D}^{*} α ∈ D ∗
This is a slight refinement of Kingman’s classification of the ergodic S ∞ S_{\infty} S ∞ [19 ] . We choose to reduce the proposition to Vershik’s classification of invariant measures on signed partitions [35 ] . A signed partition of the set N \mathbb{N} N N \mathbb{N} N E ± ( N ) \mathcal{E}^{\pm}(\mathbb{N}) E ± ( N ) α ∈ [ 0 , 1 ] Z \alpha\in[0,1]^{\mathbb{Z}} α ∈ [ 0 , 1 ] Z ν α ± \nu_{\alpha}^{\pm} ν α ± E ± ( N ) \mathcal{E}^{\pm}(\mathbb{N}) E ± ( N ) λ α ∗ \lambda_{\alpha}^{*} λ α ∗ λ α \lambda_{\alpha} λ α c ∈ Z N c\in\mathbb{Z}^{\mathbb{N}} c ∈ Z N α \alpha α P c P_{c} P c
P c = { B k ( c ) : k ∈ Z ∖ { 0 } } ∪ { { i } : i ∈ B 0 ( c ) } . P_{c}=\{B_{k}(c):k\in\mathbb{Z}\setminus\{0\}\}\cup\{\{i\}:i\in B_{0}(c)\}. P c = { B k ( c ) : k ∈ Z ∖ { 0 }} ∪ {{ i } : i ∈ B 0 ( c )} . The positive blocks are the B k ( c ) B_{k}(c) B k ( c ) k > 0 k>0 k > 0 B k ( c ) B_{k}(c) B k ( c ) k < 0 k<0 k < 0 S ∞ S_{\infty} S ∞ E ± ( N ) \mathcal{E}^{\pm}(\mathbb{N}) E ± ( N ) λ α ± \lambda_{\alpha}^{\pm} λ α ± α \alpha α A ∞ A_{\infty} A ∞
Proof of Theorem 4.10. Let λ \lambda λ A ∞ A_{\infty} A ∞ E ∗ ( N ) \mathcal{E}^{*}(\mathbb{N}) E ∗ ( N ) λ \lambda λ λ \lambda λ λ \lambda λ E ∗ ( N ) → N \mathcal{E}^{*}(\mathbb{N}) \to\mathbb{N} E ∗ ( N ) → N A ∞ A_{\infty} A ∞ N \mathbb{N} N
It follows that the map s : E ∗ ( N ) → E ± ( N ) s:\mathcal{E}^{*}(\mathbb{N}) \to\mathcal{E}^{\pm}(\mathbb{N}) s : E ∗ ( N ) → E ± ( N ) λ \lambda λ A ∞ A_{\infty} A ∞ s ∗ λ = λ α ∗ s_{*}\lambda=\lambda_{\alpha}^{*} s ∗ λ = λ α ∗ α ∈ [ 0 , 1 ] Z \alpha\in[0,1]^{\mathbb{Z}} α ∈ [ 0 , 1 ] Z α \alpha α Z ≥ − 1 \mathbb{Z}_{\geq-1} Z ≥ − 1 λ = λ α ∗ \lambda=\lambda_{\alpha}^{*} λ = λ α ∗
Given a pointed partition P ∈ E ∗ ( N ) P\in\mathcal{E}^{*}(\mathbb{N}) P ∈ E ∗ ( N ) k ∈ N ≥ − 1 k\in\mathbb{N}_{\geq-1} k ∈ N ≥ − 1 d k ( P ) ∈ [ 0 , 1 ] d_k(P)\in[0,1] d k ( P ) ∈ [ 0 , 1 ] k k k P P P k ≥ 1 k\geq1 k ≥ 1 k = − 1 k=-1 k = − 1 D ( P ) = ( d k ( P ) ) k ≥ − 1 D(P)=(d_k(P))_{k\geq-1} D ( P ) = ( d k ( P ) ) k ≥ − 1
Consider any fixed sequence β ∈ D ∗ \beta\in D^{*} β ∈ D ∗ λ β ∗ \lambda_{\beta}^{*} λ β ∗ P ∈ E ∗ ( N ) P\in\mathcal{E}^{*}(\mathbb{N}) P ∈ E ∗ ( N ) D ( P ) D(P) D ( P ) β \beta β c ∈ ( Z ≥ − 1 ) N c\in(\mathbb{Z}_{\geq-1})^{\mathbb{N}} c ∈ ( Z ≥ − 1 ) N P P P μ β \mu_{\beta} μ β
Lemma 4.11. Let λ \lambda λ A ∞ A_{\infty} A ∞ E ∗ ( N ) \mathcal{E}^{*}(\mathbb{N}) E ∗ ( N ) α ∈ D ∗ \alpha\in D^{*} α ∈ D ∗ D ( P ) D(P) D ( P ) λ \lambda λ α \alpha α λ = λ α ∗ \lambda=\lambda_{\alpha}^{*} λ = λ α ∗ λ \lambda λ
Proof. By Theorem 4.10, the probability measure λ \lambda λ λ = ∫ D ∗ λ β ∗ d m ( β ) \lambda=\int_{D^{*}}\lambda_{\beta}^{*}\,dm(\beta) λ = ∫ D ∗ λ β ∗ d m ( β ) m m m D ∗ D^{*} D ∗ D ∗ λ = ∫ δ β d m ( β ) = m D_{*}\lambda=\int\delta_{\beta}\,dm(\beta)=m D ∗ λ = ∫ δ β d m ( β ) = m D ( P ) D(P) D ( P ) λ \lambda λ α \alpha α m = δ α m=\delta_{\alpha} m = δ α λ = λ α ∗ \lambda=\lambda_{\alpha}^{*} λ = λ α ∗ λ \lambda λ
The Vershik trace of an invariant random subgroup of S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
Let F q \mathbb{F}_q F q q q q S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) μ \mu μ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) μ \mu μ χ μ \chi_{\mu} χ μ
Example. Observe that any character of S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) δ I \delta_I δ I S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) μ ≠ δ { I } \mu\ne\delta_{\{I\}} μ = δ { I } χ μ ≠ δ I \chi_{\mu}\ne\delta_I χ μ = δ I χ μ \chi_{\mu} χ μ every element of S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) μ \mu μ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) μ \mu μ
Here is a more elaborate use of the Vershik trace. Regard S L n ( F q ) \mathrm{SL}_n(\mathbb{F}_q) SL n ( F q ) S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) μ \mu μ H H H S L n ( F q ) \mathrm{SL}_n(\mathbb{F}_q) SL n ( F q ) g g g rk ~ ( g ) = rk ( g − I ) = n \widetilde{\operatorname{rk}}(g)=\operatorname{rk}(g-I)=n rk ( g ) = rk ( g − I ) = n H H H μ \mu μ S L n ( F q ) \mathrm{SL}_n(\mathbb{F}_q) SL n ( F q ) § 5.2 for more details on this direction.
In § 5.3 , we use the theory of invariant random subgroups and Thoma characters of the group A ∞ A_{\infty} A ∞ § 4.3 to identify the image of the invariant random subgroup μ \mu μ IRS ( S L ∞ ( F q ) ) → IRS ( A ∞ ) \operatorname{IRS}(\mathrm{SL}_{\infty}(\mathbb{F}_q))\to\operatorname{IRS}(A_{\infty}) IRS ( SL ∞ ( F q )) → IRS ( A ∞ ) A ∞ A_{\infty} A ∞ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
The degree of an invariant random subgroup
Let χ μ \chi_\mu χ μ μ \mu μ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) χ μ \chi_\mu χ μ
χ μ = ∑ k = 0 ∞ a k ( μ ) χ k + a ∞ ( μ ) χ ∞ , (10) \chi_\mu=\sum_{k=0}^{\infty}a_k(\mu)\chi_k+a_\infty(\mu)\chi_\infty, \tag*{(10)} χ μ = k = 0 ∑ ∞ a k ( μ ) χ k + a ∞ ( μ ) χ ∞ , (10) where a k ( μ ) ≥ 0 a_k(\mu)\geq0 a k ( μ ) ≥ 0 ∑ k = 0 ∞ a k ( μ ) + a ∞ ( μ ) = 1 \sum_{k=0}^{\infty}a_k(\mu)+a_\infty(\mu)=1 ∑ k = 0 ∞ a k ( μ ) + a ∞ ( μ ) = 1
χ k ( g ) = q − k ⋅ rk ~ g ∀ g ∈ S L ∞ ( F q ) \chi_k(g)=q^{-k\cdot\widetilde{\operatorname{rk}}g}\quad\forall g\in\mathrm{SL}_{\infty}(\mathbb{F}_q) χ k ( g ) = q − k ⋅ rk g ∀ g ∈ SL ∞ ( F q ) for all k ∈ N k\in\mathbb{N} k ∈ N χ ∞ = δ I \chi_\infty=\delta_I χ ∞ = δ I
Definition 5.1. The degree of the invariant random subgroup μ \mu μ r ( μ ) ∈ N ∪ { ∞ } r(\mu)\in\mathbb{N}\cup\{\infty\} r ( μ ) ∈ N ∪ { ∞ } a r ( μ ) ( μ ) > 0 a_{r(\mu)}(\mu)>0 a r ( μ ) ( μ ) > 0
Note that the degree r ( μ ) r(\mu) r ( μ ) χ μ = δ I \chi_\mu=\delta_I χ μ = δ I S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) μ = δ { I } \mu=\delta_{\{I\}} μ = δ { I }
We observe several additional useful properties of the Vershik trace χ μ \chi_\mu χ μ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) g ∈ S L ∞ ( F q ) g\in\mathrm{SL}_{\infty}(\mathbb{F}_q) g ∈ SL ∞ ( F q ) rk ~ g \widetilde{\operatorname{rk}}g rk g χ μ \chi_\mu χ μ
Definition 5.2. Given any n ∈ N 0 n\in\mathbb{N}_0 n ∈ N 0 χ μ ( n ) \chi_\mu(n) χ μ ( n ) χ μ \chi_\mu χ μ g ∈ S L ∞ ( F q ) g\in\mathrm{SL}_{\infty}(\mathbb{F}_q) g ∈ SL ∞ ( F q ) rk ~ ( g ) = n \widetilde{\operatorname{rk}}(g)=n rk ( g ) = n
The second property is χ μ \chi_\mu χ μ n = rk ~ g n=\widetilde{\operatorname{rk}}g n = rk g n = rk ~ g → ∞ n=\widetilde{\operatorname{rk}}g\to\infty n = rk g → ∞
χ μ ( n ) = a r ( μ ) ( μ ) ⋅ q − r ( μ ) n ( 1 + o ( 1 ) ) . (11) \chi_\mu(n)=a_{r(\mu)}(\mu)\cdot q^{-r(\mu)n}(1+o(1)). \tag*{(11)} χ μ ( n ) = a r ( μ ) ( μ ) ⋅ q − r ( μ ) n ( 1 + o ( 1 )) . (11) In particular, we see that χ μ ( n ) \chi_\mu(n) χ μ ( n ) n n n q − r ( μ ) q^{-r(\mu)} q − r ( μ )
Invariant random subgroups contain SLn(Fq)
Singer cycles were discussed in § 2.4 . We observe that the size of the conjugacy class of a Singer cycle in S L n ( F q ) \mathrm{SL}_n(\mathbb{F}_q) SL n ( F q ) μ \mu μ μ \mu μ μ \mu μ S L n ( F q ) \mathrm{SL}_n(\mathbb{F}_q) SL n ( F q )
Proposition 5.3. Let μ \mu μ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) f f f
μ ( { H ∈ Sub ( S L ∞ ( F q ) ) : S L n ( f ) ≤ H } ) = χ μ ( n ) ( 1 − o ( 1 ) ) . (12) \mu\bigl(\{H\in\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)):\mathrm{SL}_n(f)\leq H\}\bigr)=\chi_\mu(n)(1-o(1)). \tag*{(12)} μ ( { H ∈ Sub ( SL ∞ ( F q )) : SL n ( f ) ≤ H } ) = χ μ ( n ) ( 1 − o ( 1 )) . (12) Proof. For every n ∈ N n\in\mathbb{N} n ∈ N s n ∈ S L n ( f ) s_n\in\mathrm{SL}_n(f) s n ∈ SL n ( f ) 1 1 1 s n s_n s n rk ~ ( s n ) = n \widetilde{\operatorname{rk}}(s_n)=n rk ( s n ) = n
μ ( { H : S L n ( f ) ≤ H } ) ≤ μ ( { H : s n ∈ H } ) = χ μ ( n ) . \mu\bigl(\{H:\mathrm{SL}_n(f)\leq H\}\bigr)\leq\mu\bigl(\{H:s_n\in H\}\bigr)=\chi_\mu(n). μ ( { H : SL n ( f ) ≤ H } ) ≤ μ ( { H : s n ∈ H } ) = χ μ ( n ) . We need to show that the other inequality holds, up to a factor of ( 1 − o ( 1 ) ) (1-o(1)) ( 1 − o ( 1 )) μ = δ { I } \mu=\delta_{\{I\}} μ = δ { I } n ∈ N n\in\mathbb{N} n ∈ N r ( μ ) < ∞ r(\mu)<\infty r ( μ ) < ∞
Let N n N_n N n s n s_n s n S L n ( f ) \mathrm{SL}_n(f) SL n ( f ) N n N_n N n q n 2 + O ( n ) q^{n^2+O(n)} q n 2 + O ( n ) (3) . By Theorem 2.2 any k n = q 3 4 n 2 k_n=q^{\frac{3}{4}n^2} k n = q 4 3 n 2 s n s_n s n S L n ( f ) \mathrm{SL}_n(f) SL n ( f ) n n n
μ ( { H : SL n ( f ) ≤ H } ) ≥ μ ( { k n + 1 distinct conjugates of s n belong to H } ) ≥ N n χ μ ( s n ) − k n N n − k n ≥ χ μ ( n ) − k n N n = χ μ ( n ) − q − 1 4 n 2 + O ( n ) = χ μ ( n ) ( 1 − o ( 1 ) ) . \begin{aligned}
\mu\left(\{H:\operatorname{SL}_{n}(f)\leq H\}\right)\geq\mu\left(\{k_{n}+1\text{ distinct conjugates of }s_{n}\text{ belong to }H\}\right) \\
&\geq\frac{N_{n}\chi_{\mu}(s_{n})-k_{n}}{N_{n}-k_{n}}\geq\chi_{\mu}(n)-\frac{k_{n}}{N_{n}} \\
&=\chi_{\mu}(n)-q^{-\frac{1}{4}n^{2}+O(n)}=\chi_{\mu}(n)(1-o(1)).
\end{aligned} μ ( { H : SL n ( f ) ≤ H } ) ≥ μ ( { k n + 1 distinct conjugates of s n belong to H } ) ≥ N n − k n N n χ μ ( s n ) − k n ≥ χ μ ( n ) − N n k n = χ μ ( n ) − q − 4 1 n 2 + O ( n ) = χ μ ( n ) ( 1 − o ( 1 )) . In the last equality we use the fact that χ μ ( n ) \chi_{\mu}(n) χ μ ( n ) n n n (11) ). This concludes the proof. □
In the above proof, we have used the following quantitative version of the pigeonhole principle for probability spaces.
Lemma 5.4. Let ( X , B , P ) (X,\mathcal{B},\mathbb{P}) ( X , B , P ) N ∈ N N\in\mathbb{N} N ∈ N E 1 , … , E N E_{1},\ldots,E_{N} E 1 , … , E N B \mathcal{B} B 0 ≤ k < N 0\leq k<N 0 ≤ k < N
P ( ∑ i = 1 N 1 E i > k ) ≥ ∑ i = 1 N P ( E i ) − k N − k . (13) \mathbb{P}\left(\sum_{i=1}^{N}\mathbf{1}_{E_{i}}>k\right)\geq\frac{\sum_{i=1}^{N}\mathbb{P}(E_{i})-k}{N-k}. \tag*{(13)} P ( i = 1 ∑ N 1 E i > k ) ≥ N − k ∑ i = 1 N P ( E i ) − k . (13) Proof. Let F = { x ∈ X : ∑ i = 1 N 1 E i ( x ) > k } F=\left\{x\in X:\sum_{i=1}^{N}\mathbf{1}_{E_{i}}(x)>k\right\} F = { x ∈ X : ∑ i = 1 N 1 E i ( x ) > k }
∑ i = 1 N P ( E i ) = E [ ∑ i = 1 N 1 E i ] = E [ ∑ i = 1 N 1 E i ∩ F + ∑ i = 1 N 1 E i ∩ F c ] \sum_{i=1}^{N}\mathbb{P}(E_{i})=\mathbb{E}\left[\sum_{i=1}^{N}\mathbf{1}_{E_{i}}\right]=\mathbb{E}\left[\sum_{i=1}^{N}\mathbf{1}_{E_{i}\cap F}+\sum_{i=1}^{N}\mathbf{1}_{E_{i}\cap F^{c}}\right] i = 1 ∑ N P ( E i ) = E [ i = 1 ∑ N 1 E i ] = E [ i = 1 ∑ N 1 E i ∩ F + i = 1 ∑ N 1 E i ∩ F c ] ≤ E [ N 1 F + k 1 F c ] = N P ( F ) + k ( 1 − P ( F ) ) . \leq\mathbb{E}[N\mathbf{1}_{F}+k\mathbf{1}_{F^{c}}]=N\mathbb{P}(F)+k(1-\mathbb{P}(F)). ≤ E [ N 1 F + k 1 F c ] = N P ( F ) + k ( 1 − P ( F )) . The desired eq. (13) follows by rearranging. □
We proceed to consider some of the consequences of Theorem 5.3.
Proposition 5.5. Let μ ∈ IRS ( SL ∞ ( F q ) ) \mu\in\operatorname{IRS}(\operatorname{SL}_{\infty}(\mathbb{F}_{q})) μ ∈ IRS ( SL ∞ ( F q )) non-trivial ergodic invariant random subgroup . Then for every n ∈ N n\in\mathbb{N} n ∈ N μ \mu μ H ∈ Sub ( SL ∞ ( F q ) ) H\in\operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_{q})) H ∈ Sub ( SL ∞ ( F q )) f f f SL n ( f ) ≤ H \operatorname{SL}_{n}(f)\leq H SL n ( f ) ≤ H
Proof. For each n ∈ N n\in\mathbb{N} n ∈ N
E n = { H ∈ Sub ( SL ∞ ( F q ) ) : there exists a finitary basis f such that SL n ( f ) ≤ H } E_{n}=\{H\in\operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_{q})):\text{ there exists a finitary basis }f\text{ such that }\operatorname{SL}_{n}(f)\leq H\} E n = { H ∈ Sub ( SL ∞ ( F q )) : there exists a finitary basis f such that SL n ( f ) ≤ H } is SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_{q}) SL ∞ ( F q ) μ ( E n ) ∈ { 0 , 1 } \mu(E_{n})\in\{0,1\} μ ( E n ) ∈ { 0 , 1 } μ ( { H : SL n ( F q ) ≤ H } ) > 0 \mu(\{H:\operatorname{SL}_{n}(\mathbb{F}_{q})\leq H\})>0 μ ({ H : SL n ( F q ) ≤ H }) > 0 n n n n 0 ∈ N n_{0}\in\mathbb{N} n 0 ∈ N n < n 0 n<n_{0} n < n 0
μ ( { H : SL n ( F q ) ≤ H } ) ≥ μ ( { H : SL n 0 ( F q ) ≤ H } ) > 0. \mu(\{H:\operatorname{SL}_{n}(\mathbb{F}_{q})\leq H\})\geq\mu(\{H:\operatorname{SL}_{n_{0}}(\mathbb{F}_{q})\leq H\})>0. μ ({ H : SL n ( F q ) ≤ H }) ≥ μ ({ H : SL n 0 ( F q ) ≤ H }) > 0. Hence μ ( E n ) ≥ μ ( { H : SL n ( F q ) ≤ H } ) > 0 \mu(E_{n})\geq\mu(\{H:\operatorname{SL}_{n}(\mathbb{F}_{q})\leq H\})>0 μ ( E n ) ≥ μ ({ H : SL n ( F q ) ≤ H }) > 0 μ ( E n ) = 1 \mu(E_{n})=1 μ ( E n ) = 1 n ∈ N n\in\mathbb{N} n ∈ N
Remark 5.6. We obtain the tentative result saying that a μ \mu μ H H H μ \mu μ SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_{q}) SL ∞ ( F q ) Theorem 1.1 , and can be seen as partial progress towards it.
We complete the classification of ergodic invariant random subgroups of degree 0.
Proposition 5.7. Let μ ∈ IRS ( SL ∞ ( F q ) ) \mu\in\operatorname{IRS}(\operatorname{SL}_{\infty}(\mathbb{F}_{q})) μ ∈ IRS ( SL ∞ ( F q )) ergodic invariant random subgroup . If r ( μ ) = 0 r(\mu)=0 r ( μ ) = 0 μ = δ SL ∞ ( F q ) \mu=\delta_{\operatorname{SL}_{\infty}(\mathbb{F}_{q})} μ = δ SL ∞ ( F q )
Proof. Assume that the degree r ( μ ) r(\mu) r ( μ ) 0 0 0
μ ( { H : S L ∞ ( F q ) ≤ H } ) = lim n → ∞ μ ( { H : S L n ( F q ) ≤ H } ) = lim n → ∞ a 0 ( μ ) ( 1 − o ( 1 ) ) = a 0 > 0. \mu\left(\left\{H:\mathrm{SL}_{\infty}(\mathbb{F}_q)\leq H\right\}\right)=\lim_{n\to\infty}\mu\left(\left\{H:\mathrm{SL}_{n}(\mathbb{F}_q)\leq H\right\}\right)=\lim_{n\to\infty}a_0(\mu)(1-o(1))=a_0>0. μ ( { H : SL ∞ ( F q ) ≤ H } ) = n → ∞ lim μ ( { H : SL n ( F q ) ≤ H } ) = n → ∞ lim a 0 ( μ ) ( 1 − o ( 1 )) = a 0 > 0. Because the event { H : S L ∞ ( F q ) ≤ H } \left\{H:\mathrm{SL}_{\infty}(\mathbb{F}_q)\leq H\right\} { H : SL ∞ ( F q ) ≤ H } S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
The intersection map from S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) A ∞ A_{\infty} A ∞
We use the Vershik trace of the given invariant random subgroup μ \mu μ ν f \nu_f ν f IRS ( S L ∞ ( F q ) ) → IRS ( A ∞ ( f ) ) \operatorname{IRS}(\mathrm{SL}_{\infty}(\mathbb{F}_q))\to\operatorname{IRS}(A_{\infty}(f)) IRS ( SL ∞ ( F q )) → IRS ( A ∞ ( f )) f f f
We use notations from § 4.3 . For each k ∈ N k\in\mathbb{N} k ∈ N α k ∈ D \alpha^k\in\mathcal{D} α k ∈ D
α i k = { 1 / q k , 1 ≤ i ≤ q k , 0 , i = 0 or i > q k . \alpha_i^k=
\begin{cases}
1/q^k, & 1\leq i\leq q^k,\\
0, & i=0\text{ or }i>q^k.
\end{cases} α i k = { 1/ q k , 0 , 1 ≤ i ≤ q k , i = 0 or i > q k . Define α ∞ ∈ D \alpha^\infty\in\mathcal{D} α ∞ ∈ D α 0 ∞ = 1 \alpha_0^\infty=1 α 0 ∞ = 1 α i ∞ = 0 \alpha_i^\infty=0 α i ∞ = 0 i ∈ N i\in\mathbb{N} i ∈ N k ∈ N 0 ∪ { ∞ } k\in\mathbb{N}_0\cup\{\infty\} k ∈ N 0 ∪ { ∞ }
ν k = ν α k E ∈ IRS ( A ∞ ) . \nu_k=\nu_{\alpha^k}^{E}\in\operatorname{IRS}(A_\infty). ν k = ν α k E ∈ IRS ( A ∞ ) . In other words ν k \nu_k ν k N \mathbb{N} N q k q^k q k
Proposition 5.8. Let μ \mu μ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) r ( μ ) < ∞ r(\mu)<\infty r ( μ ) < ∞ f f f ν f \nu_f ν f μ \mu μ IRS ( S L ∞ ( F q ) ) → IRS ( A ∞ ( f ) ) \operatorname{IRS}(\mathrm{SL}_{\infty}(\mathbb{F}_q))\to\operatorname{IRS}(A_{\infty}(f)) IRS ( SL ∞ ( F q )) → IRS ( A ∞ ( f ))
ν f = ∑ k = r ( μ ) ∞ a k ( μ ) ν k + a ∞ ( μ ) ν ∞ . (14) \nu_f=\sum_{k=r(\mu)}^\infty a_k(\mu)\nu_k+a_\infty(\mu)\nu_\infty. \tag*{(14)} ν f = k = r ( μ ) ∑ ∞ a k ( μ ) ν k + a ∞ ( μ ) ν ∞ . (14) The coefficients a k ( μ ) a_k(\mu) a k ( μ ) (14) correspond to the ergodic decomposition of the Vershik trace χ μ \chi_\mu χ μ (10) .
Proof of Theorem 5.8. Fix a finitary basis f f f r = r ( μ ) r=r(\mu) r = r ( μ ) a k = a k ( μ ) a_k=a_k(\mu) a k = a k ( μ ) k k k ν = ν f \nu=\nu_f ν = ν f χ ν \chi_\nu χ ν ν \nu ν χ ν \chi_\nu χ ν A ∞ ( f ) A_\infty(f) A ∞ ( f ) χ μ \chi_\mu χ μ μ \mu μ σ ∈ A ∞ \sigma\in A_\infty σ ∈ A ∞ σ 1 , … , σ t \sigma_1,\ldots,\sigma_t σ 1 , … , σ t σ \sigma σ ℓ i \ell_i ℓ i σ i \sigma_i σ i rk ~ σ i = ℓ i − 1 \widetilde{\operatorname{rk}}\sigma_i=\ell_i-1 rk σ i = ℓ i − 1
χ ν ( σ ) = χ μ ( σ ) = ∑ k = r ∞ a k q − k rk ~ σ = ∑ k = r ∞ a k ∏ ℓ = 2 ∞ ( q − k ( ℓ − 1 ) ) c ℓ ( σ ) = ∑ k = r ∞ a k ∏ ℓ = 2 ∞ ( ∑ i = 1 q k ( 1 q k ) ℓ ) c ℓ ( σ ) = ∑ k = r ∞ a k χ ( α k ; 0 ) ( σ ) = ∑ k = r ∞ a k χ ( α k ; 0 ) ( σ ) + a ∞ χ ( α ∞ ; 0 ) ( σ ) . \begin{aligned}
\chi_\nu(\sigma)=\chi_\mu(\sigma)&=\sum_{k=r}^{\infty}a_kq^{-k\widetilde{\operatorname{rk}}\sigma}
=\sum_{k=r}^{\infty}a_k\prod_{\ell=2}^{\infty}\left(q^{-k(\ell-1)}\right)^{c_\ell(\sigma)}\\
&=\sum_{k=r}^{\infty}a_k\prod_{\ell=2}^{\infty}\left(\sum_{i=1}^{q^k}\left(\frac{1}{q^k}\right)^\ell\right)^{c_\ell(\sigma)}
=\sum_{k=r}^{\infty}a_k\chi_{(\alpha^k;0)}(\sigma)\\
&=\sum_{k=r}^{\infty}a_k\chi_{(\alpha^k;0)}(\sigma)+a_\infty\chi_{(\alpha^\infty;0)}(\sigma).
\end{aligned} χ ν ( σ ) = χ μ ( σ ) = k = r ∑ ∞ a k q − k rk σ = k = r ∑ ∞ a k ℓ = 2 ∏ ∞ ( q − k ( ℓ − 1 ) ) c ℓ ( σ ) = k = r ∑ ∞ a k ℓ = 2 ∏ ∞ i = 1 ∑ q k ( q k 1 ) ℓ c ℓ ( σ ) = k = r ∑ ∞ a k χ ( α k ; 0 ) ( σ ) = k = r ∑ ∞ a k χ ( α k ; 0 ) ( σ ) + a ∞ χ ( α ∞ ; 0 ) ( σ ) . This relation clearly holds also for the trivial permutation σ = id \sigma=\operatorname{id} σ = id
χ ν = ∑ k = r ∞ a k χ ( α k ; 0 ) + a ∞ χ ( α ∞ ; 0 ) . (15) \chi_\nu=\sum_{k=r}^{\infty}a_k\chi_{(\alpha^k;0)}+a_\infty\chi_{(\alpha^\infty;0)}. \tag*{(15)} χ ν = k = r ∑ ∞ a k χ ( α k ; 0 ) + a ∞ χ ( α ∞ ; 0 ) . (15) By Choquet theory, there exists a probability measure m m m A ∞ A_{\infty} A ∞
ν = ∫ Extr ( IRS ( A ∞ ) ) λ d m ( λ ) . \nu= \int_{\operatorname{Extr}(\operatorname{IRS}(A_{\infty}))} \lambda\, \mathrm{d}m(\lambda). ν = ∫ Extr ( IRS ( A ∞ )) λ d m ( λ ) . Let tr : IRS ( A ∞ ) → Tr ( A ∞ ) \operatorname{tr} : \operatorname{IRS}(A_{\infty}) \to\operatorname{Tr}(A_{\infty}) tr : IRS ( A ∞ ) → Tr ( A ∞ ) tr ( λ ) = χ λ \operatorname{tr}(\lambda) = \chi_{\lambda} tr ( λ ) = χ λ
tr ( ν ) = χ ν = ∫ Extr ( IRS ( A ∞ ) ) χ λ d m ( λ ) = ∫ Tr ( A ∞ ) χ d ( tr ∗ m ) ( χ ) . (16) \operatorname{tr}(\nu) = \chi_{\nu} = \int_{\operatorname{Extr}(\operatorname{IRS}(A_{\infty}))} \chi_{\lambda} \, \mathrm{d}m(\lambda) = \int_{\operatorname{Tr}(A_{\infty})} \chi\, \mathrm{d}(\operatorname{tr}_{*}m)(\chi). \tag*{(16)} tr ( ν ) = χ ν = ∫ Extr ( IRS ( A ∞ )) χ λ d m ( λ ) = ∫ Tr ( A ∞ ) χ d ( tr ∗ m ) ( χ ) . (16) Recall from § 4.2 that the set of traces of a group forms a Choquet simplex. That is, the ergodic decomposition of a trace is unique. Comparing eq. (16) to eq. (15) , we see that tr ∗ m \operatorname{tr}_{*}m tr ∗ m { χ ( α k ; 0 ) : k ∈ N ≥ r ∪ { ∞ } } \{\chi_{(\alpha_k;0)} : k \in\mathbb{N}_{\ge r} \cup\{\infty\}\} { χ ( α k ; 0 ) : k ∈ N ≥ r ∪ { ∞ }} A ∞ A_{\infty} A ∞ Theorem 4.6 and the formula preceding it, we must have ν = ∑ k = r ∞ a k ν α k + a ∞ ν α ∞ \nu= \sum_{k=r}^{\infty} a_k \nu_{\alpha_k} + a_{\infty}\nu_{\alpha_{\infty}} ν = ∑ k = r ∞ a k ν α k + a ∞ ν α ∞
Let us work out some consequences of the previous proposition.
Proposition 5.9. Let μ ∈ IRS ( SL ∞ ( F q ) ) \mu\in\operatorname{IRS}(\operatorname{SL}_{\infty}(\mathbb{F}_q)) μ ∈ IRS ( SL ∞ ( F q )) μ \mu μ H ∈ Sub ( SL ∞ ( F q ) ) H \in\operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_q)) H ∈ Sub ( SL ∞ ( F q )) H ∩ A ∞ ( f ) ≠ { I } H \cap A_{\infty}(f) \ne\{I\} H ∩ A ∞ ( f ) = { I } f f f
In particular, if μ \mu μ μ \mu μ H H H H ∩ A ∞ ( f ) ≠ { I } H \cap A_{\infty}(f) \ne\{I\} H ∩ A ∞ ( f ) = { I } f f f
Proof of Theorem 5.9. Let f f f μ \mu μ μ ≠ δ { I } \mu\ne\delta_{\{I\}} μ = δ { I } μ \mu μ H H H f ′ f' f ′ SL 3 ( f ′ ) ≤ H \operatorname{SL}_{3}(f') \le H SL 3 ( f ′ ) ≤ H SL 3 ( f ′ ) ∩ A ∞ ( f ′ ) \operatorname{SL}_{3}(f') \cap A_{\infty}(f') SL 3 ( f ′ ) ∩ A ∞ ( f ′ ) H ∩ A ∞ ( f ′ ) H \cap A_{\infty}(f') H ∩ A ∞ ( f ′ ) Theorem 5.8 implies that μ \mu μ c ⊆ N c \subseteq\mathbb{N} c ⊆ N A c ( f ′ ) ≤ H A_c(f') \le H A c ( f ′ ) ≤ H § 6 ). The two bases f f f f ′ f' f ′ f n = f ′ n f^n = f'{}^n f n = f ′ n n ≥ n 0 n \ge n_0 n ≥ n 0 n 0 ∈ N n_0 \in\mathbb{N} n 0 ∈ N
A c ∩ N ≥ n 0 ( f ) = A c ∩ N ≥ n 0 ( f ′ ) ≤ H . A_{c \cap\mathbb{N}_{\ge n_0}}(f) = A_{c \cap\mathbb{N}_{\ge n_0}}(f') \le H. A c ∩ N ≥ n 0 ( f ) = A c ∩ N ≥ n 0 ( f ′ ) ≤ H . This is a non-trivial subgroup containing permutation matrices in the basis f f f A ∞ ( f ) ∩ H ≠ { I } A_{\infty}(f) \cap H \ne\{I\} A ∞ ( f ) ∩ H = { I }
Corollary 5.10. Let μ ∈ IRS ( SL ∞ ( F q ) ) \mu\in\operatorname{IRS}(\operatorname{SL}_{\infty}(\mathbb{F}_q)) μ ∈ IRS ( SL ∞ ( F q )) μ \mu μ a ∞ ( μ ) = 0 a_{\infty}(\mu) = 0 a ∞ ( μ ) = 0
Proof. The ergodic decomposition of the image of the intersection map (see Theorem 5.8 ) shows that a ∞ ( μ ) a_{\infty}(\mu) a ∞ ( μ ) H ∩ A ∞ ( e ) = { I } H \cap A_{\infty}(e) = \{I\} H ∩ A ∞ ( e ) = { I } e e e Theorem 5.9 this condition is μ \mu μ H = { I } H = \{I\} H = { I } μ \mu μ μ ( { { I } } ) ∈ { 0 , 1 } \mu(\{\{I\}\}) \in\{0,1\} μ ({{ I }}) ∈ { 0 , 1 } μ \mu μ
The coloring equivalence relation
In this section we define the coloring equivalence relation corresponding to a subgroup of SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) f f f § 5 , will allow us to show in Theorem 6.8 that the Vershik trace of a non-trivial ergodic invariant random subgroup μ \mu μ χ r ( μ ) \chi_{r(\mu)} χ r ( μ ) r ( μ ) r(\mu) r ( μ ) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) Theorem 3.2 , thereby allowing us to equivariantly identify the space of color vectors Col ( H ) N \operatorname{Col}(H)^{\mathbb{N}} Col ( H ) N V k , ℓ V^{k,\ell} V k , ℓ
Definition of colors
The primary focus of this section is the following object. We use notations from § 4.6 .
Definition 6.1. Let f f f
R f : Sub ( S L ∞ ( F q ) ) → E ( N ) , R f ( H ) = O ( H ∩ A ∞ ( f ) ) R_f \colon\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) \to\mathcal{E}(\mathbb{N}), \qquad R_f(H)=\mathcal{O}(H\cap A_{\infty}(f)) R f : Sub ( SL ∞ ( F q )) → E ( N ) , R f ( H ) = O ( H ∩ A ∞ ( f )) where O \mathcal{O} O R f R_f R f S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) f f f
The coloring partition of H H H f f f N \mathbb{N} N R f ( H ) R_f(H) R f ( H ) Col f ( H ) \operatorname{Col}_f(H) Col f ( H ) Col f ( H ) \operatorname{Col}_f(H) Col f ( H ) colors of H H H f f f
An index i ∈ N i\in\mathbb{N} i ∈ N colored in the color c c c H H H f f f if c ∈ Col f ( H ) c\in\operatorname{Col}_f(H) c ∈ Col f ( H ) i ∈ c i\in c i ∈ c
Given a non-trivial ergodic invariant random subgroup μ ∈ IRS ( S L ∞ ( F q ) ) \mu\in\operatorname{IRS}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) μ ∈ IRS ( SL ∞ ( F q )) f f f R f R_f R f k ∈ N k\in\mathbb{N} k ∈ N α k ∈ D \alpha^k\in\mathcal{D} α k ∈ D § 5.3 . Let λ k = λ α k \lambda_k=\lambda_{\alpha^k} λ k = λ α k A ∞ A_{\infty} A ∞ E ( N ) \mathcal{E}(\mathbb{N}) E ( N ) α k \alpha^k α k § 4.6 ). Combining Theorem 4.8, the information from § 5.3 and taking into account all of the involved definitions gives the decomposition
( R f ) ∗ μ = ∑ k = r ( μ ) ∞ a k ( μ ) λ k . (17) (R_f)_*\mu=\sum_{k=r(\mu)}^{\infty}a_k(\mu)\lambda_k. \tag*{(17)} ( R f ) ∗ μ = k = r ( μ ) ∑ ∞ a k ( μ ) λ k . (17) We immediately obtain the following.
Proposition 6.2. Let μ \mu μ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) f f f μ \mu μ H ∈ Sub ( S L ∞ ( F q ) ) H\in\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) H ∈ Sub ( SL ∞ ( F q ))
(1) any color in Col f ( H ) \operatorname{Col}_f(H) Col f ( H ) N \mathbb{N} N
(2) a permutation matrix σ ∈ A ∞ ( f ) \sigma\in A_{\infty}(f) σ ∈ A ∞ ( f ) H H H σ \sigma σ f f f H H H c ∈ Col f ( H ) c\in\operatorname{Col}_f(H) c ∈ Col f ( H ) A c ( f ) ≤ H A_c(f)\leq H A c ( f ) ≤ H
Any finitary basis f f f e e e μ \mu μ H H H Col f ( H ) \operatorname{Col}_f(H) Col f ( H ) Col e ( H ) \operatorname{Col}_e(H) Col e ( H ) Col e ( H ) \operatorname{Col}_e(H) Col e ( H ) Col f ( H ) \operatorname{Col}_f(H) Col f ( H )
Definition 6.3. Let μ ∈ IRS ( S L ∞ ( F q ) ) \mu\in\operatorname{IRS}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) μ ∈ IRS ( SL ∞ ( F q )) H H H μ \mu μ
The colors Col ( H ) \operatorname{Col}(H) Col ( H ) H H H
The color vector of the subgroup H H H f f f is the point in ∏ i ∈ N Col ( H ) \prod_{i\in\mathbb{N}}\operatorname{Col}(H) ∏ i ∈ N Col ( H ) i i i i ∈ N i\in\mathbb{N} i ∈ N f f f color f ( H ) \operatorname{color}_f(H) color f ( H )
Note that the color vector of a subgroup does depend on the choice of a finitary basis (even if its set of colors does not). The number of colors ∣ Col ( H ) ∣ |\operatorname{Col}(H)| ∣ Col ( H ) ∣ μ \mu μ q q q
The number of colors
We will now define a σ \sigma σ Sub ( SL ∞ ( F q ) ) \operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_{q})) Sub ( SL ∞ ( F q )) σ \sigma σ χ r ( μ ) \chi_{r}(\mu) χ r ( μ )
Definition 6.5. Fix an arbitrary finitary basis f f f R f > n : Sub ( SL ∞ ( F q ) ) → E ( N > n ) R_{f}^{>n} : \operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_{q})) \to\mathcal{E}(\mathbb{N}_{>n}) R f > n : Sub ( SL ∞ ( F q )) → E ( N > n ) H ∈ Sub ( SL ∞ ( F q ) ) H \in\operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_{q})) H ∈ Sub ( SL ∞ ( F q )) R f ( H ) R_{f}(H) R f ( H ) N > n \mathbb{N}_{>n} N > n σ \sigma σ T n \mathcal{T}_{n} T n σ \sigma σ Sub ( SL ∞ ( F q ) ) \operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_{q})) Sub ( SL ∞ ( F q )) R f > n R_{f}^{>n} R f > n σ \sigma σ T = ⋂ n ∈ N T n \mathcal{T}=\bigcap_{n\in\mathbb{N}}\mathcal{T}_{n} T = ⋂ n ∈ N T n
Note that the tail color σ \sigma σ f f f
Proposition 6.6. Let μ \mu μ SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_{q}) SL ∞ ( F q ) n ∈ N n\in\mathbb{N} n ∈ N σ \sigma σ T n \mathcal{T}_{n} T n SL n ( F q ) \operatorname{SL}_{n}(\mathbb{F}_{q}) SL n ( F q ) μ \mu μ E ∈ T n E\in\mathcal{T}_{n} E ∈ T n g ∈ SL n ( F q ) g\in\operatorname{SL}_{n}(\mathbb{F}_{q}) g ∈ SL n ( F q ) μ ( E △ g E g − 1 ) = 0 \mu(E\mathbin{\triangle}gEg^{-1})=0 μ ( E △ g E g − 1 ) = 0
It follows that the σ \sigma σ T \mathcal{T} T SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_{q}) SL ∞ ( F q ) μ \mu μ μ \mu μ T \mathcal{T} T μ \mu μ μ ( E ) ∈ { 0 , 1 } \mu(E)\in\{0,1\} μ ( E ) ∈ { 0 , 1 } E ∈ T E\in\mathcal{T} E ∈ T
Proof of Theorem 6.6. Fix an arbitrary finitary basis f f f μ \mu μ H ∈ Sub ( SL ∞ ( F q ) ) H\in\operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_{q})) H ∈ Sub ( SL ∞ ( F q )) n < i < j ∈ N n<i<j\in\mathbb{N} n < i < j ∈ N g ∈ SL n ( F q ) g\in\operatorname{SL}_{n}(\mathbb{F}_{q}) g ∈ SL n ( F q ) ( i , j ) ∈ R f ( H ) (i,j)\in R_{f}(H) ( i , j ) ∈ R f ( H ) ( i , j ) ∈ R f ( g H g − 1 ) (i,j)\in R_{f}(gHg^{-1}) ( i , j ) ∈ R f ( g H g − 1 ) H H H μ \mu μ i i i j j j f f f H H H H ∩ A ∞ ( f ) H\cap A_{\infty}(f) H ∩ A ∞ ( f ) c c c i i i j j j A c ( f ) ⊆ H A_{c}(f)\subseteq H A c ( f ) ⊆ H σ ∈ A c ∩ N > n ( f ) \sigma\in A_{c\cap\mathbb{N}_{>n}}(f) σ ∈ A c ∩ N > n ( f ) i i i j j j σ \sigma σ g g g σ = g σ g − 1 ∈ g H g − 1 \sigma=g\sigma g^{-1}\in gHg^{-1} σ = g σ g − 1 ∈ g H g − 1 ( i , j ) ∈ R f ( g H g − 1 ) (i,j)\in R_{f}(gHg^{-1}) ( i , j ) ∈ R f ( g H g − 1 ) g − 1 H g g^{-1}Hg g − 1 H g μ \mu μ
We now exhibit a π \pi π σ \sigma σ T n \mathcal{T}_{n} T n SL n ( F q ) \operatorname{SL}_{n}(\mathbb{F}_{q}) SL n ( F q ) μ \mu μ ( i 1 , j 1 ) , … , ( i a , j a ) (i_{1},j_{1}),\ldots,(i_{a},j_{a}) ( i 1 , j 1 ) , … , ( i a , j a ) ( s 1 , t 1 ) , … , ( s b , t b ) (s_{1},t_{1}),\ldots,(s_{b},t_{b}) ( s 1 , t 1 ) , … , ( s b , t b ) a , b ∈ N a,b\in\mathbb{N} a , b ∈ N n n n
E = { H ∈ Sub ( SL ∞ ( F q ) ) : ∀ 1 ≤ k ≤ a , ( i k , j k ) ∈ R f ( H ) ∧ ∀ 1 ≤ ℓ ≤ b , ( s ℓ , t ℓ ) ∉ R f ( H ) } . E=\{H\in\operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_{q})):\ \forall1\leq k\leq a,\ (i_{k},j_{k})\in R_{f}(H)\land\forall1\leq\ell\leq b,\ (s_{\ell},t_{\ell})\notin R_{f}(H)\}. E = { H ∈ Sub ( SL ∞ ( F q )) : ∀1 ≤ k ≤ a , ( i k , j k ) ∈ R f ( H ) ∧ ∀1 ≤ ℓ ≤ b , ( s ℓ , t ℓ ) ∈ / R f ( H )} . The set of all such events E E E π \pi π T n \mathcal{T}_{n} T n μ ( g E g − 1 △ E ) = 0 \mu(gEg^{-1}\mathbin{\triangle}E)=0 μ ( g E g − 1 △ E ) = 0 g ∈ SL n ( F q ) g\in\operatorname{SL}_{n}(\mathbb{F}_{q}) g ∈ SL n ( F q ) □ \square □
Proposition 6.7. Let μ ∈ IRS ( SL ∞ ( F q ) ) \mu\in\operatorname{IRS}(\operatorname{SL}_{\infty}(\mathbb{F}_{q})) μ ∈ IRS ( SL ∞ ( F q )) ∣ Col ( H ) ∣ |\operatorname{Col}(H)| ∣ Col ( H ) ∣ H H H μ \mu μ
Proof. We claim that the number-of-colors map
N C : Sub ( SL ∞ ( F q ) ) ⟶ N ∪ { ∞ } , N C : H ↦ ∣ Col ( H ) ∣ \mathcal{N}\mathcal{C}:\operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_{q}))\longrightarrow\mathbb{N}\cup\{\infty\},\qquad\mathcal{N}\mathcal{C}:H\mapsto|\operatorname{Col}(H)| N C : Sub ( SL ∞ ( F q )) ⟶ N ∪ { ∞ } , N C : H ↦ ∣ Col ( H ) ∣ is T \mathcal{T} T N : E ( N ) → N ∪ { ∞ } \mathcal{N}:\mathcal{E}(\mathbb{N})\to\mathbb{N}\cup\{\infty\} N : E ( N ) → N ∪ { ∞ } f f f N \mathcal{N} N R f : Sub ( S L ∞ ( F q ) ) → E ( N ) R_f : \operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) \to\mathcal{E}(\mathbb{N}) R f : Sub ( SL ∞ ( F q )) → E ( N ) number-of-colors in the basis f f f
N f : Sub ( S L ∞ ( F q ) ) → N 0 ∪ { ∞ } , N f = N ∘ R f . \mathcal{N}_f : \operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) \to\mathbb{N}_0 \cup\{ \infty\}, \qquad\mathcal{N}_f = \mathcal{N} \circ R_f . N f : Sub ( SL ∞ ( F q )) → N 0 ∪ { ∞ } , N f = N ∘ R f . Note that the map N C \mathcal{N}C N C μ \mu μ N f \mathcal{N}_f N f
For a μ \mu μ H H H R f ( H ) R_f(H) R f ( H ) R f > n ( H ) R_f^{>n}(H) R f > n ( H ) n ∈ N n \in\mathbb{N} n ∈ N N f = N f > n \mathcal{N}_f = \mathcal{N}_f^{>n} N f = N f > n N f > n : Sub ( S L ∞ ( F q ) ) → N 0 ∪ { ∞ } \mathcal{N}_f^{>n} : \operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) \to\mathbb{N}_0 \cup\{\infty\} N f > n : Sub ( SL ∞ ( F q )) → N 0 ∪ { ∞ } R f > n ( H ) R_f^{>n}(H) R f > n ( H ) T n \mathcal{T}_n T n N f \mathcal{N}_f N f T n \mathcal{T}_n T n n ∈ N n \in\mathbb{N} n ∈ N N f \mathcal{N}_f N f N C \mathcal{N}C N C T \mathcal{T} T
Lastly, since T \mathcal{T} T μ \mu μ N C \mathcal{N}C N C μ \mu μ
Proposition 6.8. Let μ \mu μ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) r = r ( μ ) r = r(\mu) r = r ( μ ) μ \mu μ a r ( μ ) = 1 a_r(\mu) = 1 a r ( μ ) = 1 μ \mu μ χ r \chi_r χ r μ \mu μ q r q^r q r
Proof. Propositions 4.8 and 5.8 together imply that the value a k ( μ ) a_k(\mu) a k ( μ ) μ \mu μ q k q^k q k μ \mu μ a r ( μ ) = 1 a_r(\mu) = 1 a r ( μ ) = 1 χ μ = χ r \chi_\mu= \chi_r χ μ = χ r μ \mu μ q r q^r q r
The color red
Fix throughout the current § 6.3 a non-trivial ergodic invariant random subgroup μ ∈ IRS ( S L ∞ ( F q ) ) \mu\in\operatorname{IRS}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) μ ∈ IRS ( SL ∞ ( F q ))
Proposition 6.9. For μ \mu μ H ∈ Sub ( S L ∞ ( F q ) ) H \in\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) H ∈ Sub ( SL ∞ ( F q )) f f f c ∈ Col f ( H ) c \in\operatorname{Col}_f(H) c ∈ Col f ( H ) S L c ( f ) ≤ H \mathrm{SL}_c(f) \le H SL c ( f ) ≤ H c c c Col ( H ) \operatorname{Col}(H) Col ( H ) f f f
Once established, this result will allow us to introduce the following key notion.
Definition 6.10. The unique color red ( H ) ∈ Col ( H ) \operatorname{red}(H) \in\operatorname{Col}(H) red ( H ) ∈ Col ( H ) red color of the μ \mu μ H H H
The proof of Theorem 6.9 will require some preparation.
Claim 6.11. Let f f f μ \mu μ H H H I ⊆ N I \subseteq\mathbb{N} I ⊆ N ∣ I ∣ ≥ 3 |I| \ge3 ∣ I ∣ ≥ 3 S L I ( f ) ≤ H \mathrm{SL}_I(f) \le H SL I ( f ) ≤ H I I I c ∈ Col f ( H ) c \in\operatorname{Col}_f(H) c ∈ Col f ( H ) S L c ( f ) ≤ H \mathrm{SL}_c(f) \le H SL c ( f ) ≤ H
Proof. Let H ∈ Sub ( S L ∞ ( F q ) ) H \in\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) H ∈ Sub ( SL ∞ ( F q )) I ⊂ N I \subset\mathbb{N} I ⊂ N ∣ I ∣ ≥ 3 |I| \ge3 ∣ I ∣ ≥ 3 S L I ( f ) ≤ H \mathrm{SL}_I(f) \le H SL I ( f ) ≤ H S L I ( f ) \mathrm{SL}_I(f) SL I ( f ) i ∈ I i \in I i ∈ I c ∈ Col f ( H ) c \in\operatorname{Col}_f(H) c ∈ Col f ( H )
We wish to show that S L c ( f ) ≤ H \mathrm{SL}_c(f) \le H SL c ( f ) ≤ H J J J c c c ∣ J ∣ ≥ 3 |J| \ge3 ∣ J ∣ ≥ 3 S L J ( f ) ≤ H \mathrm{SL}_J(f) \le H SL J ( f ) ≤ H i ∈ c ∖ J i \in c \setminus J i ∈ c ∖ J S L J ∪ { i } ( f ) ≤ H \mathrm{SL}_{J \cup\{i\}}(f) \le H SL J ∪ { i } ( f ) ≤ H
We prove the above statement. Let σ 0 \sigma_0 σ 0 f f f J ∪ { i } J \cup\{i\} J ∪ { i } σ 0 \sigma_0 σ 0 σ = σ 0 ∈ A c ( f ) \sigma= \sigma_0 \in A_c(f) σ = σ 0 ∈ A c ( f ) σ 0 \sigma_0 σ 0 c ∖ ( J ∪ { i } ) c \setminus(J \cup\{i\}) c ∖ ( J ∪ { i }) σ ∈ A c ( f ) \sigma\in A_c(f) σ ∈ A c ( f ) A c ( f ) ⊆ A ∞ ( f ) ∩ H A_c(f) \subseteq A_{\infty}(f) \cap H A c ( f ) ⊆ A ∞ ( f ) ∩ H σ ∈ H \sigma\in H σ ∈ H S L J ( f ) \mathrm{SL}_J(f) SL J ( f ) σ \sigma σ S L J ′ ( f ) \mathrm{SL}_{J'}(f) SL J ′ ( f ) H H H J ′ = ( J ∖ { σ ( i ) } ) ∪ { i } J' = (J \setminus\{\sigma(i)\}) \cup\{i\} J ′ = ( J ∖ { σ ( i )}) ∪ { i } S L J ( f ) \mathrm{SL}_J(f) SL J ( f ) S L J ′ ( f ) \mathrm{SL}_{J'}(f) SL J ′ ( f ) H H H S L J ∪ J ′ ( f ) = S L J ∪ { i } ( f ) ≤ H \mathrm{SL}_{J \cup J'}(f) = \mathrm{SL}_{J \cup\{i\}}(f) \le H SL J ∪ J ′ ( f ) = SL J ∪ { i } ( f ) ≤ H
From what we have established already, the existence of the desired color is easy to show.
Claim 6.12. For μ \mu μ H ∈ Sub ( SL ∞ ( F q ) ) H \in\operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_q)) H ∈ Sub ( SL ∞ ( F q )) f f f c ∈ Col f ( H ) c \in\operatorname{Col}_{f}(H) c ∈ Col f ( H ) SL c ( f ) ≤ H \operatorname{SL}_{c}(f) \le H SL c ( f ) ≤ H
Proof. Fix a finitary basis f f f Claim 6.11 , for μ \mu μ H ∈ Sub ( SL ∞ ( F q ) ) H \in\operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_q)) H ∈ Sub ( SL ∞ ( F q )) f ′ f' f ′ c ∈ Col ( H ) c \in\operatorname{Col}(H) c ∈ Col ( H ) SL c ( f ′ ) ≤ H \operatorname{SL}_{c}(f') \le H SL c ( f ′ ) ≤ H f f f f ′ f' f ′ f i = f i ′ f_i=f'_i f i = f i ′ i > n i>n i > n
SL c ∩ N > n ( f ) = SL c ∩ N > n ( f ′ ) ≤ H . \operatorname{SL}_{c\cap\mathbb{N}_{>n}}(f)=\operatorname{SL}_{c\cap\mathbb{N}_{>n}}(f')\le H. SL c ∩ N > n ( f ) = SL c ∩ N > n ( f ′ ) ≤ H . Another application of Claim 6.11 implies that SL c ( f ) ≤ H \operatorname{SL}_{c}(f)\le H SL c ( f ) ≤ H
The argument to show the uniqueness of the desired color will be quantitative. The rough idea is the following: assume we had two distinct colors c 1 , c 2 ∈ Col ( H ) c_1,c_2 \in\operatorname{Col}(H) c 1 , c 2 ∈ Col ( H ) SL c i ( f ) ≤ H \operatorname{SL}_{c_i}(f)\le H SL c i ( f ) ≤ H i = 1 , 2 i=1,2 i = 1 , 2 SL n ( f ) ≤ H \operatorname{SL}_{n}(f)\le H SL n ( f ) ≤ H 2 q − r n 2q^{-rn} 2 q − r n n n n c 1 c_1 c 1 c 2 c_2 c 2 q − r n q^{-rn} q − r n
To make this quantitative argument rigorous, we make use of the notion of pointed partitions from Section 4.5. For a μ \mu μ H ∈ Sub ( SL ∞ ( F q ) ) H \in\operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_q)) H ∈ Sub ( SL ∞ ( F q )) f f f
Col f r e d ( H ) = { c ∈ Col f ( H ) : SL c ( f ) ≤ H } . \operatorname{Col}^{\mathrm{red}}_{f}(H)=\{c\in\operatorname{Col}_{f}(H):\operatorname{SL}_{c}(f)\le H\}. Col f red ( H ) = { c ∈ Col f ( H ) : SL c ( f ) ≤ H } . We form the pointed partition P f r e d ( H ) ∈ E ∗ ( N ) P^{\mathrm{red}}_{f}(H)\in E^*(\mathbb{N}) P f red ( H ) ∈ E ∗ ( N )
P f r e d ( H ) = { { c : c ∈ Col f r e d ( H ) } ∪ { N ∖ ⋃ c ∈ Col f r e d ( H ) c } } ∈ E ∗ ( N ) P^{\mathrm{red}}_{f}(H)=\left\{ \{c:c\in\operatorname{Col}^{\mathrm{red}}_{f}(H)\}\cup\left\{\mathbb{N}\setminus\bigcup_{c\in\operatorname{Col}^{\mathrm{red}}_{f}(H)}c\right\}\right\}\in E^*(\mathbb{N}) P f red ( H ) = ⎩ ⎨ ⎧ { c : c ∈ Col f red ( H )} ∪ ⎩ ⎨ ⎧ N ∖ c ∈ Col f red ( H ) ⋃ c ⎭ ⎬ ⎫ ⎭ ⎬ ⎫ ∈ E ∗ ( N ) with N ∖ ⋃ c ∈ Col f r e d ( H ) c \mathbb{N}\setminus\bigcup_{c\in\operatorname{Col}^{\mathrm{red}}_{f}(H)}c N ∖ ⋃ c ∈ Col f red ( H ) c
Claim 6.13. Let f f f μ \mu μ P f r e d : Sub ( SL ∞ ( F q ) ) → E ∗ ( N ) P^{\mathrm{red}}_{f}:\operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_q))\to E^*(\mathbb{N}) P f red : Sub ( SL ∞ ( F q )) → E ∗ ( N ) A ∞ A_{\infty} A ∞ A ∞ A_{\infty} A ∞ Sub ( SL ∞ ( F q ) ) \operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_q)) Sub ( SL ∞ ( F q )) f f f A ∞ → A ∞ ( f ) ⊆ SL ∞ ( F q ) A_{\infty}\to A_{\infty}(f)\subseteq\operatorname{SL}_{\infty}(\mathbb{F}_q) A ∞ → A ∞ ( f ) ⊆ SL ∞ ( F q )
Proof. Let H H H μ \mu μ σ ∈ A ∞ \sigma\in A_{\infty} σ ∈ A ∞ σ f \sigma_f σ f A ∞ ( f ) A_{\infty}(f) A ∞ ( f ) σ \sigma σ c ∈ Col f r e d ( H ) c\in\operatorname{Col}^{\mathrm{red}}_{f}(H) c ∈ Col f red ( H ) σ ⋅ c ∈ Col f r e d ( σ f H σ f − 1 ) \sigma\cdot c\in\operatorname{Col}^{\mathrm{red}}_{f}(\sigma_fH\sigma_f^{-1}) σ ⋅ c ∈ Col f red ( σ f H σ f − 1 ) σ ⋅ c \sigma\cdot c σ ⋅ c σ f H σ f − 1 \sigma_fH\sigma_f^{-1} σ f H σ f − 1 SL c ( f ) ≤ H \operatorname{SL}_{c}(f)\le H SL c ( f ) ≤ H
SL σ ⋅ c ( f ) = σ f SL c ( f ) σ f − 1 ≤ σ f H σ f − 1 . \operatorname{SL}_{\sigma\cdot c}(f)=\sigma_f\operatorname{SL}_{c}(f)\sigma_f^{-1}\le\sigma_fH\sigma_f^{-1}. SL σ ⋅ c ( f ) = σ f SL c ( f ) σ f − 1 ≤ σ f H σ f − 1 . Hence σ ⋅ c ∈ Col f r e d ( σ f H σ f − 1 ) \sigma\cdot c\in\operatorname{Col}^{\mathrm{red}}_{f}(\sigma_fH\sigma_f^{-1}) σ ⋅ c ∈ Col f red ( σ f H σ f − 1 )
We are now ready to conclude the proof concerning the uniqueness of the red color.
Proof of Theorem 6.9. Let μ ∈ IRS ( SL ∞ ( F q ) ) \mu\in\operatorname{IRS}(\operatorname{SL}_{\infty}(\mathbb{F}_q)) μ ∈ IRS ( SL ∞ ( F q )) f f f
N f r e d : Sub ( SL ∞ ( F q ) ) → N 0 ∪ { ∞ } \mathcal{N}^{\mathrm{red}}_{f}:\operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_q))\to\mathbb{N}_0\cup\{\infty\} N f red : Sub ( SL ∞ ( F q )) → N 0 ∪ { ∞ } be the map given by N f r e d ( H ) = ∣ Col f r e d ( H ) ∣ \mathcal{N}^{\mathrm{red}}_{f}(H)=\left|\operatorname{Col}^{\mathrm{red}}_{f}(H)\right| N f red ( H ) = Col f red ( H ) H ∈ Sub ( SL ∞ ( F q ) ) H\in\operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_q)) H ∈ Sub ( SL ∞ ( F q )) N f r e d \mathcal{N}^{\mathrm{red}}_{f} N f red T \mathcal{T} T T \mathcal{T} T σ \sigma σ N f r e d \mathcal{N}^{\mathrm{red}}_{f} N f red μ \mu μ N N N Claim 6.12 implies that N > 0 N>0 N > 0 N < ∞ N<\infty N < ∞ N = 1 N=1 N = 1 λ = ( P f r e d ) ∗ μ \lambda=(P_f^{\mathrm{red}})_*\mu λ = ( P f red ) ∗ μ E ∗ ( N ) \mathcal{E}^*(\mathbb{N}) E ∗ ( N ) A ∞ A_\infty A ∞ N N N ( R f ) ∗ μ = λ α r (R_f)_*\mu=\lambda_{\alpha^r} ( R f ) ∗ μ = λ α r α r ∈ D \alpha^r\in\mathcal{D} α r ∈ D q r q^r q r μ \mu μ f f f q − r ( μ ) q^{-r(\mu)} q − r ( μ ) N N N Col f r e d ( H ) \operatorname{Col}^{\mathrm{red}}_f(H) Col f red ( H ) λ \lambda λ P ∈ E ∗ ( N ) P\in\mathcal{E}^*(\mathbb{N}) P ∈ E ∗ ( N ) D ( P ) D(P) D ( P ) α ∈ D ∗ \alpha\in\mathcal{D}^* α ∈ D ∗
α n = { q − r ( μ ) , 1 ≤ n ≤ N , 1 − N q − r ( μ ) , n = − 1 , 0 , otherwise . \alpha_n=
\begin{cases}
q^{-r(\mu)}, & 1\le n\le N,\\
1-Nq^{-r(\mu)}, & n=-1,\\
0, & \text{otherwise}.
\end{cases} α n = ⎩ ⎨ ⎧ q − r ( μ ) , 1 − N q − r ( μ ) , 0 , 1 ≤ n ≤ N , n = − 1 , otherwise . By Theorem 4.11 we have λ = λ α ∗ \lambda=\lambda_{\alpha}^* λ = λ α ∗ λ \lambda λ n ∈ N n\in\mathbb{N} n ∈ N
q − r n ( 1 − o ( 1 ) ) = μ ( { SL n ( f ) ≤ H } ) = μ ( { 1 , … , n have the same color from Col f r e d ( H ) } ) = λ α ∗ ( { 1 , … , n belong to the same non-distinguished block } ) = N q − r n . \begin{aligned}
q^{-rn}(1-o(1))=\mu(\{\operatorname{SL}_n(f)\le H\})=\mu\left(\{1,\ldots,n\text{ have the same color from }\operatorname{Col}^{\mathrm{red}}_f(H)\}\right) \\
&=\lambda_\alpha^*\left(\{1,\ldots,n\text{ belong to the same non-distinguished block}\}\right)=Nq^{-rn}.
\end{aligned} q − r n ( 1 − o ( 1 )) = μ ({ SL n ( f ) ≤ H }) = μ ( { 1 , … , n have the same color from Col f red ( H )} ) = λ α ∗ ( { 1 , … , n belong to the same non-distinguished block } ) = N q − r n . This implies N = 1 N=1 N = 1
Lastly, let f f f f ′ f' f ′ μ \mu μ H H H c ∈ Col f ( H ) c\in\operatorname{Col}_f(H) c ∈ Col f ( H ) c ′ ∈ Col f ′ ( H ) c'\in\operatorname{Col}_{f'}(H) c ′ ∈ Col f ′ ( H ) SL c ( f ) ≤ H \operatorname{SL}_c(f)\le H SL c ( f ) ≤ H SL c ′ ( f ′ ) ≤ H \operatorname{SL}_{c'}(f')\le H SL c ′ ( f ′ ) ≤ H n 0 ∈ N n_0\in\mathbb{N} n 0 ∈ N f n = ( f ′ ) n f^n=(f')^n f n = ( f ′ ) n n > n 0 n>n_0 n > n 0
SL c ∩ N > n 0 ( f ′ ) = SL c ∩ N > n 0 ( f ) ≤ SL c ( f ) ≤ H . \operatorname{SL}_{c\cap\mathbb{N}_{>n_0}}(f')=\operatorname{SL}_{c\cap\mathbb{N}_{>n_0}}(f)\le\operatorname{SL}_c(f)\le H. SL c ∩ N > n 0 ( f ′ ) = SL c ∩ N > n 0 ( f ) ≤ SL c ( f ) ≤ H . It follows from Claim 6.11 and the uniqueness of the color c ′ c' c ′ c ∩ N > n 0 c\cap\mathbb{N}_{>n_0} c ∩ N > n 0 c ′ c' c ′ c c c c ′ c' c ′ Col ( H ) \operatorname{Col}(H) Col ( H ) □ \square □
GLn(Fq)-action on color vectors
Let μ ∈ IRS ( SL ∞ ( F q ) ) \mu\in\operatorname{IRS}(\operatorname{SL}_\infty(\mathbb{F}_q)) μ ∈ IRS ( SL ∞ ( F q )) ( R e ) ∗ μ = λ r ( μ ) (R_e)_*\mu=\lambda_{r(\mu)} ( R e ) ∗ μ = λ r ( μ ) μ \mu μ H ∈ Sub ( SL ∞ ( F q ) ) H\in\operatorname{Sub}(\operatorname{SL}_\infty(\mathbb{F}_q)) H ∈ Sub ( SL ∞ ( F q )) n ∈ N n\in\mathbb{N} n ∈ N Col ( H ) n \operatorname{Col}(H)^n Col ( H ) n color e ( H ) I \operatorname{color}_e(H)_I color e ( H ) I I ⊆ N I\subseteq\mathbb{N} I ⊆ N ∣ I ∣ = n |I|=n ∣ I ∣ = n min I \min I min I
Recall that the color vector color f ( H ) ∈ Col ( H ) N \operatorname{color}_f(H)\in\operatorname{Col}(H)^{\mathbb{N}} color f ( H ) ∈ Col ( H ) N GL ∞ ( F q ) \operatorname{GL}_\infty(\mathbb{F}_q) GL ∞ ( F q ) Col ( H ) N \operatorname{Col}(H)^{\mathbb{N}} Col ( H ) N f = A − 1 e f=A^{-1}e f = A − 1 e A ∈ GL ∞ ( F q ) A\in\operatorname{GL}_\infty(\mathbb{F}_q) A ∈ GL ∞ ( F q ) color f ( H ) = A ⋅ color e ( H ) \operatorname{color}_f(H)=A\cdot\operatorname{color}_e(H) color f ( H ) = A ⋅ color e ( H )
Definition 6.14 (Action on color vectors). Let H H H μ \mu μ Θ H \Theta_H Θ H GL n ( F q ) \operatorname{GL}_n(\mathbb{F}_q) GL n ( F q ) Col ( H ) n \operatorname{Col}(H)^n Col ( H ) n A ∈ GL n ( F q ) A\in\operatorname{GL}_n(\mathbb{F}_q) A ∈ GL n ( F q ) γ ∈ Col ( H ) n \gamma\in\operatorname{Col}(H)^n γ ∈ Col ( H ) n I ⊂ N I\subset\mathbb{N} I ⊂ N ∣ I ∣ = n |I|=n ∣ I ∣ = n γ = color e ( H ) I \gamma=\operatorname{color}_e(H)_I γ = color e ( H ) I f f f f = [ A − 1 ] I e f=[A^{-1}]_I e f = [ A − 1 ] I e
Θ H ( A ) γ = color f ( H ) I ∈ Col ( H ) n . \Theta_H(A)\gamma=\operatorname{color}_f(H)_I\in\operatorname{Col}(H)^n. Θ H ( A ) γ = color f ( H ) I ∈ Col ( H ) n . When the context is clear, we shall suppress the notation Θ H \Theta_H Θ H A ⋅ γ A\cdot\gamma A ⋅ γ Θ H ( A ) γ \Theta_H(A)\gamma Θ H ( A ) γ
The basic properties of the action Θ H \Theta_H Θ H
Proposition 6.15. Let H H H μ \mu μ Θ H \Theta_H Θ H GL n ( F q ) \operatorname{GL}_n(\mathbb{F}_q) GL n ( F q ) Col ( H ) n \operatorname{Col}(H)^n Col ( H ) n n ∈ N n\in\mathbb{N} n ∈ N
(1) Θ H \Theta_H Θ H I I I
(2) Θ H \Theta_H Θ H GL n ( F q ) ↷ Col ( H ) n \operatorname{GL}_n(\mathbb{F}_q)\curvearrowright\operatorname{Col}(H)^n GL n ( F q ) ↷ Col ( H ) n
(3) Fix n 1 , n 2 ∈ N n_1,n_2 \in\mathbb{N} n 1 , n 2 ∈ N γ 1 ∈ Col ( H ) n 1 \gamma_1 \in\operatorname{Col}(H)^{n_1} γ 1 ∈ Col ( H ) n 1 γ 2 ∈ Col ( H ) n 2 \gamma_2 \in\operatorname{Col}(H)^{n_2} γ 2 ∈ Col ( H ) n 2 A 1 ∈ GL n 1 ( F q ) A_1 \in\operatorname{GL}_{n_1}(\mathbb{F}_q) A 1 ∈ GL n 1 ( F q ) A 2 ∈ GL n 2 ( F q ) A_2 \in\operatorname{GL}_{n_2}(\mathbb{F}_q) A 2 ∈ GL n 2 ( F q )
Θ H ( A 1 0 0 A 2 ) ( γ 1 , γ 2 ) = ( A 1 γ 1 , A 2 γ 2 ) . \Theta_H \begin{pmatrix} A_1 & 0 \\ 0 & A_2 \end{pmatrix}(\gamma_1,\gamma_2)=(A_1\gamma_1,A_2\gamma_2). Θ H ( A 1 0 0 A 2 ) ( γ 1 , γ 2 ) = ( A 1 γ 1 , A 2 γ 2 ) . The proof of Theorem 6.15 will require the following Theorem 6.16, which is the basic tool to show that two color vectors are equal. It uses the notion of extension by permutations and of the subgroup H ∗ ≤ GL ∞ ( F q ) H^* \leq\operatorname{GL}_{\infty}(\mathbb{F}_q) H ∗ ≤ GL ∞ ( F q ) H ≤ GL ∞ ( F q ) H \leq\operatorname{GL}_{\infty}(\mathbb{F}_q) H ≤ GL ∞ ( F q ) § 2.3 . In most situations where Theorem 6.16 is invoked, we will only need conditions (1) and (2). The more complicated condition (3) will be seldom used.
Lemma 6.16. Let H H H μ \mu μ f f f f = A ⋅ e f=A\cdot e f = A ⋅ e A ∈ GL ∞ ( F q ) A\in\operatorname{GL}_{\infty}(\mathbb{F}_q) A ∈ GL ∞ ( F q ) I , J ⊆ N I,J\subseteq\mathbb{N} I , J ⊆ N ∣ I ∣ = ∣ J ∣ = n |I|=|J|=n ∣ I ∣ = ∣ J ∣ = n max I < min J \max I<\min J max I < min J
Ω = ( 0 I n I n 0 ) ∈ SL 2 n ( F q ) . \Omega=\begin{pmatrix}0&I_n\\I_n&0\end{pmatrix}\in\operatorname{SL}_{2n}(\mathbb{F}_q). Ω = ( 0 I n I n 0 ) ∈ SL 2 n ( F q ) . The following three conditions are equivalent:
color f ( H ) I = color f ( H ) J \operatorname{color}_f(H)_I=\operatorname{color}_f(H)_J color f ( H ) I = color f ( H ) J
A [ Ω ] I ∪ J A − 1 ∈ H ∗ A[\Omega]_{I\cup J}A^{-1}\in H^* A [ Ω ] I ∪ J A − 1 ∈ H ∗
Let ω ∈ GL 2 ( F q ) \omega\in\operatorname{GL}_2(\mathbb{F}_q) ω ∈ GL 2 ( F q ) ( 1 0 0 1 ) \begin{pmatrix}1&0\\0&1\end{pmatrix} ( 1 0 0 1 ) n n n ( 0 1 1 0 ) \begin{pmatrix}0&1\\1&0\end{pmatrix} ( 0 1 1 0 ) n n n m > max J m>\max J m > max J A ∈ GL m ( F q ) A\in\operatorname{GL}_m(\mathbb{F}_q) A ∈ GL m ( F q ) m < i < j m<i<j m < i < j color f ( H ) i = color f ( H ) j \operatorname{color}_f(H)_i=\operatorname{color}_f(H)_j color f ( H ) i = color f ( H ) j
A [ ( Ω 0 0 ω ) ] I ∪ J ∪ { i , j } A − 1 ∈ H . A\left[\begin{pmatrix}\Omega&0\\0&\omega\end{pmatrix}\right]_{I\cup J\cup\{i,j\}}A^{-1}\in H. A [ ( Ω 0 0 ω ) ] I ∪ J ∪ { i , j } A − 1 ∈ H . Proof. Condition (2) implies condition (1) by the definition of the subgroup H ∗ H^* H ∗
Assume condition (1). Let m > max J m>\max J m > max J m < i < j m<i<j m < i < j f f f A ∞ ( f ) A_{\infty}(f) A ∞ ( f ) I I I J J J ( i , j ) (i,j) ( i , j ) n n n i , j i,j i , j H H H
Assume condition (3). Let m > max J m>\max J m > max J A ∈ GL m ( F q ) A\in\operatorname{GL}_m(\mathbb{F}_q) A ∈ GL m ( F q ) m < i < j m<i<j m < i < j color f ( H ) i = color f ( H ) j \operatorname{color}_f(H)_i=\operatorname{color}_f(H)_j color f ( H ) i = color f ( H ) j A ∈ GL m ( F q ) A\in\operatorname{GL}_m(\mathbb{F}_q) A ∈ GL m ( F q ) A [ Ω ] I ∪ J A − 1 A[\Omega]_{I\cup J}A^{-1} A [ Ω ] I ∪ J A − 1 GL m ( F q ) \operatorname{GL}_m(\mathbb{F}_q) GL m ( F q ) ω \omega ω { i , j } \{i,j\} { i , j } H H H H ∗ H^* H ∗ □ \square □
Proof of Theorem 6.15. We start by showing item (1). Let I I I I ′ I' I ′ n n n γ = color e ( H ) I = color e ( H ) I ′ \gamma=\operatorname{color}_e(H)_I=\operatorname{color}_e(H)_{I'} γ = color e ( H ) I = color e ( H ) I ′ f = [ A − 1 ] I e f=[A^{-1}]_I e f = [ A − 1 ] I e f ′ = [ A ′ − 1 ] I ′ e f'=[A'^{-1}]_{I'}e f ′ = [ A ′ − 1 ] I ′ e color f ( H ) I = color f ′ ( H ) I ′ \operatorname{color}_f(H)_I=\operatorname{color}_{f'}(H)_{I'} color f ( H ) I = color f ′ ( H ) I ′ γ = color e ( H ) I ′ ′ \gamma=\operatorname{color}_e(H)_{I''} γ = color e ( H ) I ′′ I ′ ′ I'' I ′′ min I ′ ′ \min I'' min I ′′ I ∩ I ′ = ∅ I\cap I'=\varnothing I ∩ I ′ = ∅ max I < min I ′ \max I<\min I' max I < min I ′
Let δ = color f ( H ) I \delta=\operatorname{color}_f(H)_I δ = color f ( H ) I J ⊆ N J\subseteq\mathbb{N} J ⊆ N min J > max I ′ \min J>\max I' min J > max I ′ n n n δ = color e ( H ) J \delta=\operatorname{color}_e(H)_J δ = color e ( H ) J δ = color f ( H ) J \delta=\operatorname{color}_f(H)_J δ = color f ( H ) J
δ = color f ( H ) I = color f ( H ) J \delta=\operatorname{color}_f(H)_I=\operatorname{color}_f(H)_J δ = color f ( H ) I = color f ( H ) J and
γ = color e ( H ) I = color e ( H ) I ′ . \gamma=\operatorname{color}_{e}(H)_I=\operatorname{color}_{e}(H)_{I'}. γ = color e ( H ) I = color e ( H ) I ′ . The first equality gives
B 1 = [ A − 1 ] I [ Ω ] I ∪ J [ A ] I ∈ H ∗ B_1=[A^{-1}]_I[\Omega]_{I\cup J}[A]_I\in H^* B 1 = [ A − 1 ] I [ Ω ] I ∪ J [ A ] I ∈ H ∗ and the second
C = [ Ω ] I ∪ I ′ ∈ H ∗ . C=[\Omega]_{I\cup I'}\in H^*. C = [ Ω ] I ∪ I ′ ∈ H ∗ . What we are trying to show is δ = color f ′ ( H ) I ′ \delta=\operatorname{color}_{f'}(H)_{I'} δ = color f ′ ( H ) I ′
B 2 = [ A − 1 ] I ′ [ Ω ] I ′ ∪ J [ A ] I ′ ∈ H ∗ . B_2=[A^{-1}]_{I'}[\Omega]_{I'\cup J}[A]_{I'}\in H^*. B 2 = [ A − 1 ] I ′ [ Ω ] I ′ ∪ J [ A ] I ′ ∈ H ∗ . One may check directly the matrix equation B 2 = B 1 C B 1 − 1 B_2=B_1CB_1^{-1} B 2 = B 1 C B 1 − 1 B 2 ∈ H ∗ B_2\in H^* B 2 ∈ H ∗
We next show item (3). Let I 1 , I 2 ⊆ N I_1,I_2\subseteq\mathbb{N} I 1 , I 2 ⊆ N γ 1 = color e ( H ) I 1 \gamma_1=\operatorname{color}_{e}(H)_{I_1} γ 1 = color e ( H ) I 1 γ 2 = color e ( H ) I 2 \gamma_2=\operatorname{color}_{e}(H)_{I_2} γ 2 = color e ( H ) I 2 max I 1 < min I 2 \max I_1<\min I_2 max I 1 < min I 2
A ⋅ γ = color f ( H ) I 1 ∪ I 2 , A\cdot\gamma=\operatorname{color}_{f}(H)_{I_1\cup I_2}, A ⋅ γ = color f ( H ) I 1 ∪ I 2 , where A = ( A 1 0 0 A 2 ) A=\begin{pmatrix} A_1 & 0 \\ 0 & A_2 \end{pmatrix} A = ( A 1 0 0 A 2 ) γ = ( γ 1 , γ 2 ) \gamma=(\gamma_1,\gamma_2) γ = ( γ 1 , γ 2 ) f = [ A − 1 ] I 1 ∪ I 2 ⋅ e f=[A^{-1}]_{I_1\cup I_2}\cdot e f = [ A − 1 ] I 1 ∪ I 2 ⋅ e
A 1 ⋅ γ 1 = color f ( H ) I 1 and A 2 ⋅ γ 2 = color f ( H ) I 2 . A_1\cdot\gamma_1=\operatorname{color}_{f}(H)_{I_1}
\qquad\text{and}\qquad
A_2\cdot\gamma_2=\operatorname{color}_{f}(H)_{I_2}. A 1 ⋅ γ 1 = color f ( H ) I 1 and A 2 ⋅ γ 2 = color f ( H ) I 2 . Let δ 1 = A 1 ⋅ γ 1 \delta_1=A_1\cdot\gamma_1 δ 1 = A 1 ⋅ γ 1 δ 1 = color f 1 ( H ) I 1 \delta_1=\operatorname{color}_{f_1}(H)_{I_1} δ 1 = color f 1 ( H ) I 1 f 1 = [ A 1 − 1 ] I 1 e f_1=[A_1^{-1}]_{I_1}e f 1 = [ A 1 − 1 ] I 1 e J ⊆ N J\subseteq\mathbb{N} J ⊆ N min J > max I 2 \min J>\max I_2 min J > max I 2 δ 1 = color e ( H ) J \delta_1=\operatorname{color}_{e}(H)_J δ 1 = color e ( H ) J δ 1 = color f 1 ( H ) J = color f ( H ) J \delta_1=\operatorname{color}_{f_1}(H)_J=\operatorname{color}_{f}(H)_J δ 1 = color f 1 ( H ) J = color f ( H ) J
color f 1 ( H ) I 1 = color f 1 ( H ) J . \operatorname{color}_{f_1}(H)_{I_1}=\operatorname{color}_{f_1}(H)_J. color f 1 ( H ) I 1 = color f 1 ( H ) J . This gives
[ A 1 − 1 ] I 1 [ Ω ] I 1 ∪ J [ A 1 ] I 1 ∈ H ∗ . [A_1^{-1}]_{I_1}[\Omega]_{I_1\cup J}[A_1]_{I_1}\in H^*. [ A 1 − 1 ] I 1 [ Ω ] I 1 ∪ J [ A 1 ] I 1 ∈ H ∗ . The above matrix also equals
[ A − 1 ] I 1 ∪ I 2 [ Ω ] I 1 ∪ J [ A ] I 1 ∪ I 2 . [A^{-1}]_{I_1\cup I_2}[\Omega]_{I_1\cup J}[A]_{I_1\cup I_2}. [ A − 1 ] I 1 ∪ I 2 [ Ω ] I 1 ∪ J [ A ] I 1 ∪ I 2 . Hence
color f ( H ) I 1 = color f ( H ) J = δ 1 , \operatorname{color}_{f}(H)_{I_1}=\operatorname{color}_{f}(H)_J=\delta_1, color f ( H ) I 1 = color f ( H ) J = δ 1 , as required. The proof for the counterpart action A 2 ⋅ γ 2 A_2\cdot\gamma_2 A 2 ⋅ γ 2
For item (2), the group action property, let γ ∈ Col ( H ) n \gamma\in\operatorname{Col}(H)^n γ ∈ Col ( H ) n A , B ∈ GL n ( F q ) A,B\in\operatorname{GL}_n(\mathbb{F}_q) A , B ∈ GL n ( F q ) δ = B ⋅ γ \delta=B\cdot\gamma δ = B ⋅ γ A ⋅ δ = A B ⋅ γ A\cdot\delta=AB\cdot\gamma A ⋅ δ = A B ⋅ γ I , J ⊆ N I,J\subseteq\mathbb{N} I , J ⊆ N max I < min J \max I<\min J max I < min J γ = color e ( H ) I \gamma=\operatorname{color}_{e}(H)_I γ = color e ( H ) I δ = color e ( H ) J \delta=\operatorname{color}_{e}(H)_J δ = color e ( H ) J T = ( A B 0 0 A ) T=\begin{pmatrix} AB & 0 \\ 0 & A \end{pmatrix} T = ( A B 0 0 A ) f = [ T − 1 ] I ∪ J ⋅ e f=[T^{-1}]_{I\cup J}\cdot e f = [ T − 1 ] I ∪ J ⋅ e
A B ⋅ γ = color f ( H ) I , A ⋅ δ = color f ( H ) J . AB\cdot\gamma=\operatorname{color}_{f}(H)_I,
\qquad
A\cdot\delta=\operatorname{color}_{f}(H)_J. A B ⋅ γ = color f ( H ) I , A ⋅ δ = color f ( H ) J . It suffices to show that
[ T − 1 ] I ∪ J [ Ω ] I ∪ J [ T ] I ∪ J ∈ H ∗ , [T^{-1}]_{I\cup J}[\Omega]_{I\cup J}[T]_{I\cup J}\in H^*, [ T − 1 ] I ∪ J [ Ω ] I ∪ J [ T ] I ∪ J ∈ H ∗ , where the matrix Ω \Omega Ω
C = [ B − 1 ] I [ Ω ] I ∪ J [ B ] I . C=[B^{-1}]_I[\Omega]_{I\cup J}[B]_I. C = [ B − 1 ] I [ Ω ] I ∪ J [ B ] I . We have C ∈ H ∗ C\in H^* C ∈ H ∗ δ = B ⋅ γ \delta=B\cdot\gamma δ = B ⋅ γ
GL(F q \mathbb{F}_q F q
Given a non-trivial ergodic invariant random subgroup μ ∈ IRS ( SL ∞ ( F q ) ) \mu\in\operatorname{IRS}(\operatorname{SL}_{\infty}(\mathbb{F}_q)) μ ∈ IRS ( SL ∞ ( F q )) μ \mu μ H H H Θ H : GL n ( F q ) ↷ Col ( H ) n \Theta_H : \operatorname{GL}_n(\mathbb{F}_q) \curvearrowright\operatorname{Col}(H)^n Θ H : GL n ( F q ) ↷ Col ( H ) n n ∈ N n \in\mathbb{N} n ∈ N Θ H : GL ∞ ( F q ) ↷ Col ( H ) N \Theta_H : \operatorname{GL}_{\infty}(\mathbb{F}_q) \curvearrowright\operatorname{Col}(H)^{\mathbb{N}} Θ H : GL ∞ ( F q ) ↷ Col ( H ) N
Proposition 6.17. Let H H H μ \mu μ Θ H : GL ∞ ( F q ) ↷ Col ( H ) N \Theta_H : \operatorname{GL}_{\infty}(\mathbb{F}_q) \curvearrowright\operatorname{Col}(H)^{\mathbb{N}} Θ H : GL ∞ ( F q ) ↷ Col ( H ) N
The constant vector ρ = ( red ( H ) ) n ∈ N ∈ Col ( H ) N \rho= (\operatorname{red}(H))_{n\in\mathbb{N}} \in\operatorname{Col}(H)^{\mathbb{N}} ρ = ( red ( H ) ) n ∈ N ∈ Col ( H ) N
The action is permutational, in the sense of Theorem 3.1.
Proof. We first show statement (1). Fix n ∈ N n \in\mathbb{N} n ∈ N A ∈ GL n ( F q ) A \in\operatorname{GL}_n(\mathbb{F}_q) A ∈ GL n ( F q ) ρ n ∈ Col ( H ) n \rho_n \in\operatorname{Col}(H)^n ρ n ∈ Col ( H ) n ( ρ n ) i = red ( H ) (\rho_n)_i = \operatorname{red}(H) ( ρ n ) i = red ( H ) i ∈ { 1 , … , n } i \in\{1,\ldots,n\} i ∈ { 1 , … , n } I ⊂ N I \subset\mathbb{N} I ⊂ N ∣ I ∣ = n |I|=n ∣ I ∣ = n colore ( H ) I = ρ n \operatorname{colore}(H)_I=\rho_n colore ( H ) I = ρ n A ⋅ ρ n = color [ A − 1 ] I e ( H ) I A\cdot\rho_n=\operatorname{color}_{[A^{-1}]_I e}(H)_I A ⋅ ρ n = color [ A − 1 ] I e ( H ) I ρ n \rho_n ρ n SL I ( [ A − 1 ] I e ) ≤ H \operatorname{SL}_I([A^{-1}]_I e)\leq H SL I ([ A − 1 ] I e ) ≤ H [ A − 1 ] I SL I ( e ) [ A ] I = SL I ( e ) [A^{-1}]_I\operatorname{SL}_I(e)[A]_I=\operatorname{SL}_I(e) [ A − 1 ] I SL I ( e ) [ A ] I = SL I ( e ) H H H I I I
We now show statement (2). Let σ ∈ S ∞ \sigma\in S_{\infty} σ ∈ S ∞ P σ ∈ GL ∞ ( F q ) P_\sigma\in\operatorname{GL}_{\infty}(\mathbb{F}_q) P σ ∈ GL ∞ ( F q )
( P σ ⋅ γ ) i = γ σ ( i ) ∀ γ ∈ Col ( H ) N , i ∈ N . (P_\sigma\cdot\gamma)_i=\gamma_{\sigma(i)}\qquad\forall\gamma\in\operatorname{Col}(H)^{\mathbb{N}},\ i\in\mathbb{N}. ( P σ ⋅ γ ) i = γ σ ( i ) ∀ γ ∈ Col ( H ) N , i ∈ N . Fix some n ∈ N n\in\mathbb{N} n ∈ N σ ∈ S n \sigma\in S_n σ ∈ S n δ = P σ ⋅ γ \delta=P_\sigma\cdot\gamma δ = P σ ⋅ γ I , J ⊆ N I,J\subseteq\mathbb{N} I , J ⊆ N ∣ I ∣ = ∣ J ∣ = n |I|=|J|=n ∣ I ∣ = ∣ J ∣ = n max I < min J \max I<\min J max I < min J γ = colore ( H ) I \gamma=\operatorname{colore}(H)_I γ = colore ( H ) I δ = colore ( H ) J \delta=\operatorname{colore}(H)_J δ = colore ( H ) J δ = color f ( H ) J \delta=\operatorname{color}_f(H)_J δ = color f ( H ) J f = [ P σ − 1 ] I ⋅ e f=[P_\sigma^{-1}]_I\cdot e f = [ P σ − 1 ] I ⋅ e color f ( H ) I = color f ( H ) J \operatorname{color}_f(H)_I=\operatorname{color}_f(H)_J color f ( H ) I = color f ( H ) J Ω \Omega Ω
[ ( 0 P σ − 1 P σ 0 ) ] I ∪ J = [ P σ − 1 ] I [ Ω ] I ∪ J [ P σ ] I ∈ H ∗ . \left[\begin{pmatrix}0&P_\sigma^{-1}\\P_\sigma&0\end{pmatrix}\right]_{I\cup J}=[P_\sigma^{-1}]_I[\Omega]_{I\cup J}[P_\sigma]_I\in H^*. [ ( 0 P σ P σ − 1 0 ) ] I ∪ J = [ P σ − 1 ] I [ Ω ] I ∪ J [ P σ ] I ∈ H ∗ . This is a permutation matrix. Therefore Theorem 6.2 implies that δ k = γ σ ( k ) \delta_k=\gamma_{\sigma(k)} δ k = γ σ ( k ) □ \square □
We now wish to relate the action Θ H \Theta_H Θ H μ \mu μ H H H SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) g ∈ SL ∞ ( F q ) g\in\operatorname{SL}_{\infty}(\mathbb{F}_q) g ∈ SL ∞ ( F q ) g g g Col ( H ) \operatorname{Col}(H) Col ( H ) Col ( g H g − 1 ) \operatorname{Col}(gHg^{-1}) Col ( g H g − 1 ) Col ( H ) \operatorname{Col}(H) Col ( H ) Col ( g H g − 1 ) \operatorname{Col}(gHg^{-1}) Col ( g H g − 1 )
Proposition 6.18. Let g ∈ SL ∞ ( F q ) g\in\operatorname{SL}_{\infty}(\mathbb{F}_q) g ∈ SL ∞ ( F q ) μ \mu μ H H H
Θ H ( g ) color ( H ) = color ( g H g − 1 ) . \Theta_H(g)\operatorname{color}(H)=\operatorname{color}(gHg^{-1}). Θ H ( g ) color ( H ) = color ( g H g − 1 ) . Proof. Let n ∈ N n\in\mathbb{N} n ∈ N g ∈ SL n ( F q ) g\in\operatorname{SL}_n(\mathbb{F}_q) g ∈ SL n ( F q ) I = { 1 , … , n } I=\{1,\ldots,n\} I = { 1 , … , n } γ = color ( H ) I \gamma=\operatorname{color}(H)_I γ = color ( H ) I δ = color ( g H g − 1 ) I \delta=\operatorname{color}(gHg^{-1})_I δ = color ( g H g − 1 ) I J ⊆ N J\subseteq\mathbb{N} J ⊆ N min J > max I \min J>\max I min J > max I δ = color ( H ) J \delta=\operatorname{color}(H)_J δ = color ( H ) J Col ( H ) \operatorname{Col}(H) Col ( H ) Col ( g H g − 1 ) \operatorname{Col}(gHg^{-1}) Col ( g H g − 1 ) δ = color ( g H g − 1 ) J \delta=\operatorname{color}(gHg^{-1})_J δ = color ( g H g − 1 ) J
[ Ω ] I ∪ J ∈ ( g H g − 1 ) ∗ = g H ∗ g − 1 . [\Omega]_{I\cup J}\in(gHg^{-1})^*=gH^*g^{-1}. [ Ω ] I ∪ J ∈ ( g H g − 1 ) ∗ = g H ∗ g − 1 . This is equivalent to
g − 1 [ Ω ] I ∪ J g ∈ H ∗ , g^{-1}[\Omega]_{I\cup J}g\in H^*, g − 1 [ Ω ] I ∪ J g ∈ H ∗ , which shows that Θ H ( g ) γ = δ . □ \Theta_H(g)\gamma=\delta.\quad\square Θ H ( g ) γ = δ . □
Fixed points for the G L n ( F q ) \mathrm{GL}_n(\mathbb{F}_q) GL n ( F q )
There is one last property of the action Θ H \Theta_H Θ H
Proposition 6.19. The action Θ H \Theta_H Θ H μ \mu μ H H H Fix n ∈ N n \in\mathbb{N} n ∈ N A ∈ G L n ( F q ) A \in\mathrm{GL}_n(\mathbb{F}_q) A ∈ GL n ( F q ) 1 1 1 A A A rk A = n \operatorname{rk} A=n rk A = n A A A Col ( H ) n \operatorname{Col}(H)^n Col ( H ) n ρ n = ( red ( H ) ) i = 1 n \rho_n=(\operatorname{red}(H))_{i=1}^n ρ n = ( red ( H ) ) i = 1 n
Before proceeding with the proof of the above, we note that it is not apriori clear that the event described is measurable. We start by showing something stronger.
Claim 6.20. Fix n ∈ N n \in\mathbb{N} n ∈ N A ∈ G L n ( F q ) A \in\mathrm{GL}_n(\mathbb{F}_q) A ∈ GL n ( F q ) Φ n A : Sub ( S L ∞ ( F q ) ) → N \Phi_n^A:\operatorname{Sub}(\mathrm{SL}_\infty(\mathbb{F}_q))\to\mathbb{N} Φ n A : Sub ( SL ∞ ( F q )) → N μ \mu μ μ \mu μ H H H A A A Col ( H ) n \operatorname{Col}(H)^n Col ( H ) n Θ H \Theta_H Θ H Φ n A \Phi_n^A Φ n A T \mathcal{T} T T \mathcal{T} T σ \sigma σ μ \mu μ
The tail color σ \sigma σ ρ n \rho_n ρ n { H : Φ n A ( H ) = 1 } \{H:\Phi_n^A(H)=1\} { H : Φ n A ( H ) = 1 } Claim 6.20 .
Proof of Claim 6.20. For each m ∈ N m\in\mathbb{N} m ∈ N Φ n A \Phi_n^A Φ n A T m \mathcal{T}_m T m I , J ⊆ N > m I,J\subseteq\mathbb{N}_{>m} I , J ⊆ N > m ∣ I ∣ = ∣ J ∣ = n |I|=|J|=n ∣ I ∣ = ∣ J ∣ = n max I < min J \max I<\min J max I < min J B ∈ G L n ( F q ) B\in\mathrm{GL}_n(\mathbb{F}_q) B ∈ GL n ( F q )
E I , J B = { H : Θ H ( B ) color e ( H ) I = color e ( H ) J } . E_{I,J}^{B}=\{H:\Theta_H(B)\operatorname{color}_{e}(H)_I=\operatorname{color}_{e}(H)_J\}. E I , J B = { H : Θ H ( B ) color e ( H ) I = color e ( H ) J } . Making use of Theorem 6.16 and the notations therein, this event can be written as
E I , J B = { H : [ B − 1 ] I [ Ω ] I ∪ J [ B ] I ∈ H ∗ } . E_{I,J}^{B}=\{H:[B^{-1}]_I[\Omega]_{I\cup J}[B]_I\in H^*\}. E I , J B = { H : [ B − 1 ] I [ Ω ] I ∪ J [ B ] I ∈ H ∗ } . This equation and statement (3) of Theorem 6.16 show that for any such pair I I I J J J E I , J B E_{I,J}^{B} E I , J B T m \mathcal{T}_m T m I n I_n I n B B B
C I , J = { H : color e ( H ) I = color e ( H ) J } = E I , J I n ∈ T m , C_{I,J}=\{H:\operatorname{color}_{e}(H)_I=\operatorname{color}_{e}(H)_J\}=E_{I,J}^{I_n}\in\mathcal{T}_m, C I , J = { H : color e ( H ) I = color e ( H ) J } = E I , J I n ∈ T m , as well as the event
F I A = { H : A ⋅ color e ( H ) I = color e ( H ) I } . F_I^A=\{H:A\cdot\operatorname{color}_{e}(H)_I=\operatorname{color}_{e}(H)_I\}. F I A = { H : A ⋅ color e ( H ) I = color e ( H ) I } . The latter event F I A F_I^A F I A T m \mathcal{T}_m T m
F I A = { H : ∃ J ⊆ N > m , ∣ J ∣ = n , min J > max I , H ∈ E I , J A ∩ C I , J } ∈ T m . F_I^A=\{H:\exists J\subseteq\mathbb{N}_{>m},|J|=n,\min J>\max I,H\in E_{I,J}^{A}\cap C_{I,J}\}\in\mathcal{T}_m. F I A = { H : ∃ J ⊆ N > m , ∣ J ∣ = n , min J > max I , H ∈ E I , J A ∩ C I , J } ∈ T m . Finally, the event { H : Φ n A ≥ k } \{H:\Phi_n^A\geq k\} { H : Φ n A ≥ k } H H H I 1 , … , I k ⊆ N > m I_1,\ldots,I_k\subseteq\mathbb{N}_{>m} I 1 , … , I k ⊆ N > m n n n max I s < min I s + 1 \max I_s<\min I_{s+1} max I s < min I s + 1 1 ≤ s < k 1\leq s<k 1 ≤ s < k H ∈ F I s A H\in F_{I_s}^A H ∈ F I s A 1 ≤ s ≤ k 1\leq s\leq k 1 ≤ s ≤ k H ∉ C I s , I t H\notin C_{I_s,I_t} H ∈ / C I s , I t Φ n A \Phi_n^A Φ n A T m \mathcal{T}_m T m m ∈ N m\in\mathbb{N} m ∈ N Φ n A \Phi_n^A Φ n A T \mathcal{T} T □ \square □
Proof of Theorem 6.19. Fix n ∈ N n\in\mathbb{N} n ∈ N A ∈ G L n ( F q ) A\in\mathrm{GL}_n(\mathbb{F}_q) A ∈ GL n ( F q ) H ∈ Sub ( S L ∞ ( F q ) ) H\in\operatorname{Sub}(\mathrm{SL}_\infty(\mathbb{F}_q)) H ∈ Sub ( SL ∞ ( F q )) Claim 6.20 , the cardinality of the set Fix n ( A ) \operatorname{Fix}_n(A) Fix n ( A ) A A A Col ( H ) n \operatorname{Col}(H)^n Col ( H ) n Θ H \Theta_H Θ H μ \mu μ 2 2 2 A A A 1 1 1
Suppose γ ∈ Col ( H ) n \gamma\in\operatorname{Col}(H)^n γ ∈ Col ( H ) n A A A γ = color e ( H ) { 1 , … , n } = color e ( H ) { n + 1 , … , 2 n } \gamma=\operatorname{color}_{e}(H)_{\{1,\ldots,n\}}=\operatorname{color}_{e}(H)_{\{n+1,\ldots,2n\}} γ = color e ( H ) { 1 , … , n } = color e ( H ) { n + 1 , … , 2 n }
[ Ω ] { 1 , … , 2 n } ∈ H ∗ and [ A − 1 ] { 1 , … , n } [ Ω ] { 1 , … , 2 n } [ A ] { 1 , … , n } ∈ H ∗ . [\Omega]_{\{1,\ldots,2n\}}\in H^*\quad\text{and}\quad[A^{-1}]_{\{1,\ldots,n\}}[\Omega]_{\{1,\ldots,2n\}}[A]_{\{1,\ldots,n\}}\in H^*. [ Ω ] { 1 , … , 2 n } ∈ H ∗ and [ A − 1 ] { 1 , … , n } [ Ω ] { 1 , … , 2 n } [ A ] { 1 , … , n } ∈ H ∗ . Denote by B B B
B = [ ( A − 1 0 0 A ) ] { 1 , … , 2 n } . B=\left[\begin{pmatrix} A^{-1} & 0 \\ 0 & A \end{pmatrix}\right]_{\{1,\ldots,2n\}}. B = [ ( A − 1 0 0 A ) ] { 1 , … , 2 n } . We have B ∈ H B\in H B ∈ H B ∈ H ∗ B\in H^* B ∈ H ∗ 2 n < i < j 2n<i<j 2 n < i < j H H H ω \omega ω { i , j } \{i,j\} { i , j } ω 2 = I 2 \omega^2=I_2 ω 2 = I 2
The probability that B ∈ H B\in H B ∈ H q − r ⋅ rk ~ B = q − 2 r ⋅ rk ~ A q^{-r\cdot\widetilde{\operatorname{rk}}B}=q^{-2r\cdot\widetilde{\operatorname{rk}}A} q − r ⋅ rk B = q − 2 r ⋅ rk A r = r ( μ ) r=r(\mu) r = r ( μ ) μ \mu μ δ = colore ( H ) { 1 , … , n } = colore ( H ) { n + 1 , … , 2 n } \delta=\operatorname{colore}(H)_{\{1,\ldots,n\}}=\operatorname{colore}(H)_{\{n+1,\ldots,2n\}} δ = colore ( H ) { 1 , … , n } = colore ( H ) { n + 1 , … , 2 n } δ ∈ Fix n ( A ) \delta\in\operatorname{Fix}_n(A) δ ∈ Fix n ( A ) ∣ Fix n ( A ) ∣ q − 2 r n |\operatorname{Fix}_n(A)|q^{-2rn} ∣ Fix n ( A ) ∣ q − 2 r n § 6.3 .
We convert the coloring partition Col ( H ) \operatorname{Col}(H) Col ( H )
N [ n ] = { ( n ( m − 1 ) + 1 , … , n m ) : m ∈ N } . \mathbb{N}^{[n]}=\{(n(m-1)+1,\ldots,nm):m\in\mathbb{N}\}. N [ n ] = {( n ( m − 1 ) + 1 , … , nm ) : m ∈ N } . We define an equivalence relation L ( H ) L(H) L ( H ) N [ n ] \mathbb{N}^{[n]} N [ n ] ( b 1 , b 2 ) ∈ L ( H ) (b_1,b_2)\in L(H) ( b 1 , b 2 ) ∈ L ( H )
colore ( H ) b 1 = colore ( H ) b 2 . \operatorname{colore}(H)_{b_1}=\operatorname{colore}(H)_{b_2}. colore ( H ) b 1 = colore ( H ) b 2 . Each tuple of colors γ ∈ Col ( H ) n \gamma\in\operatorname{Col}(H)^n γ ∈ Col ( H ) n L ( H ) L(H) L ( H ) c γ c_\gamma c γ L ( H ) L(H) L ( H ) N [ n ] \mathbb{N}^{[n]} N [ n ] P A ( H ) P^A(H) P A ( H ) § 4.7 ). The blocks of P A ( H ) P^A(H) P A ( H )
P A ( H ) = { c γ : γ ∈ Fix n ( A ) } ∪ { N [ n ] ∖ ⋃ γ ∈ Fix n ( A ) c γ } , P^A(H)=\left\{c_\gamma:\gamma\in\operatorname{Fix}_n(A)\right\}\cup\left\{\mathbb{N}^{[n]}\setminus\bigcup_{\gamma\in\operatorname{Fix}_n(A)}c_\gamma\right\}, P A ( H ) = { c γ : γ ∈ Fix n ( A ) } ∪ ⎩ ⎨ ⎧ N [ n ] ∖ γ ∈ Fix n ( A ) ⋃ c γ ⎭ ⎬ ⎫ , with the distinguished block being N [ n ] ∖ ⋃ γ ∈ Fix n ( A ) c γ \mathbb{N}^{[n]}\setminus\bigcup_{\gamma\in\operatorname{Fix}_n(A)}c_\gamma N [ n ] ∖ ⋃ γ ∈ Fix n ( A ) c γ A ∞ ( N [ n ] ) A_\infty(\mathbb{N}^{[n]}) A ∞ ( N [ n ] ) N [ n ] \mathbb{N}^{[n]} N [ n ] SL ∞ ( F q ) \operatorname{SL}_\infty(\mathbb{F}_q) SL ∞ ( F q ) n n n P A : Sub ( SL ∞ ( F q ) ) → E ∗ ( N [ n ] ) P^A:\operatorname{Sub}(\operatorname{SL}_\infty(\mathbb{F}_q))\to E^*(\mathbb{N}^{[n]}) P A : Sub ( SL ∞ ( F q )) → E ∗ ( N [ n ] ) μ \mu μ A ∞ ( N [ n ] ) A_\infty(\mathbb{N}^{[n]}) A ∞ ( N [ n ] ) E ∗ ( N [ n ] ) E^*(\mathbb{N}^{[n]}) E ∗ ( N [ n ] ) N [ n ] \mathbb{N}^{[n]} N [ n ]
Set λ = P ∗ A μ \lambda=P^A_*\mu λ = P ∗ A μ λ \lambda λ A ∞ ( N [ n ] ) A_\infty(\mathbb{N}^{[n]}) A ∞ ( N [ n ] ) α r ∈ D \alpha^r\in\mathcal{D} α r ∈ D q r q^r q r ( R e ) ∗ μ = λ α r (R_e)_*\mu=\lambda\alpha^r ( R e ) ∗ μ = λ α r μ \mu μ γ ∈ Col ( H ) n \gamma\in\operatorname{Col}(H)^n γ ∈ Col ( H ) n q − r n q^{-rn} q − r n N = ∣ Fix n ( A ) ∣ N=|\operatorname{Fix}_n(A)| N = ∣ Fix n ( A ) ∣ μ \mu μ Claim 6.20 . We group the N N N Fix n ( A ) \operatorname{Fix}_n(A) Fix n ( A ) Fix n ( A ) \operatorname{Fix}_n(A) Fix n ( A ) q − r n q^{-rn} q − r n 1 − N q − r n 1-Nq^{-rn} 1 − N q − r n D ( P A ( H ) ) D(P^A(H)) D ( P A ( H )) μ \mu μ α = ( α m ) m ∈ Z ≥ − 1 \alpha=(\alpha_m)_{m\in\mathbb{Z}_{\geq-1}} α = ( α m ) m ∈ Z ≥ − 1
α m = { q − r n , 1 ≤ m ≤ ∣ Fix n ( A ) ∣ , 1 − ∣ Fix n ( A ) ∣ q − r n , m = − 1 , 0 , otherwise . \alpha_m=
\begin{cases}
q^{-rn}, & 1\leq m\leq|\operatorname{Fix}_n(A)|,\\
1-|\operatorname{Fix}_n(A)|q^{-rn}, & m=-1,\\
0, & \text{otherwise}.
\end{cases} α m = ⎩ ⎨ ⎧ q − r n , 1 − ∣ Fix n ( A ) ∣ q − r n , 0 , 1 ≤ m ≤ ∣ Fix n ( A ) ∣ , m = − 1 , otherwise . By Theorem 4.11 we deduce that λ = λ α ∗ \lambda=\lambda_{\alpha}^{*} λ = λ α ∗ λ \lambda λ P ∗ A μ = λ α ∗ P_{*}^{A}\mu=\lambda_{\alpha}^{*} P ∗ A μ = λ α ∗
q − r ⋅ rk ~ B = μ ( { H : B ∈ H } ) ≥ μ ( { H : color e ( H ) { 1 , … , n } = color e ( H ) { n + 1 , … , 2 n } = δ for some δ ∈ Fix n ( A ) } ) = λ α ∗ ( { H : ( 1 , … , n ) , ( n + 1 , … , 2 n ) ∈ N [ n ] are in the same non-distinguished block of P A ( H ) } ) = ∣ Fix n ( A ) ∣ ⋅ q − 2 r n . \begin{aligned}
q^{-r\cdot\widetilde{\operatorname{rk}}B}=\mu(\{H:B\in H\})\geq\mu\left(\left\{H:\begin{array}{c}\operatorname{color}_{e}(H)_{\{1,\ldots,n\}}=\operatorname{color}_{e}(H)_{\{n+1,\ldots,2n\}}=\delta\\ \text{for some }\delta\in\operatorname{Fix}_{n}(A)\end{array}\right\}\right) \\
&=\lambda_{\alpha}^{*}\left(\left\{H:\begin{array}{c}(1,\ldots,n),(n+1,\ldots,2n)\in\mathbb{N}^{[n]}\text{ are in}\\ \text{the same non-distinguished block of }P^{A}(H)\end{array}\right\}\right)=|\operatorname{Fix}_{n}(A)|\cdot q^{-2rn}.
\end{aligned} q − r ⋅ rk B = μ ({ H : B ∈ H }) ≥ μ ( { H : color e ( H ) { 1 , … , n } = color e ( H ) { n + 1 , … , 2 n } = δ for some δ ∈ Fix n ( A ) } ) = λ α ∗ ( { H : ( 1 , … , n ) , ( n + 1 , … , 2 n ) ∈ N [ n ] are in the same non-distinguished block of P A ( H ) } ) = ∣ Fix n ( A ) ∣ ⋅ q − 2 r n . Since rk ~ B = 2 rk ~ A \widetilde{\operatorname{rk}}B=2\widetilde{\operatorname{rk}}A rk B = 2 rk A ∣ Fix n ( A ) ∣ ≤ q 2 r ( n − rk ~ A ) |\operatorname{Fix}_{n}(A)|\leq q^{2r(n-\widetilde{\operatorname{rk}}A)} ∣ Fix n ( A ) ∣ ≤ q 2 r ( n − rk A ) ∣ Fix n ( A ) ∣ > 1 |\operatorname{Fix}_{n}(A)|>1 ∣ Fix n ( A ) ∣ > 1 rk ~ A < n \widetilde{\operatorname{rk}}A<n rk A < n 1 1 1 A A A □ \square □
Summary
Given a non-trivial ergodic invariant random subgroup μ \mu μ SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_{q}) SL ∞ ( F q ) μ \mu μ H H H GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_{q}) GL ∞ ( F q ) Col ( H ) N \operatorname{Col}(H)^{\mathbb{N}} Col ( H ) N Col ( H ) \operatorname{Col}(H) Col ( H ) H H H Col ( H ) \operatorname{Col}(H) Col ( H ) q r q^{r} q r r = r ( μ ) r=r(\mu) r = r ( μ ) μ \mu μ GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_{q}) GL ∞ ( F q ) GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_{q}) GL ∞ ( F q ) § 8 .
Invariant measures on compact abelian groups
Fix a pair k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_{0} k , ℓ ∈ N 0 § 2 that V k , ℓ V^{k,\ell} V k , ℓ V k × V ^ ℓ V^{k}\times\widehat{V}^{\ell} V k × V ℓ GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_{q}) GL ∞ ( F q ) X k , ℓ ⊆ V k , ℓ X^{k,\ell}\subseteq V^{k,\ell} X k , ℓ ⊆ V k , ℓ ( v 1 , … , v k ) , ( φ 1 , … , φ ℓ ) ∈ V k × V ^ ℓ (v^{1},\ldots,v^{k}),(\varphi^{1},\ldots,\varphi^{\ell})\in V^{k}\times\widehat{V}^{\ell} ( v 1 , … , v k ) , ( φ 1 , … , φ ℓ ) ∈ V k × V ℓ { v 1 , … , v k } \{v^{1},\ldots,v^{k}\} { v 1 , … , v k } V V V { φ 1 , … , φ ℓ } \{\varphi^{1},\ldots,\varphi^{\ell}\} { φ 1 , … , φ ℓ } V ^ \widehat{V} V X k , ℓ X^{k,\ell} X k , ℓ GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_{q}) GL ∞ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ Haar k , ℓ \operatorname{Haar}^{k,\ell} Haar k , ℓ GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_{q}) GL ∞ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ X k , ℓ X^{k,\ell} X k , ℓ
Recall that a Borel action of a group G G G uniquely ergodic if it admits exactly one G G G SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_{q}) SL ∞ ( F q ) X k , ℓ X^{k,\ell} X k , ℓ Haar k , ℓ \operatorname{Haar}^{k,\ell} Haar k , ℓ SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_{q}) SL ∞ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ F q \mathbb{F}_{q} F q § 7.5 ).
Ergodic theory and dynamics
We employ the pointwise ergodic theorem for amenable groups. The most general case was established by Lindenstrauss in [21 ] . However, our specific setting involves the locally finite group G = SL ∞ ( F q ) G=\operatorname{SL}_{\infty}(\mathbb{F}_{q}) G = SL ∞ ( F q ) G = ⋃ n = 1 ∞ G n G=\bigcup_{n=1}^{\infty}G_{n} G = ⋃ n = 1 ∞ G n G n = SL n ( F q ) G_{n}=\operatorname{SL}_{n}(\mathbb{F}_{q}) G n = SL n ( F q ) [34 ] . The version of the theorem that we will use states the following.
Theorem 7.1. Let G G G G = ⋃ n = 1 ∞ G n G=\bigcup_{n=1}^{\infty}G_{n} G = ⋃ n = 1 ∞ G n G G G ( X , μ ) (X,\mu) ( X , μ ) E ⊆ X E\subseteq X E ⊆ X μ \mu μ x ∈ X x\in X x ∈ X
μ ( E ) = lim n → ∞ 1 ∣ G n ∣ ∣ { g ∈ G n : g ⋅ x ∈ E } ∣ . \mu(E)=\lim_{n\to\infty}\frac{1}{|G_{n}|}|\{g\in G_{n}:g\cdot x\in E\}|. μ ( E ) = n → ∞ lim ∣ G n ∣ 1 ∣ { g ∈ G n : g ⋅ x ∈ E } ∣. We use this theorem to reduce the question of classifying invariant measures to that of counting matrices with specific properties. In § 7.2 , we use the ergodic theorem to show that SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_{q}) SL ∞ ( F q ) X k , 0 X^{k,0} X k , 0 X 0 , ℓ X^{0,\ell} X 0 , ℓ § 7.1 we reduce the problem of showing that SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_{q}) SL ∞ ( F q ) X k , ℓ = X k , 0 × X 0 , ℓ X^{k,\ell}=X^{k,0}\times X^{0,\ell} X k , ℓ = X k , 0 × X 0 , ℓ Haar 0 , ℓ \operatorname{Haar}^{0,\ell} Haar 0 , ℓ φ ∈ X 0 , ℓ \varphi\in X^{0,\ell} φ ∈ X 0 , ℓ Stab SL ∞ ( F q ) ( φ ) \operatorname{Stab}_{\operatorname{SL}_{\infty}(\mathbb{F}_{q})}(\varphi) Stab SL ∞ ( F q ) ( φ ) X k , 0 X^{k,0} X k , 0 § 7.5 , we classify the ergodic measures on V k , ℓ V^{k,\ell} V k , ℓ Gr k ( V ) × Gr ℓ ( V ^ ) \operatorname{Gr}_k(V) \times\operatorname{Gr}_{\ell}(\widehat{V}) Gr k ( V ) × Gr ℓ ( V )
Joins of measured systems
We briefly recall some notions in the ergodic theory of dynamical systems. A reference for these definitions is [12 Chapter 6] .
Let G G G ( X , μ ) (X,\mu) ( X , μ ) ( Y , ν ) (Y,\nu) ( Y , ν ) ( X , μ ) (X,\mu) ( X , μ ) ( Y , ν ) (Y,\nu) ( Y , ν ) G G G λ \lambda λ X × Y X \times Y X × Y μ \mu μ ν \nu ν proj X ( λ ) = μ \operatorname{proj}_X(\lambda) = \mu proj X ( λ ) = μ proj Y ( λ ) = ν \operatorname{proj}_Y(\lambda) = \nu proj Y ( λ ) = ν ( X , μ ) (X,\mu) ( X , μ ) ( Y , ν ) (Y,\nu) ( Y , ν ) μ × ν \mu\times\nu μ × ν
Lemma 7.2. Let G G G ( X , μ ) (X,\mu) ( X , μ ) ( Y , ν ) (Y,\nu) ( Y , ν ) μ \mu μ x ∈ X x \in X x ∈ X Stab G ( x ) \operatorname{Stab}_G(x) Stab G ( x ) Y Y Y ( X , μ ) (X,\mu) ( X , μ ) ( Y , ν ) (Y,\nu) ( Y , ν )
Proof. Let λ \lambda λ ( X , μ ) (X,\mu) ( X , μ ) ( Y , ν ) (Y,\nu) ( Y , ν ) G x = Stab G ( x ) G_x = \operatorname{Stab}_G(x) G x = Stab G ( x ) x ∈ X x \in X x ∈ X λ \lambda λ proj X \operatorname{proj}_X proj X { λ x } x ∈ X \{\lambda_x\}_{x\in X} { λ x } x ∈ X Y Y Y
λ = ∫ X ( δ x × λ x ) d μ ( x ) . \lambda= \int_X (\delta_x \times\lambda_x)\,d\mu(x). λ = ∫ X ( δ x × λ x ) d μ ( x ) . We claim that μ \mu μ λ x \lambda_x λ x G x G_x G x g ∈ G g \in G g ∈ G
λ = g ∗ λ = ∫ X ( δ g x × g ∗ λ x ) d μ ( x ) . \lambda= g_*\lambda= \int_X (\delta_{gx} \times g_*\lambda_x)\,d\mu(x). λ = g ∗ λ = ∫ X ( δ g x × g ∗ λ x ) d μ ( x ) . By the uniqueness of the disintegration, we have g ∗ λ x = λ g x g_*\lambda_x = \lambda_{gx} g ∗ λ x = λ g x μ \mu μ x ∈ X x \in X x ∈ X G G G μ \mu μ x ∈ X x \in X x ∈ X g ∈ G g \in G g ∈ G g ∗ λ x = λ g x g_*\lambda_x = \lambda_{gx} g ∗ λ x = λ g x x ∈ X x \in X x ∈ X λ x \lambda_x λ x G x G_x G x G x ↷ Y G_x \curvearrowright Y G x ↷ Y μ \mu μ ν \nu ν G x G_x G x G G G G x G_x G x Y Y Y ν \nu ν λ x = ν \lambda_x = \nu λ x = ν μ \mu μ λ = μ × ν \lambda= \mu\times\nu λ = μ × ν □ \square □
Invariant measures on X k , 0 X^{k,0} X k , 0
Fix k ∈ N k \in\mathbb{N} k ∈ N ℓ \ell ℓ SL ∞ ( F q ) ↷ X k , 0 \operatorname{SL}_{\infty}(\mathbb{F}_q) \curvearrowright X^{k,0} SL ∞ ( F q ) ↷ X k , 0 SL ∞ ( F q ) ↷ X 0 , k \operatorname{SL}_{\infty}(\mathbb{F}_q) \curvearrowright X^{0,k} SL ∞ ( F q ) ↷ X 0 , k V k V^k V k V ^ k \widehat{V}^k V k g ↦ ( g − 1 ) t g \mapsto(g^{-1})^t g ↦ ( g − 1 ) t SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) X k , 0 X^{k,0} X k , 0
Given m ∈ N m \in\mathbb{N} m ∈ N a 1 , … , a k ∈ V m a^1,\ldots,a^k \in V_m a 1 , … , a k ∈ V m
[ a 1 , … , a k ] = { v ∈ X k , 0 : v ≤ m i = a i ∀ 1 ≤ i ≤ k } . [a^1,\ldots,a^k] = \{v \in X^{k,0}: v_{\leq m}^i = a^i \ \forall1 \leq i \leq k\}. [ a 1 , … , a k ] = { v ∈ X k , 0 : v ≤ m i = a i ∀1 ≤ i ≤ k } . We refer to a set of this form as a box and to the integer m m m π \pi π σ \sigma σ X k , 0 X^{k,0} X k , 0
A straightforward application of the pointwise ergodic theorem (Theorem 7.1 ) allows us to classify the ergodic invariant measures on X k , 0 X^{k,0} X k , 0
Proposition 7.3. The only SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) X k , 0 X^{k,0} X k , 0 X 0 , ℓ X^{0,\ell} X 0 , ℓ
Remark 7.4. In the next sections, we will show the stronger statement that an action of some specific subgroup of SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) X k , 0 X^{k,0} X k , 0
Proof of Theorem 7.3. Let μ \mu μ SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) X k , 0 X^{k,0} X k , 0 μ \mu μ E = [ a 1 , … , a k ] E=[a^1,\ldots,a^k] E = [ a 1 , … , a k ] m m m a i ∈ V m a^i\in V_m a i ∈ V m v = ( v 1 , … , v k ) ∈ X k , 0 v=(v^1,\ldots,v^k)\in X^{k,0} v = ( v 1 , … , v k ) ∈ X k , 0
μ ( E ) = lim n → ∞ ∣ { g ∈ SL n ( F q ) : ( g v 1 , … , g v k ) ∈ E } ∣ ∣ SL n ( F q ) ∣ . \mu(E)=\lim_{n\to\infty}\frac{\left|\left\{g\in\operatorname{SL}_n(\mathbb{F}_q):(gv^1,\ldots,gv^k)\in E\right\}\right|}{|\operatorname{SL}_n(\mathbb{F}_q)|}. μ ( E ) = n → ∞ lim ∣ SL n ( F q ) ∣ { g ∈ SL n ( F q ) : ( g v 1 , … , g v k ) ∈ E } . as per Theorem 7.1 . Let n > m + k n>m+k n > m + k { v ≤ n 1 , … , v ≤ n k } \{v^1_{\leq n},\ldots,v^k_{\leq n}\} { v ≤ n 1 , … , v ≤ n k } g ∈ SL n ( F q ) g\in\operatorname{SL}_n(\mathbb{F}_q) g ∈ SL n ( F q ) ( g v i ) ∈ E (gv^i)\in E ( g v i ) ∈ E ( g v i ) ≤ m = a i (gv^i)_{\leq m}=a^i ( g v i ) ≤ m = a i 1 ≤ i ≤ k 1\leq i\leq k 1 ≤ i ≤ k v ≤ n 1 , … , v ≤ n k v^1_{\leq n},\ldots,v^k_{\leq n} v ≤ n 1 , … , v ≤ n k f = { v ≤ n 1 , … , v ≤ n k , u k + 1 , … , u n } f=\{v^1_{\leq n},\ldots,v^k_{\leq n},u^{k+1},\ldots,u^n\} f = { v ≤ n 1 , … , v ≤ n k , u k + 1 , … , u n } V n V_n V n g g g f f f g ∈ SL n ( F q ) g\in\operatorname{SL}_n(\mathbb{F}_q) g ∈ SL n ( F q ) g j i = a j i g_{ji}=a^i_j g j i = a j i 1 ≤ i ≤ m 1\leq i\leq m 1 ≤ i ≤ m 1 ≤ j ≤ k 1\leq j\leq k 1 ≤ j ≤ k
This is a standard counting argument. We proceed as follows. Count the number of possibilities for each column of the matrix g g g a 1 a^1 a 1 q n − m − 1 q^{n-m}-1 q n − m − 1 q n − m q^{n-m} q n − m a 1 a^1 a 1 a 2 a^2 a 2 q n − m q^{n-m} q n − m q q q a 1 a^1 a 1 a 2 a^2 a 2 i ≤ k i\leq k i ≤ k i i i g g g q n − m − β i q^{n-m}-\beta_i q n − m − β i β i ≤ q i − 1 \beta_i\leq q^{i-1} β i ≤ q i − 1 n − m > k n-m>k n − m > k k k k k k k i i i k < i < n k<i<n k < i < n i i i q n − q i − 1 q^n-q^{i-1} q n − q i − 1 det g = 1 \det g=1 det g = 1 1 q − 1 ( q n − q n − 1 ) \frac{1}{q-1}(q^n-q^{n-1}) q − 1 1 ( q n − q n − 1 ) Theorem 7.1 :
μ ( E ) = lim n → ∞ 1 q − 1 ( q n − m − β 1 ) ⋯ ( q n − m − β k ) ( q n − q k ) ⋯ ( q n − q n − 1 ) 1 q − 1 ( q n − 1 ) ⋯ ( q n − q k − 1 ) ( q n − q k ) ⋯ ( q n − q n − 1 ) = lim n → ∞ q − m k ( 1 − β 1 q m − n 1 − q − n ) ⋯ ( 1 − β k q m − n 1 − q k − 1 − n ) = q − m k = Haar k , 0 ( E ) . \begin{aligned}
\mu(E)=\lim_{n\to\infty}\frac{\frac{1}{q-1}(q^{n-m}-\beta_1)\cdots(q^{n-m}-\beta_k)(q^n-q^k)\cdots(q^n-q^{n-1})}{\frac{1}{q-1}(q^n-1)\cdots(q^n-q^{k-1})(q^n-q^k)\cdots(q^n-q^{n-1})} \\
&=\lim_{n\to\infty}q^{-mk}\left(\frac{1-\beta_1q^{m-n}}{1-q^{-n}}\right)\cdots\left(\frac{1-\beta_kq^{m-n}}{1-q^{k-1-n}}\right)=q^{-mk}=\operatorname{Haar}^{k,0}(E).
\end{aligned} μ ( E ) = n → ∞ lim q − 1 1 ( q n − 1 ) ⋯ ( q n − q k − 1 ) ( q n − q k ) ⋯ ( q n − q n − 1 ) q − 1 1 ( q n − m − β 1 ) ⋯ ( q n − m − β k ) ( q n − q k ) ⋯ ( q n − q n − 1 ) = n → ∞ lim q − mk ( 1 − q − n 1 − β 1 q m − n ) ⋯ ( 1 − q k − 1 − n 1 − β k q m − n ) = q − mk = Haar k , 0 ( E ) . Since the boxes comprise a π \pi π μ = Haar k , 0 \mu=\operatorname{Haar}^{k,0} μ = Haar k , 0 X 0 , k X^{0,k} X 0 , k X k , 0 X^{k,0} X k , 0 g ↦ g − t g\mapsto g^{-t} g ↦ g − t SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
Regular linear independence
For each vector v ∈ V v\in V v ∈ V m , t ∈ N m,t\in\mathbb{N} m , t ∈ N v ( m , m + t ] v_{(m,m+t]} v ( m , m + t ] ( v i ) i = m + 1 m + t ∈ F q t (v_i)_{i=m+1}^{m+t}\in\mathbb{F}_q^t ( v i ) i = m + 1 m + t ∈ F q t V ^ \widehat{V} V V V V V ^ \widehat{V} V
Definition 7.5. Fix r ∈ N r\in\mathbb{N} r ∈ N v ∈ V r v\in V^r v ∈ V r m ∈ N m\in\mathbb{N} m ∈ N t ∈ N t\in\mathbb{N} t ∈ N
v ( m , m + t ] 1 , … , v ( m , m + t ] r ∈ F q t v^1_{(m,m+t]},\ldots,v^r_{(m,m+t]}\in\mathbb{F}_q^t v ( m , m + t ] 1 , … , v ( m , m + t ] r ∈ F q t is linearly independent is called the m m m v v v I m ( v ) I_m(v) I m ( v ) t t t I m ( v ) = ∞ I_m(v)=\infty I m ( v ) = ∞ T ∈ N T\in\mathbb{N} T ∈ N
lim inf m → ∞ I m ( v ) = T , \liminf_{m\to\infty}I_m(v)=T, m → ∞ lim inf I m ( v ) = T , then v v v T T T T T T v v v T T T v v v
Notice that a generic element of X 0 , ℓ X^{0,\ell} X 0 , ℓ Haar 0 , ℓ \operatorname{Haar}^{0,\ell} Haar 0 , ℓ ℓ \ell ℓ k ≤ ℓ k \le\ell k ≤ ℓ φ ∈ X 0 , ℓ \varphi\in X^{0,\ell} φ ∈ X 0 , ℓ X k , 0 X^{k,0} X k , 0 Haar k , 0 \operatorname{Haar}^{k,0} Haar k , 0
Lemma 7.6. Let φ 1 , … , φ r ∈ V ^ n \varphi^{1},\ldots,\varphi^{r} \in\widehat{V}_{n} φ 1 , … , φ r ∈ V n 1 ≤ m ≤ n 1 \le m \le n 1 ≤ m ≤ n φ > m 1 , … , φ > m r \varphi^{1}_{>m},\ldots,\varphi^{r}_{>m} φ > m 1 , … , φ > m r
Π m n : V n → V m , v ↦ v ≤ m \Pi^{n}_{m}: V_{n} \to V_{m}, \qquad v \mapsto v_{\le m} Π m n : V n → V m , v ↦ v ≤ m to the joint kernel { φ 1 , … , φ r } ⊥ ⊆ V n \{\varphi^{1},\ldots,\varphi^{r}\}^{\perp} \subseteq V_{n} { φ 1 , … , φ r } ⊥ ⊆ V n
Proof. Follows from basic linear algebra. □ \square □
Lemma 7.7. Fix k , ℓ ∈ N k,\ell\in\mathbb{N} k , ℓ ∈ N φ ∈ X 0 , ℓ \varphi\in X^{0,\ell} φ ∈ X 0 , ℓ m ∈ N m \in\mathbb{N} m ∈ N φ > m 1 , … , φ > m ℓ \varphi^{1}_{>m},\ldots,\varphi^{\ell}_{>m} φ > m 1 , … , φ > m ℓ
K = max { Im ( φ ) , ℓ + 2 k } . K = \max\{ \operatorname{Im}(\varphi), \ell+ 2k \}. K = max { Im ( φ ) , ℓ + 2 k } . Then for any n > m + K n > m + K n > m + K v ∈ V n k v \in V_{n}^{k} v ∈ V n k w ∈ V m k w \in V_{m}^{k} w ∈ V m k g ∈ Stab S L n ( F q ) ( φ ) g \in\operatorname{Stab}_{\mathrm{SL}_{n}(\mathbb{F}_{q})}(\varphi) g ∈ Stab SL n ( F q ) ( φ ) ( g v ) ≤ m = w (gv)_{\le m} = w ( g v ) ≤ m = w
Proof. Fix some n > m + K n > m + K n > m + K v ∈ V n k v \in V_{n}^{k} v ∈ V n k w ∈ V m k w \in V_{m}^{k} w ∈ V m k g ∈ Stab S L n ( F q ) ( φ ) g \in\operatorname{Stab}_{\mathrm{SL}_{n}(\mathbb{F}_{q})}(\varphi) g ∈ Stab SL n ( F q ) ( φ ) ( g v ) ≤ m = w (gv)_{\le m} = w ( g v ) ≤ m = w
For each i = 1 , … , k i = 1,\ldots,k i = 1 , … , k a i = v ≤ m i − w i ∈ V m a^{i} = v^{i}_{\le m} - w^{i} \in V_{m} a i = v ≤ m i − w i ∈ V m a i a^{i} a i a ~ i \widetilde{a}^{i} a i { φ 1 , … , φ ℓ } ⊥ ⊆ V n \{\varphi^{1},\ldots,\varphi^{\ell}\}^{\perp} \subseteq V_{n} { φ 1 , … , φ ℓ } ⊥ ⊆ V n Π m n \Pi^{n}_{m} Π m n Π m n \Pi^{n}_{m} Π m n n − ℓ − m n-\ell-m n − ℓ − m 2 k 2k 2 k a ~ i \widetilde{a}^{i} a i { v i } i = 1 k ∪ { a ~ i } i = 1 k \{v^{i}\}_{i=1}^{k} \cup\{\widetilde{a}^{i}\}_{i=1}^{k} { v i } i = 1 k ∪ { a i } i = 1 k
v 1 , … , v k , a ~ 1 , … , a ~ k , f 2 k + 1 , … , f n v^{1},\ldots,v^{k},\widetilde{a}^{1},\ldots,\widetilde{a}^{k},f^{2k+1},\ldots,f^{n} v 1 , … , v k , a 1 , … , a k , f 2 k + 1 , … , f n of V n V_{n} V n g ∈ S L n ( F q ) g \in\mathrm{SL}_{n}(\mathbb{F}_{q}) g ∈ SL n ( F q ) v i v^{i} v i v i − a ~ i v^{i}-\widetilde{a}^{i} v i − a i 1 ≤ i ≤ k 1 \le i \le k 1 ≤ i ≤ k ( g v i ) ≤ m = v ≤ m i − a i = w i (gv^{i})_{\le m}=v^{i}_{\le m}-a^{i}=w^{i} ( g v i ) ≤ m = v ≤ m i − a i = w i g g g
Showing that the element g g g φ \varphi φ g − 1 g^{-1} g − 1 φ \varphi φ
⟨ g v , φ j ⟩ n = ⟨ v , φ j ⟩ n for all v ∈ V n and j = 1 , … , ℓ . \langle gv,\varphi^{j}\rangle_{n}=\langle v,\varphi^{j}\rangle_{n}\qquad\text{for all }v\in V_{n}\text{ and }j=1,\ldots,\ell. ⟨ g v , φ j ⟩ n = ⟨ v , φ j ⟩ n for all v ∈ V n and j = 1 , … , ℓ . These equations are satisfied for each of the basis vectors given in ( ∗ ) (*) ( ∗ ) □ \square □
The following proposition is the main technical tool that will help us apply the ergodic theorem to Stab S L ∞ ( F q ) ( φ ) \operatorname{Stab}_{\mathrm{SL}_{\infty}(\mathbb{F}_{q})}(\varphi) Stab SL ∞ ( F q ) ( φ ) § 2.2 .
Proposition 7.8. Let k , ℓ ∈ N k,\ell\in\mathbb{N} k , ℓ ∈ N φ ∈ X 0 , ℓ \varphi\in X^{0,\ell} φ ∈ X 0 , ℓ μ \mu μ Stab S L ∞ ( F q ) ( φ ) \operatorname{Stab}_{\mathrm{SL}_{\infty}(\mathbb{F}_{q})}(\varphi) Stab SL ∞ ( F q ) ( φ ) X k , 0 X^{k,0} X k , 0
μ ( { v ∈ X k , 0 : M n ( v , φ ) has full rank for infinitely many n } ) = 1. \mu\left(\left\{v\in X^{k,0}:M_{n}(v,\varphi)\text{ has full rank for infinitely many }n\right\}\right)=1. μ ( { v ∈ X k , 0 : M n ( v , φ ) has full rank for infinitely many n } ) = 1. Proof. Define P = Stab S L ∞ ( F q ) ( φ ) P=\operatorname{Stab}_{\mathrm{SL}_{\infty}(\mathbb{F}_{q})}(\varphi) P = Stab SL ∞ ( F q ) ( φ ) P n = P ∩ S L n ( F q ) P_{n}=P\cap\mathrm{SL}_{n}(\mathbb{F}_{q}) P n = P ∩ SL n ( F q ) n ∈ N n\in\mathbb{N} n ∈ N E E E ⟨ ⋅ , ⋅ ⟩ n \langle\cdot,\cdot\rangle_{n} ⟨ ⋅ , ⋅ ⟩ n P m P_{m} P m n ≥ m n\ge m n ≥ m E E E P P P μ \mu μ μ ( E ) > 0 \mu(E)>0 μ ( E ) > 0
Let T ∈ N T \in\mathbb{N} T ∈ N φ \varphi φ T T T J ⊆ N J \subseteq\mathbb{N} J ⊆ N m ∈ N m \in\mathbb{N} m ∈ N φ ( m , m + T ] j \varphi^{j}_{(m,m+T]} φ ( m , m + T ] j j ∈ { 1 , … , ℓ } j \in\{1,\ldots,\ell\} j ∈ { 1 , … , ℓ } J J J E E E E = ⋂ m ∈ J S m E = \bigcap_{m \in J} S_m E = ⋂ m ∈ J S m
S m = { v ∈ X k , 0 : ∃ n ≥ m such that M n ( v , φ ) has full rank } . S_m = \{v \in X^{k,0} : \exists n \ge m \text{ such that } M_n(v,\varphi) \text{ has full rank}\}. S m = { v ∈ X k , 0 : ∃ n ≥ m such that M n ( v , φ ) has full rank } . We will show that μ ( S m ) \mu(S_m) μ ( S m ) δ > 0 \delta> 0 δ > 0 m m m
A box [ a 1 , … , a k ] ⊆ X k , 0 [a^1,\ldots,a^k] \subseteq X^{k,0} [ a 1 , … , a k ] ⊆ X k , 0 m m m independent if the vectors a 1 , … , a k a^1,\ldots,a^k a 1 , … , a k V m V_m V m X k , 0 = ⋃ { B an independent box } X^{k,0} = \bigcup\{B \text{ an independent box}\} X k , 0 = ⋃ { B an independent box } m m m m m m 1 / 2 1/2 1/2 m ∈ J m \in J m ∈ J B 1 , … , B K B_1,\ldots,B_K B 1 , … , B K m m m φ ≤ m \varphi_{\le m} φ ≤ m
Define Y ⊆ V m k Y \subseteq V_m^k Y ⊆ V m k w ∈ V m k w \in V_m^k w ∈ V m k M m ( w , φ ) M_m(w,\varphi) M m ( w , φ ) ∣ Y ∣ / q m k |Y|/q^{mk} ∣ Y ∣/ q mk M ℓ × k ( F q ) M_{\ell\times k}(\mathbb{F}_q) M ℓ × k ( F q ) γ > 0 \gamma> 0 γ > 0 m m m
Now, let n = m + C n = m+C n = m + C C = max { T , ℓ + 2 k } + 1 C = \max\{T,\ell+2k\}+1 C = max { T , ℓ + 2 k } + 1 m m m q k C q^{kC} q k C n n n i = 1 , … , K i = 1,\ldots,K i = 1 , … , K B i ′ ⊆ B i B'_i \subseteq B_i B i ′ ⊆ B i n n n μ ( B i ′ ) ≥ μ ( B i ) q − k C \mu(B'_i) \ge\mu(B_i)q^{-kC} μ ( B i ′ ) ≥ μ ( B i ) q − k C μ ( B i ′ ) \mu(B'_i) μ ( B i ′ ) w 1 , … , w ∣ Y ∣ w_1,\ldots,w_{|Y|} w 1 , … , w ∣ Y ∣ Y Y Y ( B 1 ′ , w 1 ) , … , ( B N ′ , w N ) (B'_1,w_1),\ldots,(B'_N,w_N) ( B 1 ′ , w 1 ) , … , ( B N ′ , w N ) N = min ( K , ∣ Y ∣ ) N = \min(K,|Y|) N = min ( K , ∣ Y ∣ ) K ≤ q m k K \le q^{mk} K ≤ q mk N / K ≥ γ N/K \ge\gamma N / K ≥ γ
For each pair ( B i ′ , w i ) (B'_i,w_i) ( B i ′ , w i ) i ≤ N i \le N i ≤ N g i ∈ P n g_i \in P_n g i ∈ P n ( g i v ) ≤ m = w i (g_iv)_{\le m} = w_i ( g i v ) ≤ m = w i v ∈ B i ′ v \in B'_i v ∈ B i ′ g i B i ′ g_iB'_i g i B i ′ i = 1 , … , N i = 1,\ldots,N i = 1 , … , N g i B i ′ ⊆ S m g_iB'_i \subseteq S_m g i B i ′ ⊆ S m
μ ( S m ) ≥ ∑ i = 1 N μ ( g i B i ′ ) ≥ N K ∑ i = 1 K μ ( B i ′ ) ≥ γ q − k C ∑ i = 1 K μ ( B i ) ≥ 1 2 γ q − k C . \mu(S_m) \ge\sum_{i=1}^{N}\mu(g_iB'_i) \ge\frac{N}{K}\sum_{i=1}^{K}\mu(B'_i) \ge\gamma q^{-kC}\sum_{i=1}^{K}\mu(B_i) \ge\frac{1}{2}\gamma q^{-kC}. μ ( S m ) ≥ i = 1 ∑ N μ ( g i B i ′ ) ≥ K N i = 1 ∑ K μ ( B i ′ ) ≥ γ q − k C i = 1 ∑ K μ ( B i ) ≥ 2 1 γ q − k C . Since the right-hand side is independent of m m m m ∈ J m \in J m ∈ J
Invariant measures on X k , ℓ X^{k,\ell} X k , ℓ
We turn to our second use of the ergodic theorem.
Proposition 7.9. Fix k , ℓ ∈ N k,\ell\in\mathbb{N} k , ℓ ∈ N φ ∈ X 0 , ℓ \varphi\in X^{0,\ell} φ ∈ X 0 , ℓ P = Stab S L ∞ ( F q ) ( φ ) P = \operatorname{Stab}_{\mathrm{SL}_\infty(\mathbb{F}_q)}(\varphi) P = Stab SL ∞ ( F q ) ( φ ) P P P X k , 0 X^{k,0} X k , 0 Haar k , 0 \operatorname{Haar}^{k,0} Haar k , 0
Proof. First, we may and will assume without loss of generality that k ≤ ℓ k \le\ell k ≤ ℓ φ \varphi φ φ ′ ∈ X 0 , ℓ ′ \varphi' \in X^{0,\ell'} φ ′ ∈ X 0 , ℓ ′ ℓ ′ ≥ ℓ \ell' \ge\ell ℓ ′ ≥ ℓ ℓ \ell ℓ φ \varphi φ P P P P ′ P' P ′ P P P P ′ P' P ′ P ′ P' P ′ P P P φ ′ \varphi' φ ′
Let μ \mu μ P P P X k , 0 X^{k,0} X k , 0 μ \mu μ E = [ q ] ⊆ X k , 0 E = [q] \subseteq X^{k,0} E = [ q ] ⊆ X k , 0 m m m q ∈ V m k q \in V_m^k q ∈ V m k v = ( v 1 , … , v k ) ∈ X k , 0 v = (v^1,\ldots,v^k) \in X^{k,0} v = ( v 1 , … , v k ) ∈ X k , 0 E E E
μ ( E ) = lim n → ∞ 1 ∣ P n ∣ ∣ { g ∈ P n : g v ∈ E } ∣ . (18) \mu(E) = \lim_{n\to\infty} \frac{1}{|P_n|}|\{g \in P_n : gv \in E\}|. \tag*{(18)} μ ( E ) = n → ∞ lim ∣ P n ∣ 1 ∣ { g ∈ P n : g v ∈ E } ∣. (18) as per Theorem 7.1 . Here P n = P ∩ S L n ( F q ) P_n = P \cap\mathrm{SL}_n(\mathbb{F}_q) P n = P ∩ SL n ( F q ) n ∈ N n \in\mathbb{N} n ∈ N
Choose an index n > m + ℓ n > m+\ell n > m + ℓ ( φ ≤ n j ) j = 1 ℓ (\varphi^j_{\le n})_{j=1}^{\ell} ( φ ≤ n j ) j = 1 ℓ ( v ≤ n i ) i = 1 k (v^i_{\le n})_{i=1}^{k} ( v ≤ n i ) i = 1 k f = ( f j ) j = 1 n f = (f^j)_{j=1}^{n} f = ( f j ) j = 1 n V ^ n \widehat{V}_n V n φ ≤ n 1 , … , φ ≤ n ℓ \varphi^1_{\le n},\ldots,\varphi^\ell_{\le n} φ ≤ n 1 , … , φ ≤ n ℓ f j = φ ≤ n j f^j=\varphi^j_{\leq n} f j = φ ≤ n j j = 1 , … , ℓ j=1,\ldots,\ell j = 1 , … , ℓ V ^ n \widehat{V}_n V n F F F n × n n\times n n × n j j j f j f^j f j g ∈ P n g\in P_n g ∈ P n
g = F − 1 ( I ℓ 0 ℓ , n − ℓ B A ) F (19) g=F^{-1}\begin{pmatrix} I_\ell& 0_{\ell,n-\ell}\\ B & A \end{pmatrix}F \tag*{(19)} g = F − 1 ( I ℓ B 0 ℓ , n − ℓ A ) F (19) where A ∈ S L n − ℓ ( F q ) A\in\mathrm{SL}_{n-\ell}(\mathbb{F}_q) A ∈ SL n − ℓ ( F q ) B ∈ M ( n − ℓ ) × ℓ ( F q ) B\in\mathrm{M}_{(n-\ell)\times\ell}(\mathbb{F}_q) B ∈ M ( n − ℓ ) × ℓ ( F q ) S L n − ℓ ( F q ) ⋉ M ( n − ℓ ) × ℓ ( F q ) ≅ P n \mathrm{SL}_{n-\ell}(\mathbb{F}_q)\ltimes\mathrm{M}_{(n-\ell)\times\ell}(\mathbb{F}_q)\cong P_n SL n − ℓ ( F q ) ⋉ M ( n − ℓ ) × ℓ ( F q ) ≅ P n ( A , B ) (A,B) ( A , B ) P n P_n P n
We look for conditions on the element g ∈ P n g\in P_n g ∈ P n g v ∈ E gv\in E g v ∈ E ( g v i ) ≤ m = a i (gv^i)_{\leq m}=a^i ( g v i ) ≤ m = a i i = 1 , … , k i=1,\ldots,k i = 1 , … , k v ∈ M ∞ × k ( F q ) v\in\mathrm{M}_{\infty\times k}(\mathbb{F}_q) v ∈ M ∞ × k ( F q ) a ∈ M m × k ( F q ) a\in\mathrm{M}_{m\times k}(\mathbb{F}_q) a ∈ M m × k ( F q ) v v v a a a ( ⋅ ) ≤ m (\cdot)_{\leq m} ( ⋅ ) ≤ m m m m g ∈ P n g\in P_n g ∈ P n ( g v ) ≤ m = a (gv)_{\leq m}=a ( g v ) ≤ m = a
Denote u = F v ≤ n ∈ M n × k ( F q ) u=Fv_{\leq n}\in\mathrm{M}_{n\times k}(\mathbb{F}_q) u = F v ≤ n ∈ M n × k ( F q ) ℓ \ell ℓ u = ( u ≤ ℓ u > ℓ ) u=\begin{pmatrix}u_{\leq\ell}\\u_{>\ell}\end{pmatrix} u = ( u ≤ ℓ u > ℓ ) f j = φ ≤ n j f^j=\varphi^j_{\leq n} f j = φ ≤ n j j ≤ ℓ j\leq\ell j ≤ ℓ u ≤ ℓ u_{\leq\ell} u ≤ ℓ M n ( v , φ ) M_n(v,\varphi) M n ( v , φ ) g v ≤ n gv_{\leq n} g v ≤ n ( a c ) \begin{pmatrix}a\\c\end{pmatrix} ( a c ) a ∈ M m × k ( F q ) a\in\mathrm{M}_{m\times k}(\mathbb{F}_q) a ∈ M m × k ( F q ) c ∈ M ( n − m ) × k ( F q ) c\in\mathrm{M}_{(n-m)\times k}(\mathbb{F}_q) c ∈ M ( n − m ) × k ( F q ) (19) gives
F ( a c ) = ( I ℓ 0 ℓ , n − ℓ B A ) ( u ≤ ℓ u > ℓ ) = ( u ≤ ℓ B u ≤ ℓ + A u > ℓ ) . (20) F\begin{pmatrix}a\\c\end{pmatrix}
=
\begin{pmatrix}I_\ell&0_{\ell,n-\ell}\\B&A\end{pmatrix}
\begin{pmatrix}u_{\leq\ell}\\u_{>\ell}\end{pmatrix}
=
\begin{pmatrix}u_{\leq\ell}\\Bu_{\leq\ell}+Au_{>\ell}\end{pmatrix}. \tag*{(20)} F ( a c ) = ( I ℓ B 0 ℓ , n − ℓ A ) ( u ≤ ℓ u > ℓ ) = ( u ≤ ℓ B u ≤ ℓ + A u > ℓ ) . (20) The first ℓ \ell ℓ c c c g g g
M n − m ( c , φ > m ) = u ≤ ℓ − M m ( a , φ ) . (21) M_{n-m}(c,\varphi_{>m})=u_{\leq\ell}-M_m(a,\varphi). \tag*{(21)} M n − m ( c , φ > m ) = u ≤ ℓ − M m ( a , φ ) . (21) As φ \varphi φ n n n φ ( m , n ] j \varphi^j_{(m,n]} φ ( m , n ] j j = 1 , … , ℓ j=1,\ldots,\ell j = 1 , … , ℓ n n n c c c (21) is a certain affine subspace C ( v , a ) ⊆ M ( n − m ) × k ( F q ) C(v,a)\subseteq\mathrm{M}_{(n-m)\times k}(\mathbb{F}_q) C ( v , a ) ⊆ M ( n − m ) × k ( F q ) ℓ k \ell k ℓ k
Given c ∈ C ( v , a ) c\in C(v,a) c ∈ C ( v , a ) n − ℓ n-\ell n − ℓ (20) give an equation for g g g
B u ≤ ℓ + A u > ℓ = d ( v , a , c ) , (22) Bu_{\leq\ell}+Au_{>\ell}=d(v,a,c), \tag*{(22)} B u ≤ ℓ + A u > ℓ = d ( v , a , c ) , (22) where d ( v , a , c ) ∈ M ( n − ℓ ) × k ( F q ) d(v,a,c)\in\mathrm{M}_{(n-\ell)\times k}(\mathbb{F}_q) d ( v , a , c ) ∈ M ( n − ℓ ) × k ( F q ) (22) , for each given A ∈ S L n − ℓ ( F q ) A\in\mathrm{SL}_{n-\ell}(\mathbb{F}_q) A ∈ SL n − ℓ ( F q ) B B B
B u ≤ ℓ = d ( v , a , c ) − A u > ℓ . (23) Bu_{\leq\ell}=d(v,a,c)-Au_{>\ell}. \tag*{(23)} B u ≤ ℓ = d ( v , a , c ) − A u > ℓ . (23) In this form, each row of B B B k k k u ≤ ℓ = M n ( v , φ ) u_{\leq\ell}=M_n(v,\varphi) u ≤ ℓ = M n ( v , φ )
By Theorem 7.8, we may assume that v v v n n n u ≤ ℓ u_{\leq\ell} u ≤ ℓ (18) necessarily exists, and our aim is to calculate it, we may restrict ourselves only to such n n n
In this case, we get that for each c ∈ C ( v , a ) c\in C(v,a) c ∈ C ( v , a ) A ∈ S L n − ℓ ( F q ) A\in\mathrm{SL}_{n-\ell}(\mathbb{F}_q) A ∈ SL n − ℓ ( F q ) B B B (23) is an affine subspace of M ( n − ℓ ) × ℓ ( F q ) \mathrm{M}_{(n-\ell)\times\ell}(\mathbb{F}_q) M ( n − ℓ ) × ℓ ( F q ) ( n − ℓ ) k (n-\ell)k ( n − ℓ ) k k ≤ ℓ k\leq\ell k ≤ ℓ rk u ≤ ℓ = k \operatorname{rk}u_{\leq\ell}=k rk u ≤ ℓ = k c c c ( A , B ) (A,B) ( A , B ) (20) . Thus we obtain the count
∣ { g ∈ P n : g ⋅ v ∈ E } ∣ = ∣ C ( v , a ) ∣ ⋅ ∣ S L n − ℓ ( F q ) ∣ ⋅ ∣ M ( n − ℓ ) × ℓ ( F q ) ∣ q − k ( n − ℓ ) = q k ( n − m ) − ℓ k ⋅ ∣ P n ∣ ⋅ q − k ( n − ℓ ) = ∣ P n ∣ q − k m . (24) \begin{aligned}
\left|\{g\in P_n:g\cdot v\in E\}\right|
=|C(v,a)|\cdot|\mathrm{SL}_{n-\ell}(\mathbb{F}_q)|\cdot|\mathrm{M}_{(n-\ell)\times\ell}(\mathbb{F}_q)|q^{-k(n-\ell)} \\
&=q^{k(n-m)-\ell k}\cdot|P_n|\cdot q^{-k(n-\ell)}
=|P_n|q^{-km}. \tag*{(24)}
\end{aligned} ∣ { g ∈ P n : g ⋅ v ∈ E } ∣ = ∣ C ( v , a ) ∣ ⋅ ∣ SL n − ℓ ( F q ) ∣ ⋅ ∣ M ( n − ℓ ) × ℓ ( F q ) ∣ q − k ( n − ℓ ) = q k ( n − m ) − ℓ k ⋅ ∣ P n ∣ ⋅ q − k ( n − ℓ ) = ∣ P n ∣ q − k m . (24) Substituting eq. (24) in eq. (18) we obtain
μ ( E ) = q − k m = Haar k , 0 ( E ) . \mu(E)=q^{-km}=\operatorname{Haar}^{k,0}(E). μ ( E ) = q − k m = Haar k , 0 ( E ) . The set of boxes is a π \pi π σ \sigma σ μ = Haar k , 0 \mu= \operatorname{Haar}^{k,0} μ = Haar k , 0
We establish the main theorem of this section by combining the preceding results.
Proof of Theorem 1.3 . We may assume without loss of generality that k ≤ ℓ k \le\ell k ≤ ℓ X k , ℓ X^{k,\ell} X k , ℓ X ℓ , k X^{\ell,k} X ℓ , k μ \mu μ SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) X k , ℓ = X k , 0 × X 0 , ℓ X^{k,\ell} = X^{k,0} \times X^{0,\ell} X k , ℓ = X k , 0 × X 0 , ℓ μ \mu μ ( X k , 0 , Haar k , 0 ) (X^{k,0}, \operatorname{Haar}^{k,0}) ( X k , 0 , Haar k , 0 ) ( X 0 , ℓ , Haar 0 , ℓ ) (X^{0,\ell}, \operatorname{Haar}^{0,\ell}) ( X 0 , ℓ , Haar 0 , ℓ )
Let φ ∈ X 0 , ℓ \varphi\in X^{0,\ell} φ ∈ X 0 , ℓ P = Stab SL ∞ ( F q ) ( φ ) P = \operatorname{Stab}_{\operatorname{SL}_{\infty}(\mathbb{F}_q)}(\varphi) P = Stab SL ∞ ( F q ) ( φ ) P P P X k , 0 X^{k,0} X k , 0
Since a functional φ ∈ X 0 , ℓ \varphi\in X^{0,\ell} φ ∈ X 0 , ℓ Haar 0 , ℓ \operatorname{Haar}^{0,\ell} Haar 0 , ℓ X = X 0 , ℓ X = X^{0,\ell} X = X 0 , ℓ Y = X k , 0 Y = X^{k,0} Y = X k , 0 μ = Haar k , 0 × Haar 0 , ℓ = Haar k , ℓ \mu= \operatorname{Haar}^{k,0} \times\operatorname{Haar}^{0,\ell} = \operatorname{Haar}^{k,\ell} μ = Haar k , 0 × Haar 0 , ℓ = Haar k , ℓ
Invariant measures on related spaces
In this section, we use the unique ergodicity of the system X k , ℓ X^{k,\ell} X k , ℓ V k , ℓ V^{k,\ell} V k , ℓ X k , ℓ X^{k,\ell} X k , ℓ Gr k ( V ) × Gr ℓ ( V ^ ) \operatorname{Gr}_{k}(V) \times\operatorname{Gr}_{\ell}(\widehat{V}) Gr k ( V ) × Gr ℓ ( V ) X k , ℓ X^{k,\ell} X k , ℓ
Invariant measures on V k , ℓ V^{k,\ell} V k , ℓ
We associate to each vector v ∈ V k v \in V^k v ∈ V k
T v : F q k → V , T v ( α ) = ∑ i = 1 k α i v i . T_v : \mathbb{F}_q^k \to V,\qquad T_v(\alpha) = \sum_{i=1}^{k} \alpha_i v^i. T v : F q k → V , T v ( α ) = i = 1 ∑ k α i v i . Set ker ( v ) = ker ( T v ) ≤ F q k \ker(v) = \ker(T_v) \le\mathbb{F}_q^k ker ( v ) = ker ( T v ) ≤ F q k ker ( φ ) ≤ F q ℓ \ker(\varphi) \le\mathbb{F}_q^\ell ker ( φ ) ≤ F q ℓ φ ∈ V ^ ℓ \varphi\in\widehat{V}^\ell φ ∈ V ℓ U ≤ F q k U \le\mathbb{F}_q^k U ≤ F q k W ≤ F q ℓ W \le\mathbb{F}_q^\ell W ≤ F q ℓ
X U , W k , ℓ = { ( v , φ ) ∈ V k , ℓ : ker ( v ) = U and ker ( φ ) = W } . X^{k,\ell}_{U,W} = \{(v,\varphi) \in V^{k,\ell} : \ker(v) = U \text{ and } \ker(\varphi) = W\}. X U , W k , ℓ = {( v , φ ) ∈ V k , ℓ : ker ( v ) = U and ker ( φ ) = W } . Every such X U , W k , ℓ X^{k,\ell}_{U,W} X U , W k , ℓ V k , ℓ V^{k,\ell} V k , ℓ
V k , ℓ = ⨆ U ≤ F q k , W ≤ F q ℓ X U , W k , ℓ . V^{k,\ell} = \bigsqcup_{\substack{U \le\mathbb{F}_q^k,\, W \le\mathbb{F}_q^\ell}} X^{k,\ell}_{U,W}. V k , ℓ = U ≤ F q k , W ≤ F q ℓ ⨆ X U , W k , ℓ . For example, note that X 0 , 0 k , ℓ = X k , ℓ X^{k,\ell}_{0,0} = X^{k,\ell} X 0 , 0 k , ℓ = X k , ℓ
Claim 7.10. Let k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 U ≤ F q k U \le\mathbb{F}_q^k U ≤ F q k W ≤ F q ℓ W \le\mathbb{F}_q^\ell W ≤ F q ℓ X U , W k , ℓ X^{k,\ell}_{U,W} X U , W k , ℓ SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) X d , e X^{d,e} X d , e d = k − dim U d = k - \dim U d = k − dim U e = ℓ − dim W e = \ell- \dim W e = ℓ − dim W
Proof. The invariance of X U , W k , ℓ X^{k,\ell}_{U,W} X U , W k , ℓ π 1 : F q k → F q d \pi_1 : \mathbb{F}_q^k \to\mathbb{F}_q^d π 1 : F q k → F q d π 2 : F q ℓ → F q e \pi_2 : \mathbb{F}_q^\ell\to\mathbb{F}_q^e π 2 : F q ℓ → F q e U , W U,W U , W ( π 1 , π 2 ) : V k , ℓ → V d , e (\pi_1,\pi_2) : V^{k,\ell} \to V^{d,e} ( π 1 , π 2 ) : V k , ℓ → V d , e SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) X U , W k , ℓ X^{k,\ell}_{U,W} X U , W k , ℓ X d , e X^{d,e} X d , e
The isomorphism constructed in the above claim endows X U , W k , ℓ X^{k,\ell}_{U,W} X U , W k , ℓ SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) X d , e X^{d,e} X d , e V k , ℓ V^{k,\ell} V k , ℓ τ U , W k , ℓ \tau^{k,\ell}_{U,W} τ U , W k , ℓ SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ
Theorem 7.11. Fix k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 μ \mu μ SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ U ≤ F q k U \le\mathbb{F}_q^k U ≤ F q k W ≤ F q ℓ W \le\mathbb{F}_q^\ell W ≤ F q ℓ μ \mu μ X U , W k , ℓ X^{k,\ell}_{U,W} X U , W k , ℓ μ = τ U , W k , ℓ \mu= \tau^{k,\ell}_{U,W} μ = τ U , W k , ℓ
Proof. Since each X U , W k , ℓ X^{k,\ell}_{U,W} X U , W k , ℓ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ μ \mu μ X d , e X^{d,e} X d , e d ≤ k d \le k d ≤ k e ≤ ℓ e \le\ell e ≤ ℓ X d , e X^{d,e} X d , e
Invariant measures on Grassmannians
Given a vector space W W W k k k Gr k ( W ) \operatorname{Gr}_k(W) Gr k ( W ) k ∈ N k \in\mathbb{N} k ∈ N W W W k k k W W W Gr k ( W ) \operatorname{Gr}_k(W) Gr k ( W )
We are interested in Grassmannians of the vector spaces V V V V ^ \widehat{V} V X k , ℓ X^{k,\ell} X k , ℓ V k , ℓ V^{k,\ell} V k , ℓ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) G L k ( F q ) × G L ℓ ( F q ) \mathrm{GL}_k(\mathbb{F}_q) \times\mathrm{GL}_\ell(\mathbb{F}_q) GL k ( F q ) × GL ℓ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ M ∞ × k ( F q ) × M ℓ × ∞ ( F q ) M_{\infty\times k}(\mathbb{F}_q) \times M_{\ell\times\infty}(\mathbb{F}_q) M ∞ × k ( F q ) × M ℓ × ∞ ( F q ) k k k
Gr k ( V ) = X k , 0 / ( G L k ( F q ) × 1 ) . \operatorname{Gr}_k(V) = X^{k,0}/(\mathrm{GL}_k(\mathbb{F}_q) \times1). Gr k ( V ) = X k , 0 / ( GL k ( F q ) × 1 ) . The projection map X k , 0 → Gr k ( V ) X^{k,0} \to\operatorname{Gr}_k(V) X k , 0 → Gr k ( V ) v ∈ X k , 0 v \in X^{k,0} v ∈ X k , 0 k k k X k , 0 X^{k,0} X k , 0 Gr k ( V ) \operatorname{Gr}_k(V) Gr k ( V ) X k , 0 X^{k,0} X k , 0 X k , 0 X^{k,0} X k , 0 S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) X k , 0 X^{k,0} X k , 0
We can also define the dual Grassmannian Gr ℓ ( V ^ ) = X 0 , ℓ / ( 1 × G L ℓ ( F q ) ) \operatorname{Gr}_\ell(\widehat{V}) = X^{0,\ell}/(1 \times\mathrm{GL}_\ell(\mathbb{F}_q)) Gr ℓ ( V ) = X 0 , ℓ / ( 1 × GL ℓ ( F q )) Gr ℓ ( V ) \operatorname{Gr}_\ell(V) Gr ℓ ( V ) S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
Notice that the invariant random subgroup μ k , ℓ \mu_{k,\ell} μ k , ℓ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) Gr k ( V ) × Gr ℓ ( V ^ ) \operatorname{Gr}_k(V) \times\operatorname{Gr}_\ell(\widehat{V}) Gr k ( V ) × Gr ℓ ( V ) Haar k , ℓ \operatorname{Haar}^{k,\ell} Haar k , ℓ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
Proposition 7.12. Let k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) Gr k ( V ) × Gr ℓ ( V ^ ) \operatorname{Gr}_k(V) \times\operatorname{Gr}_\ell(\widehat{V}) Gr k ( V ) × Gr ℓ ( V ) Haar k , ℓ \operatorname{Haar}^{k,\ell} Haar k , ℓ X k , ℓ X^{k,\ell} X k , ℓ
The above proposition will be an immediate corollary of the following general result in the ergodic theory of group actions.
Proposition 7.13. Let G G G X X X Y Y Y p : X → Y p : X \to Y p : X → Y G G G p ∗ : M G ( X ) → M G ( Y ) p_* : \mathcal{M}_G(X) \to\mathcal{M}_G(Y) p ∗ : M G ( X ) → M G ( Y ) G G G p p p p ∗ p_* p ∗
Proof. We use Zimmer’s theory of amenable actions [38 , 39 ] . Indeed [39 Remark 6(a)] states the analogous result, in the case where X = Y × F X = Y \times F X = Y × F Y Y Y F F F
In our case, let ν \nu ν G G G Y Y Y Y Y Y y ∈ Y y \in Y y ∈ Y C ( X y ) C(X_y) C ( X y ) X y = p − 1 ( y ) X_y = p^{-1}(y) X y = p − 1 ( y ) A y A_y A y X y X_y X y G G G A y ⊆ C ( X y ) ∗ A_y \subseteq C(X_y)^* A y ⊆ C ( X y ) ∗ Y Y Y [39 Definition 4.3.1] .
Since the group G G G ( Y , ν ) (Y,\nu) ( Y , ν ) ν \nu ν G G G λ y \lambda_y λ y λ y ∈ A y \lambda_y \in A_y λ y ∈ A y g ∈ G g \in G g ∈ G g ∗ λ y = λ g y g_*\lambda_y = \lambda_{gy} g ∗ λ y = λ g y ν \nu ν y ∈ Y y \in Y y ∈ Y μ \mu μ X X X
μ = ∫ Y μ y d ν ( y ) . \mu=\int_Y \mu_y\,d\nu(y). μ = ∫ Y μ y d ν ( y ) . Since the measure ν \nu ν G G G g ∈ G g\in G g ∈ G
g ∗ μ = ∫ Y λ g y d ν ( y ) = ∫ Y λ y d ν ( y ) = μ . g_*\mu=\int_Y\lambda_{gy}\,d\nu(y)=\int_Y\lambda_y\,d\nu(y)=\mu. g ∗ μ = ∫ Y λ g y d ν ( y ) = ∫ Y λ y d ν ( y ) = μ . So that the measure μ \mu μ G G G Y Y Y p ∗ μ = ∫ Y δ y d ν ( y ) = ν p_*\mu=\int_Y\delta_y\,d\nu(y)=\nu p ∗ μ = ∫ Y δ y d ν ( y ) = ν
In particular, in the setting of Theorem 7.13, the unique ergodicity of the action of G G G X X X Y Y Y
Proof of Theorem 7.12. The S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) X = X k , ℓ X=X^{k,\ell} X = X k , ℓ Theorem 1.3 . The fiber of the natural projection map X k , ℓ → Gr k ( V ) × Gr ℓ ( V ^ ) X^{k,\ell}\to\operatorname{Gr}_k(V)\times\operatorname{Gr}_{\ell}(\widehat{V}) X k , ℓ → Gr k ( V ) × Gr ℓ ( V ) Y = Gr k ( V ) × Gr ℓ ( V ^ ) Y=\operatorname{Gr}_k(V)\times\operatorname{Gr}_{\ell}(\widehat{V}) Y = Gr k ( V ) × Gr ℓ ( V )
Invariant random subgroups of S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q )
Fix a prime power q q q S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q )
Invariant random subgroups of S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
Recall that V = ∏ n ∈ N F q V=\prod_{n\in\mathbb{N}}\mathbb{F}_q V = ∏ n ∈ N F q G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) V ^ = V \widehat{V}=V V = V k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 V k , ℓ = V k × V ^ ℓ V^{k,\ell}=V^k\times\widehat{V}^{\ell} V k , ℓ = V k × V ℓ G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) Haar k , ℓ \operatorname{Haar}^{k,\ell} Haar k , ℓ μ k , ℓ ∈ IRS ( S L ∞ ( F q ) ) \mu_{k,\ell}\in\operatorname{IRS}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) μ k , ℓ ∈ IRS ( SL ∞ ( F q )) Haar k , ℓ \operatorname{Haar}^{k,\ell} Haar k , ℓ Stab S L ∞ ( F q ) : V k , ℓ → Sub ( S L ∞ ( F q ) ) \operatorname{Stab}_{\mathrm{SL}_{\infty}(\mathbb{F}_q)}:V^{k,\ell}\to\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) Stab SL ∞ ( F q ) : V k , ℓ → Sub ( SL ∞ ( F q )) μ ∞ = δ { I } \mu_{\infty}=\delta_{\{I\}} μ ∞ = δ { I }
In this section we calculate the Vershik trace of the invariant random subgroup μ k , ℓ \mu_{k,\ell} μ k , ℓ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) μ k , ℓ \mu_{k,\ell} μ k , ℓ μ ∞ \mu_{\infty} μ ∞ μ k , ℓ \mu_{k,\ell} μ k , ℓ Haar k , ℓ \operatorname{Haar}^{k,\ell} Haar k , ℓ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
Topology and Vershik traces of the μ k , ℓ \mu_{k,\ell} μ k , ℓ
By Skudlarek’s theorem (Theorem 1.5) the characters of the group S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
Ch ( S L ∞ ( F q ) ) = { χ r : r ∈ N 0 } ∪ { χ ∞ } . \operatorname{Ch}(\mathrm{SL}_{\infty}(\mathbb{F}_q))=\{\chi_r:r\in\mathbb{N}_0\}\cup\{\chi_{\infty}\}. Ch ( SL ∞ ( F q )) = { χ r : r ∈ N 0 } ∪ { χ ∞ } . where for each r ∈ N 0 r\in\mathbb{N}_0 r ∈ N 0 g ∈ S L ∞ ( F q ) g\in\mathrm{SL}_{\infty}(\mathbb{F}_q) g ∈ SL ∞ ( F q )
χ r ( g ) = q − r ⋅ rk g = q − r ⋅ rk ( g − I ) \chi_r(g)=q^{-r\cdot\operatorname{rk}g}=q^{-r\cdot\operatorname{rk}(g-I)} χ r ( g ) = q − r ⋅ rk g = q − r ⋅ rk ( g − I ) and χ ∞ = δ I \chi_{\infty}=\delta_I χ ∞ = δ I
Proposition 8.1. For each k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 μ k , ℓ \mu_{k,\ell} μ k , ℓ χ k + ℓ \chi_{k+\ell} χ k + ℓ
Proof. Fix k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 g ∈ S L ∞ ( F q ) g\in\mathrm{SL}_{\infty}(\mathbb{F}_q) g ∈ SL ∞ ( F q )
F g = { H ∈ Sub ( S L ∞ ( F q ) ) : g ∈ H } . F_g=\{H\in\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)):g\in H\}. F g = { H ∈ Sub ( SL ∞ ( F q )) : g ∈ H } . Choose a point ( v 1 , … , v k , φ 1 , … , φ ℓ ) ∈ V k , ℓ (v^1,\ldots,v^k,\varphi^1,\ldots,\varphi^\ell)\in V^{k,\ell} ( v 1 , … , v k , φ 1 , … , φ ℓ ) ∈ V k , ℓ μ ( F g ) \mu(F_g) μ ( F g ) g g g v i v^i v i φ j \varphi^j φ j g v i = v i gv^i=v^i g v i = v i v i ∈ ker ( g − I ) v^i\in\ker(g-I) v i ∈ ker ( g − I ) i i i
q − codim V ( ker ( g − I ) ) = q − rk ( g − I ) . q^{-\operatorname{codim}_V(\ker(g-I))}=q^{-\operatorname{rk}(g-I)}. q − codim V ( k e r ( g − I )) = q − rk ( g − I ) . The condition ( g − 1 ) t φ j = φ j (g^{-1})^t\varphi^j=\varphi^j ( g − 1 ) t φ j = φ j g t φ j = φ j g^t\varphi^j=\varphi^j g t φ j = φ j j j j
q − rk ( g t − I ) = q − rk ( g − I ) . q^{-\operatorname{rk}(g^t-I)}=q^{-\operatorname{rk}(g-I)}. q − rk ( g t − I ) = q − rk ( g − I ) . Overall, we get
χ μ k , ℓ ( g ) = μ k , ℓ ( F g ) = q − ( k + ℓ ) ⋅ rk ~ g = χ k + ℓ ( g ) (25) \chi_{\mu_{k,\ell}}(g)=\mu_{k,\ell}(F_g)=q^{-(k+\ell)\cdot\widetilde{\operatorname{rk}}g}=\chi_{k+\ell}(g) \tag*{(25)} χ μ k , ℓ ( g ) = μ k , ℓ ( F g ) = q − ( k + ℓ ) ⋅ rk g = χ k + ℓ ( g ) (25) as required. □
Consider the subset M = { μ k , ℓ : k , ℓ ∈ N 0 } ∪ { μ ∞ } ⊆ IRS ( SL ∞ ( F q ) ) M=\{\mu_{k,\ell}:k,\ell\in\mathbb{N}_0\}\cup\{\mu_\infty\}\subseteq\operatorname{IRS}(\operatorname{SL}_\infty(\mathbb{F}_q)) M = { μ k , ℓ : k , ℓ ∈ N 0 } ∪ { μ ∞ } ⊆ IRS ( SL ∞ ( F q )) Theorem 1.1 it will be established that M = Extr ( IRS ( SL ∞ ( F q ) ) ) M=\operatorname{Extr}(\operatorname{IRS}(\operatorname{SL}_\infty(\mathbb{F}_q))) M = Extr ( IRS ( SL ∞ ( F q )))
Proposition 8.2. Let X = ( N 0 × N 0 ) ∪ { ∞ } X=(\mathbb{N}_0\times\mathbb{N}_0)\cup\{\infty\} X = ( N 0 × N 0 ) ∪ { ∞ } N 0 × N 0 \mathbb{N}_0\times\mathbb{N}_0 N 0 × N 0 X → M X\to M X → M ( k , ℓ ) (k,\ell) ( k , ℓ ) μ k , ℓ \mu_{k,\ell} μ k , ℓ ∞ \infty ∞ μ ∞ \mu_\infty μ ∞
Proof. Denote by h : X → M h:X\to M h : X → M E i j E_{ij} E ij SL ∞ ( F q ) \operatorname{SL}_\infty(\mathbb{F}_q) SL ∞ ( F q )
codim V ( ker ( E 21 − I ) ∩ ker ( E 23 − I ) ) = 2 and codim V ( ker ( E 12 ) − I ) ∩ ker ( E 32 − I ) ) = 1. \operatorname{codim}_V(\ker(E_{21}-I)\cap\ker(E_{23}-I))=2\quad\text{and}\quad\operatorname{codim}_V(\ker(E_{12})-I)\cap\ker(E_{32}-I))=1. codim V ( ker ( E 21 − I ) ∩ ker ( E 23 − I )) = 2 and codim V ( ker ( E 12 ) − I ) ∩ ker ( E 32 − I )) = 1. We denote F g = { H ∈ Sub ( SL ∞ ( F q ) ) : g ∈ H } F_g=\{H\in\operatorname{Sub}(\operatorname{SL}_\infty(\mathbb{F}_q)):g\in H\} F g = { H ∈ Sub ( SL ∞ ( F q )) : g ∈ H } g ∈ SL ∞ ( F q ) g\in\operatorname{SL}_\infty(\mathbb{F}_q) g ∈ SL ∞ ( F q )
μ k , ℓ ( F t 1 ∩ F t 2 ) = q − ( 2 k + ℓ ) . (26) \mu_{k,\ell}(F_{t_1}\cap F_{t_2})=q^{-(2k+\ell)}. \tag*{(26)} μ k , ℓ ( F t 1 ∩ F t 2 ) = q − ( 2 k + ℓ ) . (26) The two equations (25) and (26) together show that the two parameters k k k ℓ \ell ℓ μ k , ℓ \mu_{k,\ell} μ k , ℓ h h h
To show that the map h h h μ ∞ = lim ( k , ℓ ) → ∞ μ k , ℓ \mu_\infty=\lim_{(k,\ell)\to\infty}\mu_{k,\ell} μ ∞ = lim ( k , ℓ ) → ∞ μ k , ℓ [3 Theorem 2.1] it will suffice to show that all open subsets U ⊆ Sub ( SL ∞ ( F q ) ) U\subseteq\operatorname{Sub}(\operatorname{SL}_\infty(\mathbb{F}_q)) U ⊆ Sub ( SL ∞ ( F q ))
lim inf ( k , ℓ ) → ∞ μ k , ℓ ( U ) ≥ μ ∞ ( U ) . \liminf_{(k,\ell)\to\infty}\mu_{k,\ell}(U)\geq\mu_\infty(U). ( k , ℓ ) → ∞ lim inf μ k , ℓ ( U ) ≥ μ ∞ ( U ) . If { I } ∉ U \{I\}\notin U { I } ∈ / U g 1 , … , g m ∈ SL ∞ ( F q ) ∖ { I } g_1,\ldots,g_m\in\operatorname{SL}_\infty(\mathbb{F}_q)\setminus\{I\} g 1 , … , g m ∈ SL ∞ ( F q ) ∖ { I } F g 1 c ∩ ⋯ ∩ F g m c ⊆ U F_{g_1}^c\cap\cdots\cap F_{g_m}^c\subseteq U F g 1 c ∩ ⋯ ∩ F g m c ⊆ U
1 − μ k , ℓ ( U ) ≤ μ k , ℓ ( ( F g 1 c ∩ ⋯ ∩ F g m c ) c ) = μ k , ℓ ( ⋃ i = 1 m F g i ) ≤ ≤ ∑ i = 1 m μ k , ℓ ( F g i ) = ∑ i = 1 m q − ( k + ℓ ) rk g i → ( k , ℓ ) → ∞ 0 \begin{aligned}
1-\mu_{k,\ell}(U)\leq\mu_{k,\ell}\left((F_{g_1}^c\cap\cdots\cap F_{g_m}^c)^c\right)=\mu_{k,\ell}\left(\bigcup_{i=1}^{m}F_{g_i}\right)\leq\\
&\leq\sum_{i=1}^{m}\mu_{k,\ell}(F_{g_i})=\sum_{i=1}^{m}q^{-(k+\ell)\operatorname{rk}g_i}\xrightarrow[(k,\ell)\to\infty]{}0
\end{aligned} 1 − μ k , ℓ ( U ) ≤ μ k , ℓ ( ( F g 1 c ∩ ⋯ ∩ F g m c ) c ) = μ k , ℓ ( i = 1 ⋃ m F g i ) ≤ ≤ i = 1 ∑ m μ k , ℓ ( F g i ) = i = 1 ∑ m q − ( k + ℓ ) rk g i ( k , ℓ ) → ∞ 0 as required. We conclude that h h h X X X M M M h h h
Remark 8.3. Once Theorem 1.1 is established, we will deduce that the Vershik trace map IRS ( SL ∞ ( F q ) ) → Tr ( SL ∞ ( F q ) ) \operatorname{IRS}(\operatorname{SL}_\infty(\mathbb{F}_q))\to\operatorname{Tr}(\operatorname{SL}_\infty(\mathbb{F}_q)) IRS ( SL ∞ ( F q )) → Tr ( SL ∞ ( F q ))
Classification of invariant random subgroups of S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
Let μ \mu μ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) r = r ( μ ) r=r(\mu) r = r ( μ ) μ \mu μ
In Section 6 we have defined for any μ \mu μ H H H Col ( H ) \operatorname{Col}(H) Col ( H ) ∣ Col ( H ) ∣ = q r \lvert\operatorname{Col}(H)\rvert=q^r ∣ Col ( H )∣ = q r μ \mu μ Propositions 6.17 and 6.19 .
Further, we have defined a local and permutational (Theorem 3.1) action of the group G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) Col ( H ) N \operatorname{Col}(H)^{\mathbb{N}} Col ( H ) N Theorem 3.2 . Hence there are k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 k + ℓ = r k+\ell=r k + ℓ = r G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) V k , ℓ V^{k,\ell} V k , ℓ μ \mu μ k k k ℓ \ell ℓ H H H μ \mu μ
Let Π \Pi Π ϖ : F q r → F q r \varpi:\mathbb{F}_q^r\to\mathbb{F}_q^r ϖ : F q r → F q r ϖ N : V k , ℓ → V k , ℓ \varpi^{\mathbb{N}}:V^{k,\ell}\to V^{k,\ell} ϖ N : V k , ℓ → V k , ℓ G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) Π \Pi Π G L k ( F q ) × G L ℓ ( F q ) \mathrm{GL}_k(\mathbb{F}_q)\times\mathrm{GL}_{\ell}(\mathbb{F}_q) GL k ( F q ) × GL ℓ ( F q ) F q r = F q k × F q ℓ \mathbb{F}_q^r=\mathbb{F}_q^k\times\mathbb{F}_q^\ell F q r = F q k × F q ℓ
For each μ \mu μ H H H Π H \Pi_H Π H π : Col ( H ) → F q k × F q ℓ = F q r \pi:\operatorname{Col}(H)\to\mathbb{F}_q^k\times\mathbb{F}_q^\ell=\mathbb{F}_q^r π : Col ( H ) → F q k × F q ℓ = F q r π N : Col ( H ) N → V k , ℓ \pi^{\mathbb{N}}:\operatorname{Col}(H)^{\mathbb{N}}\to V^{k,\ell} π N : Col ( H ) N → V k , ℓ G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) Theorem 3.2 that Π H \Pi_H Π H Π \Pi Π Π H \Pi_H Π H ∣ Π H ∣ = ∣ Π ∣ \lvert\Pi_H\rvert=\lvert\Pi\rvert ∣ Π H ∣ = ∣ Π ∣ Π H \Pi_H Π H H H H
We denote by Sub ^ ( S L ∞ ( F q ) ) \widehat{\operatorname{Sub}}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) Sub ( SL ∞ ( F q )) Π \Pi Π Sub ( S L ∞ ( F q ) ) \operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) Sub ( SL ∞ ( F q )) μ \mu μ H H H Π H \Pi_H Π H Π \Pi Π Π H \Pi_H Π H Sub ^ ( S L ∞ ( F q ) ) \widehat{\operatorname{Sub}}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) Sub ( SL ∞ ( F q )) Sub ^ ( S L ∞ ( F q ) ) \widehat{\operatorname{Sub}}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) Sub ( SL ∞ ( F q )) μ ^ \widehat{\mu} μ μ \mu μ p : Sub ^ ( S L ∞ ( F q ) ) → Sub ( S L ∞ ( F q ) ) p:\widehat{\operatorname{Sub}}(\mathrm{SL}_{\infty}(\mathbb{F}_q))\to\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) p : Sub ( SL ∞ ( F q )) → Sub ( SL ∞ ( F q ))
The action of the group S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) Sub ( S L ∞ ( F q ) ) \operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) Sub ( SL ∞ ( F q )) Sub ^ ( S L ∞ ( F q ) ) \widehat{\operatorname{Sub}}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) Sub ( SL ∞ ( F q )) μ ^ \widehat{\mu} μ g ∈ S L ∞ ( F q ) g\in\mathrm{SL}_{\infty}(\mathbb{F}_q) g ∈ SL ∞ ( F q ) ( H , π ) ∈ Sub ^ ( S L ∞ ( F q ) ) (H,\pi)\in\widehat{\operatorname{Sub}}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) ( H , π ) ∈ Sub ( SL ∞ ( F q )) H H H μ \mu μ π ∈ Π H \pi\in\Pi_H π ∈ Π H g H g − 1 gHg^{-1} g H g − 1 H H H g g g g ⋅ ( H , π ) = ( g H g − 1 , π ′ ) g\cdot(H,\pi)=(gHg^{-1},\pi') g ⋅ ( H , π ) = ( g H g − 1 , π ′ ) π ′ : Col ( g H g − 1 ) ≅ Col ( H ) → π F q r \pi':\operatorname{Col}(gHg^{-1})\cong\operatorname{Col}(H)\xrightarrow{\pi}\mathbb{F}_q^r π ′ : Col ( g H g − 1 ) ≅ Col ( H ) π F q r π \pi π
Consider the μ ^ \widehat{\mu} μ F P : Sub ^ ( S L ∞ ( F q ) ) → V k , ℓ \mathrm{FP}:\widehat{\operatorname{Sub}}(\mathrm{SL}_{\infty}(\mathbb{F}_q))\to V^{k,\ell} FP : Sub ( SL ∞ ( F q )) → V k , ℓ μ \mu μ H H H π ∈ Π H \pi\in\Pi_H π ∈ Π H
F P ( H , π ) = π N ( color e ( H ) ) . \mathrm{FP}(H,\pi)=\pi^{\mathbb{N}}(\operatorname{color}_e(H)). FP ( H , π ) = π N ( color e ( H )) . Recall that color e ( H ) \operatorname{color}_e(H) color e ( H ) H H H e e e Theorem 1.1 that the map F P \mathrm{FP} FP H H H V k , ℓ V^{k,\ell} V k , ℓ Theorem 1.3 .
Proposition 8.4. Consider the diagram
( Sub ^ ( S L ∞ ( F q ) ) , μ ^ ) → F P ( V k , ℓ , Haar k , ℓ ) p ↓ ( Sub ( S L ∞ ( F q ) ) , μ ) \begin{CD}
(\widehat{\operatorname{Sub}}(\mathrm{SL}_{\infty}(\mathbb{F}_q)),\widehat{\mu}) @>{\mathrm{FP}}>> (V^{k,\ell},\operatorname{Haar}^{k,\ell}) \\
@V{p}VV \\
(\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)),\mu)
\end{CD} ( Sub ( SL ∞ ( F q )) , μ ) p ↓ ⏐ ( Sub ( SL ∞ ( F q )) , μ ) FP ( V k , ℓ , Haar k , ℓ ) The two maps are S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) p ∗ μ ^ = μ p_*\widehat{\mu}=\mu p ∗ μ = μ F P ∗ μ ^ = Haar k , ℓ \mathrm{FP}_*\widehat{\mu}=\operatorname{Haar}^{k,\ell} FP ∗ μ = Haar k , ℓ
Proof. Both statements for the map p p p g ∈ S L ∞ ( F q ) g \in\mathrm{SL}_{\infty}(\mathbb{F}_q) g ∈ SL ∞ ( F q ) ( H , π ) ∈ Sub ( S L ∞ ( F q ) ) (H,\pi) \in\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) ( H , π ) ∈ Sub ( SL ∞ ( F q )) H H H μ \mu μ π ∈ Π H \pi\in\Pi_H π ∈ Π H
g ⋅ F P ( H , π ) = g ⋅ π N ( color e ( H ) ) = π N ( Θ H ( g ) color e ( H ) ) = π N ( color e ( g H g − 1 ) ) = F P ( g ⋅ ( H , π ) ) . \begin{aligned}
g \cdot\mathrm{FP}(H,\pi)=g\cdot\pi^{\mathbb{N}}(\operatorname{color}_e(H))=\pi^{\mathbb{N}}(\Theta_H(g)\operatorname{color}_e(H))=\pi^{\mathbb{N}}(\operatorname{color}_e(gHg^{-1})) \\
&=\mathrm{FP}(g\cdot(H,\pi)).
\end{aligned} g ⋅ FP ( H , π ) = g ⋅ π N ( color e ( H )) = π N ( Θ H ( g ) color e ( H )) = π N ( color e ( g H g − 1 )) = FP ( g ⋅ ( H , π )) . The third equality in the above uses Theorem 6.18. This shows that the map FP is S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
The last step of the proof is to calculate the pushforward measure F P ∗ μ ^ \mathrm{FP}_*\widehat{\mu} FP ∗ μ μ \mu μ H H H color e ( H ) \operatorname{color}_e(H) color e ( H ) { color e ( H ) i : i ∈ N } = Col ( H ) \{\operatorname{color}_e(H)_i:i\in\mathbb{N}\}=\operatorname{Col}(H) { color e ( H ) i : i ∈ N } = Col ( H ) μ ^ \widehat{\mu} μ π ∈ Π H \pi\in\Pi_H π ∈ Π H π N ( color e ( H ) ) ∈ V k , ℓ \pi^{\mathbb{N}}(\operatorname{color}_e(H))\in V^{k,\ell} π N ( color e ( H )) ∈ V k , ℓ X k , ℓ ⊆ V k , ℓ X^{k,\ell}\subseteq V^{k,\ell} X k , ℓ ⊆ V k , ℓ F P ∗ μ ^ \mathrm{FP}_*\widehat{\mu} FP ∗ μ X k , ℓ X^{k,\ell} X k , ℓ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) F P ∗ μ ^ = H a a r k , ℓ \mathrm{FP}_*\widehat{\mu}=\mathrm{Haar}^{k,\ell} FP ∗ μ = Haar k , ℓ Theorem 1.3 . □ \square □
We now prove our main result.
Proof of Theorem 1.1 . To ease our notations, we write G = S L ∞ ( F q ) G=\mathrm{SL}_{\infty}(\mathbb{F}_q) G = SL ∞ ( F q ) μ ∈ IRS ( G ) \mu\in\operatorname{IRS}(G) μ ∈ IRS ( G )
( Sub ^ ( G ) , μ ^ ) → F P ( V k , ℓ , H a a r k , ℓ ) p ↓ ↓ ( Sub ( G ) , μ ) → N G ( Sub ( G ) , μ ) \begin{CD}
(\widehat{\operatorname{Sub}}(G),\widehat{\mu}) @>{\mathrm{FP}}>> (V^{k,\ell},\mathrm{Haar}^{k,\ell}) \\
@V{p}VV @VVV \\
(\operatorname{Sub}(G),\mu) @>{N_G}>> (\operatorname{Sub}(G),\mu)
\end{CD} ( Sub ( G ) , μ ) p ↓ ⏐ ( Sub ( G ) , μ ) FP N G ( V k , ℓ , Haar k , ℓ ) ↓ ⏐ ( Sub ( G ) , μ ) The objects appearing in the diagram (and in particular the indices k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 N G N_G N G N G ( H ) = H N_G(H)=H N G ( H ) = H N G ∘ p = Stab G N_G\circ p=\operatorname{Stab}_G N G ∘ p = Stab G Stab G ∘ F P = Stab G \operatorname{Stab}_G\circ\mathrm{FP}=\operatorname{Stab}_G Stab G ∘ FP = Stab G
Let us take into account the following elementary Theorem 8.5 with respect to the space Y = Sub ( G ) Y=\operatorname{Sub}(G) Y = Sub ( G ) E = { F g : g ∈ G } \mathcal{E}=\{F_g:g\in G\} E = { F g : g ∈ G } F g = { H ∈ Sub ( G ) : g ∈ H } F_g=\{H\in\operatorname{Sub}(G):g\in H\} F g = { H ∈ Sub ( G ) : g ∈ H } g ∈ G g\in G g ∈ G { H ∈ Sub ( G ) : g ∈ N G ( H ) } = { H ∈ Sub ( G ) : g ∈ H } \{H\in\operatorname{Sub}(G):g\in N_G(H)\}=\{H\in\operatorname{Sub}(G):g\in H\} { H ∈ Sub ( G ) : g ∈ N G ( H )} = { H ∈ Sub ( G ) : g ∈ H } { ( H , π ) ∈ Sub ^ ( G ) : g ∈ N G ( H ) } = { ( H , π ) ∈ Sub ^ ( G ) : g ∈ Stab G ( H , π ) } \{(H,\pi)\in\widehat{\operatorname{Sub}}(G):g\in N_G(H)\}=\{(H,\pi)\in\widehat{\operatorname{Sub}}(G):g\in\operatorname{Stab}_G(H,\pi)\} {( H , π ) ∈ Sub ( G ) : g ∈ N G ( H )} = {( H , π ) ∈ Sub ( G ) : g ∈ Stab G ( H , π )} { ( H , π ) ∈ Sub ^ ( G ) : g ∈ Stab G ( F P ( H , π ) ) } = { ( H , π ) ∈ Sub ^ ( G ) : g ∈ Stab G ( H , π ) } \{(H,\pi)\in\widehat{\operatorname{Sub}}(G):g\in\operatorname{Stab}_G(\mathrm{FP}(H,\pi))\}=\{(H,\pi)\in\widehat{\operatorname{Sub}}(G):g\in\operatorname{Stab}_G(H,\pi)\} {( H , π ) ∈ Sub ( G ) : g ∈ Stab G ( FP ( H , π ))} = {( H , π ) ∈ Sub ( G ) : g ∈ Stab G ( H , π )} μ \mu μ ⊇ \supseteq ⊇ g g g ( H , π ) (H,\pi) ( H , π ) H = p ( H , π ) H=p(H,\pi) H = p ( H , π ) g ∈ N G ( H ) g\in N_G(H) g ∈ N G ( H )
q − r ⋅ rk g = Haar k , ℓ ( { v ∈ V k , ℓ : g ∈ Stab G ( v ) } ) = μ ^ ( { ( H , π ) ∈ Sub ^ ( G ) : g ∈ Stab G ( F P ( H , π ) ) } ) ≥ μ ^ ( { ( H , π ) ∈ Sub ^ ( G ) : g ∈ Stab G ( H , π ) } ) = μ ^ ( { ( H , π ) ∈ Sub ^ ( G ) : g ∈ N G ( H ) } ) = μ ( { H ∈ Sub ( G ) : g ∈ N G ( H ) } ) ≥ μ ( { H ∈ Sub ( G ) : g ∈ H } ) = q − r ⋅ rk g . \begin{aligned}
q^{-r\cdot\operatorname{rk}g}=\operatorname{Haar}^{k,\ell}\bigl(\{v\in V^{k,\ell}:g\in\operatorname{Stab}_G(v)\}\bigr)=\widehat{\mu}\bigl(\{(H,\pi)\in\widehat{\operatorname{Sub}}(G):g\in\operatorname{Stab}_G(\mathrm{FP}(H,\pi))\}\bigr) \\
&\geq\widehat{\mu}\bigl(\{(H,\pi)\in\widehat{\operatorname{Sub}}(G):g\in\operatorname{Stab}_G(H,\pi)\}\bigr)=\widehat{\mu}\bigl(\{(H,\pi)\in\widehat{\operatorname{Sub}}(G):g\in N_G(H)\}\bigr) \\
&=\mu\bigl(\{H\in\operatorname{Sub}(G):g\in N_G(H)\}\bigr)\geq\mu\bigl(\{H\in\operatorname{Sub}(G):g\in H\}\bigr)=q^{-r\cdot\operatorname{rk}g}.
\end{aligned} q − r ⋅ rk g = Haar k , ℓ ( { v ∈ V k , ℓ : g ∈ Stab G ( v )} ) = μ ( {( H , π ) ∈ Sub ( G ) : g ∈ Stab G ( FP ( H , π ))} ) ≥ μ ( {( H , π ) ∈ Sub ( G ) : g ∈ Stab G ( H , π )} ) = μ ( {( H , π ) ∈ Sub ( G ) : g ∈ N G ( H )} ) = μ ( { H ∈ Sub ( G ) : g ∈ N G ( H )} ) ≥ μ ( { H ∈ Sub ( G ) : g ∈ H } ) = q − r ⋅ rk g . The first equality follows from Theorem 8.1. The second equality on the first line and the first on the bottom line follow from Theorem 8.4. The last equality follows from Theorem 6.8. Equality must hold throughout. The remaining statements (1′) and (3′) follow and the above diagram commutes. To conclude the proof
μ k , ℓ = ( Stab G ) ∗ Haar k , ℓ = ( Stab G ∘ FP ) ∗ μ ^ = ( Stab G ) ∗ μ ^ = ( N G ∘ p ) ∗ μ ^ = ( N G ) ∗ μ = μ \mu_{k,\ell}=(\operatorname{Stab}_{G})_{*}\operatorname{Haar}^{k,\ell}=(\operatorname{Stab}_{G}\circ\operatorname{FP})_{*}\widehat{\mu}=(\operatorname{Stab}_{G})_{*}\widehat{\mu}=(N_{G}\circ p)_{*}\widehat{\mu}=(N_{G})_{*}\mu=\mu μ k , ℓ = ( Stab G ) ∗ Haar k , ℓ = ( Stab G ∘ FP ) ∗ μ = ( Stab G ) ∗ μ = ( N G ∘ p ) ∗ μ = ( N G ) ∗ μ = μ as required. □
The following elementary measure theoretic lemma was used in the above proof of Theorem 1.1 .
Lemma 8.5. Let ( X , A X , μ ) (X,\mathcal{A}_{X},\mu) ( X , A X , μ ) ( Y , A Y ) (Y,\mathcal{A}_{Y}) ( Y , A Y ) E ⊆ A Y \mathcal{E}\subseteq\mathcal{A}_{Y} E ⊆ A Y Y Y Y f , g : X → Y f,g:X\to Y f , g : X → Y f − 1 ( E ) = g − 1 ( E ) f^{-1}(E)=g^{-1}(E) f − 1 ( E ) = g − 1 ( E ) μ \mu μ E ∈ E E\in\mathcal{E} E ∈ E f f f g g g μ \mu μ
Proof. The statement follows by noting that
μ ( { x ∈ X : f ( x ) ≠ g ( x ) } ) = μ ( ⋃ E ∈ E ( f − 1 ( E ) △ g − 1 ( E ) ) ) ≤ ∑ E ∈ E μ ( f − 1 ( E ) △ g − 1 ( E ) ) = 0 \mu\bigl(\{x\in X:f(x)\ne g(x)\}\bigr)=\mu\left(\bigcup_{E\in\mathcal{E}}\left(f^{-1}(E)\mathbin{\triangle}g^{-1}(E)\right)\right)\leq\sum_{E\in\mathcal{E}}\mu\left(f^{-1}(E)\mathbin{\triangle}g^{-1}(E)\right)=0 μ ( { x ∈ X : f ( x ) = g ( x )} ) = μ ( E ∈ E ⋃ ( f − 1 ( E ) △ g − 1 ( E ) ) ) ≤ E ∈ E ∑ μ ( f − 1 ( E ) △ g − 1 ( E ) ) = 0 where the first equality comes from the hypothesis that E \mathcal{E} E
Invariant random subgroups of GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_{q}) GL ∞ ( F q )
We will show that every non-trivial ergodic invariant random subgroup of the group GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_{q}) GL ∞ ( F q )
μ k , ℓ K = ( Stab GL ∞ ( F q ) K ) ∗ Haar k , ℓ \mu^{K}_{k,\ell}=\left(\operatorname{Stab}^{K}_{\operatorname{GL}_{\infty}(\mathbb{F}_{q})}\right)_{*}\operatorname{Haar}^{k,\ell} μ k , ℓ K = ( Stab GL ∞ ( F q ) K ) ∗ Haar k , ℓ where k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_{0} k , ℓ ∈ N 0 K ≤ F q × K\leq\mathbb{F}_{q}^{\times} K ≤ F q × Stab GL ∞ ( F q ) K : V k , ℓ → Sub ( GL ∞ ( F q ) ) \operatorname{Stab}^{K}_{\operatorname{GL}_{\infty}(\mathbb{F}_{q})}:\mathbb{V}^{k,\ell}\to\operatorname{Sub}(\operatorname{GL}_{\infty}(\mathbb{F}_{q})) Stab GL ∞ ( F q ) K : V k , ℓ → Sub ( GL ∞ ( F q ))
Stab GL ∞ ( F q ) K ( x ) = { A ∈ GL ∞ ( F q ) : A ⋅ x = x and det A ∈ K } . \operatorname{Stab}^{K}_{\operatorname{GL}_{\infty}(\mathbb{F}_{q})}(x)=\{A\in\operatorname{GL}_{\infty}(\mathbb{F}_{q}):A\cdot x=x\ \text{and}\ \det A\in K\}. Stab GL ∞ ( F q ) K ( x ) = { A ∈ GL ∞ ( F q ) : A ⋅ x = x and det A ∈ K } . As the maps Stab GL ∞ ( F q ) K \operatorname{Stab}^{K}_{\operatorname{GL}_{\infty}(\mathbb{F}_{q})} Stab GL ∞ ( F q ) K GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_{q}) GL ∞ ( F q ) μ k , ℓ K \mu^{K}_{k,\ell} μ k , ℓ K
Topology and Vershik traces of the μ k , ℓ K \mu^{K}_{k,\ell} μ k , ℓ K
We return to a discussion parallel to Section 8.1.1 for the group GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_{q}) GL ∞ ( F q ) GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_{q}) GL ∞ ( F q ) F q × ^ \widehat{\mathbb{F}_{q}^{\times}} F q × F q × \mathbb{F}_{q}^{\times} F q × τ ∈ F q × ^ \tau\in\widehat{\mathbb{F}_{q}^{\times}} τ ∈ F q × r ∈ N 0 r\in\mathbb{N}_{0} r ∈ N 0 χ r τ : GL ∞ ( F q ) → C \chi^{\tau}_{r}:\operatorname{GL}_{\infty}(\mathbb{F}_{q})\to\mathbb{C} χ r τ : GL ∞ ( F q ) → C
χ r τ ( g ) = τ ( det g ) ⋅ q − r ⋅ rk ~ g , \chi^{\tau}_{r}(g)=\tau(\det g)\cdot q^{-r\cdot\widetilde{\operatorname{rk}}g}, χ r τ ( g ) = τ ( det g ) ⋅ q − r ⋅ rk g , where rk ~ g = rk ( g − I ) \widetilde{\operatorname{rk}}g=\operatorname{rk}(g-I) rk g = rk ( g − I ) χ ∞ = δ I \chi_{\infty}=\delta_{I} χ ∞ = δ I
Theorem 8.6 [27 ] . The characters of the finitary general linear group GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_{q}) GL ∞ ( F q )
Ch ( GL ∞ ( F q ) ) = { χ r τ : r ∈ N 0 , τ ∈ F q × ^ } ∪ { χ ∞ } . \operatorname{Ch}(\operatorname{GL}_{\infty}(\mathbb{F}_{q}))=\left\{\chi^{\tau}_{r}:r\in\mathbb{N}_{0},\ \tau\in\widehat{\mathbb{F}_{q}^{\times}}\right\}\cup\{\chi_{\infty}\}. Ch ( GL ∞ ( F q )) = { χ r τ : r ∈ N 0 , τ ∈ F q × } ∪ { χ ∞ } . Given a subgroup K ≤ F q × K\leq\mathbb{F}_{q}^{\times} K ≤ F q × F q × ^ / K \widehat{\mathbb{F}_{q}^{\times}}/K F q × / K F q × \mathbb{F}_{q}^{\times} F q × 1 1 1 K K K
Proposition 8.7. Fix k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 K ≤ F q × K \leq\mathbb{F}_q^\times K ≤ F q × μ k , ℓ K \mu^K_{k,\ell} μ k , ℓ K
χ μ k , ℓ K ( g ) = ∣ K ∣ q − 1 ∑ τ ∈ F q × / K χ k , ℓ τ ( g ) = { q − ( k + ℓ ) ⋅ rk ~ g det g ∈ K 0 det g ∉ K \chi_{\mu^K_{k,\ell}}(g)=\frac{|K|}{q-1}\sum_{\tau\in\mathbb{F}_q^\times/K}\chi^\tau_{k,\ell}(g)=
\begin{cases}
q^{-(k+\ell)\cdot\widetilde{\operatorname{rk}}g} & \det g\in K\\
0 & \det g\notin K
\end{cases} χ μ k , ℓ K ( g ) = q − 1 ∣ K ∣ τ ∈ F q × / K ∑ χ k , ℓ τ ( g ) = { q − ( k + ℓ ) ⋅ rk g 0 det g ∈ K det g ∈ / K for all g ∈ G g \in G g ∈ G
Proof. If det g ∉ K \det g\notin K det g ∈ / K μ k , ℓ K ( { H : g ∈ H } ) = 0 \mu^K_{k,\ell}(\{H:g\in H\})=0 μ k , ℓ K ({ H : g ∈ H }) = 0 det g ∈ K \det g\in K det g ∈ K τ ∈ F q × / K \tau\in\mathbb{F}_q^\times/K τ ∈ F q × / K
Unlike the case of the group SL ∞ ( F q ) \operatorname{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) tr : IRS ( GL ∞ ( F q ) ) → Tr ( GL ∞ ( F q ) ) \operatorname{tr}:\operatorname{IRS}(\operatorname{GL}_{\infty}(\mathbb{F}_q))\to\operatorname{Tr}(\operatorname{GL}_{\infty}(\mathbb{F}_q)) tr : IRS ( GL ∞ ( F q )) → Tr ( GL ∞ ( F q )) A ∞ A_{\infty} A ∞
The topology of the space M GL = { μ k , ℓ K : k , ℓ ∈ N 0 , K ≤ F q × } ∪ { μ ∞ } M_{\operatorname{GL}}=\{\mu^K_{k,\ell}:k,\ell\in\mathbb{N}_0,\ K\leq\mathbb{F}_q^\times\}\cup\{\mu_\infty\} M GL = { μ k , ℓ K : k , ℓ ∈ N 0 , K ≤ F q × } ∪ { μ ∞ } Theorem 1.2 is established, it will follow from it that M GL M_{\operatorname{GL}} M GL GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q )
Proposition 8.8. Let X = ( N 0 × N 0 × Sub ( F q × ) ) ∪ { ∞ } X=(\mathbb{N}_0\times\mathbb{N}_0\times\operatorname{Sub}(\mathbb{F}_q^\times))\cup\{\infty\} X = ( N 0 × N 0 × Sub ( F q × )) ∪ { ∞ } N 0 × N 0 × Sub ( F q × ) \mathbb{N}_0\times\mathbb{N}_0\times\operatorname{Sub}(\mathbb{F}_q^\times) N 0 × N 0 × Sub ( F q × ) X → M GL X\to M_{\operatorname{GL}} X → M GL ( k , ℓ , K ) (k,\ell,K) ( k , ℓ , K ) μ k , ℓ K \mu^K_{k,\ell} μ k , ℓ K ∞ \infty ∞ μ ∞ \mu_\infty μ ∞
Proof. The proof is entirely analogous to that of Theorem 8.2.
Classification of invariant random subgroups of GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q )
We will need a basic linear-algebraic property of the product action on the space V k , ℓ V^{k,\ell} V k , ℓ
Lemma 8.9. Fix k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 x = ( v , φ ) ∈ V k , ℓ x=(v,\varphi)\in V^{k,\ell} x = ( v , φ ) ∈ V k , ℓ w ∈ V w\in V w ∈ V Stab SL ∞ ( F q ) ( x ) \operatorname{Stab}_{\operatorname{SL}_{\infty}(\mathbb{F}_q)}(x) Stab SL ∞ ( F q ) ( x ) w ∈ span { v 1 , … , v k } w\in\operatorname{span}\{v^1,\ldots,v^k\} w ∈ span { v 1 , … , v k } ψ ∈ V ^ \psi\in\widehat V ψ ∈ V Stab SL ∞ ( F q ) ( x ) \operatorname{Stab}_{\operatorname{SL}_{\infty}(\mathbb{F}_q)}(x) Stab SL ∞ ( F q ) ( x ) ψ ∈ span { φ 1 , … , φ ℓ } \psi\in\operatorname{span}\{\varphi^1,\ldots,\varphi^\ell\} ψ ∈ span { φ 1 , … , φ ℓ }
Proof. The two statements are dual to each other, and we will only show the first. For each n ∈ N n\in\mathbb{N} n ∈ N V ^ n \widehat V_n V n V n V_n V n η ∈ V ^ n \eta\in\widehat V_n η ∈ V n u ∈ V n u\in V_n u ∈ V n η u ∈ F q \eta u\in\mathbb{F}_q η u ∈ F q ⟨ u , η ⟩ \langle u,\eta\rangle ⟨ u , η ⟩
We argue by contraposition. Let w ∉ span { v 1 , … , v k } w\notin\operatorname{span}\{v^1,\ldots,v^k\} w ∈ / span { v 1 , … , v k } n > ℓ + 1 n>\ell+1 n > ℓ + 1 w ≤ n ∉ U n w_{\leq n}\notin U_n w ≤ n ∈ / U n U n = span { v ≤ n 1 , … , v ≤ n k } ≤ V n U_n=\operatorname{span}\{v^1_{\leq n},\ldots,v^k_{\leq n}\}\leq V_n U n = span { v ≤ n 1 , … , v ≤ n k } ≤ V n η ∈ V ^ n \eta\in\widehat V_n η ∈ V n U n U_n U n w ≤ n w_{\leq n} w ≤ n n > ℓ + 1 n>\ell+1 n > ℓ + 1 u ∈ V n u\in V_n u ∈ V n ℓ + 1 \ell+1 ℓ + 1
φ ≤ n j u = 0 for all j = 1 , … , ℓ and η u = 0. \varphi^j_{\leq n}u=0\text{ for all }j=1,\ldots,\ell\text{ and }\eta u=0. φ ≤ n j u = 0 for all j = 1 , … , ℓ and η u = 0. Let τ = I + u η ∈ GL n ( F q ) \tau=I+u\eta\in\operatorname{GL}_n(\mathbb{F}_q) τ = I + u η ∈ GL n ( F q ) τ \tau τ GL ∞ ( F q ) \operatorname{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) GL n ( F q ) ≤ GL ∞ ( F q ) \operatorname{GL}_n(\mathbb{F}_q)\leq\operatorname{GL}_{\infty}(\mathbb{F}_q) GL n ( F q ) ≤ GL ∞ ( F q )
( τ − I ) 2 = u ( η u ) η = 0. (\tau-I)^2=u(\eta u)\eta=0. ( τ − I ) 2 = u ( η u ) η = 0. Therefore τ \tau τ 1 1 1 1 ≤ i ≤ k 1\leq i\leq k 1 ≤ i ≤ k 1 ≤ j ≤ ℓ 1\leq j\leq\ell 1 ≤ j ≤ ℓ
τ v i = v i and φ j τ = φ j . \tau v^i=v^i\qquad\text{and}\qquad\varphi^j\tau=\varphi^j. τ v i = v i and φ j τ = φ j . Hence τ ∈ Stab S L ∞ ( F q ) ( x ) \tau\in\operatorname{Stab}_{\mathrm{SL}_{\infty}(\mathbb{F}_q)}(x) τ ∈ Stab SL ∞ ( F q ) ( x )
τ w = w + u ( η w ≤ n ) ≠ w , \tau w = w + u(\eta w_{\leq n}) \ne w, τ w = w + u ( η w ≤ n ) = w , which means that τ \tau τ w w w
We extend our main result to the finitary general linear group.
Proof of Theorem 1.2 . Let μ \mu μ G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) μ \mu μ Sub ( G L ∞ ( F q ) ) → Sub ( S L ∞ ( F q ) ) \operatorname{Sub}(\mathrm{GL}_{\infty}(\mathbb{F}_q)) \to\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) Sub ( GL ∞ ( F q )) → Sub ( SL ∞ ( F q )) μ 0 \mu_0 μ 0 S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) Sub ( S L ∞ ( F q ) ) \operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) Sub ( SL ∞ ( F q )) μ 0 \mu_0 μ 0 Sub ( G L ∞ ( F q ) ) → Sub ( S L ∞ ( F q ) ) \operatorname{Sub}(\mathrm{GL}_{\infty}(\mathbb{F}_q)) \to\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) Sub ( GL ∞ ( F q )) → Sub ( SL ∞ ( F q )) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q )
For any pair of indices k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 Stab S L ∞ ( F q ) : V k , ℓ → Sub ( S L ∞ ( F q ) ) \operatorname{Stab}_{\mathrm{SL}_{\infty}(\mathbb{F}_q)}: V^{k,\ell} \to\operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) Stab SL ∞ ( F q ) : V k , ℓ → Sub ( SL ∞ ( F q )) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) Haar k , ℓ \operatorname{Haar}^{k,\ell} Haar k , ℓ G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) μ k , ℓ ∈ IRS ( S L ∞ ( F q ) ) \mu_{k,\ell} \in\operatorname{IRS}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) μ k , ℓ ∈ IRS ( SL ∞ ( F q )) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) μ ∞ = δ { I } \mu_\infty= \delta_{\{I\}} μ ∞ = δ { I } Theorem 1.1 we know that μ k , ℓ \mu_{k,\ell} μ k , ℓ μ ∞ \mu_\infty μ ∞ S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) Sub ( S L ∞ ( F q ) ) \operatorname{Sub}(\mathrm{SL}_{\infty}(\mathbb{F}_q)) Sub ( SL ∞ ( F q )) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) μ 0 \mu_0 μ 0 μ k , ℓ \mu_{k,\ell} μ k , ℓ k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 δ { I } \delta_{\{I\}} δ { I }
In case μ 0 = δ { I } \mu_0 = \delta_{\{I\}} μ 0 = δ { I } μ = δ { I } \mu= \delta_{\{I\}} μ = δ { I } S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q ) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) μ 0 = δ { I } \mu_0 = \delta_{\{I\}} μ 0 = δ { I } μ \mu μ G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) μ \mu μ μ = δ { I } \mu= \delta_{\{I\}} μ = δ { I }
We consider the remaining case, i.e. there are k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 μ 0 = μ k , ℓ \mu_0 = \mu_{k,\ell} μ 0 = μ k , ℓ det : Sub ( G L ∞ ( F q ) ) → Sub ( F q × ) \det: \operatorname{Sub}(\mathrm{GL}_{\infty}(\mathbb{F}_q)) \to\operatorname{Sub}(\mathbb{F}_q^\times) det : Sub ( GL ∞ ( F q )) → Sub ( F q × ) G L ∞ ( F q ) \mathrm{GL}_{\infty}(\mathbb{F}_q) GL ∞ ( F q ) μ \mu μ K ≤ F q × K \leq\mathbb{F}_q^\times K ≤ F q × μ = μ k , ℓ K \mu= \mu^K_{k,\ell} μ = μ k , ℓ K
The fact that μ 0 = μ k , ℓ \mu_0 = \mu_{k,\ell} μ 0 = μ k , ℓ μ \mu μ H ∈ Sub ( G L ∞ ( F q ) ) H \in\operatorname{Sub}(\mathrm{GL}_{\infty}(\mathbb{F}_q)) H ∈ Sub ( GL ∞ ( F q )) x = ( v , φ ) ∈ V k , ℓ x = (v,\varphi) \in V^{k,\ell} x = ( v , φ ) ∈ V k , ℓ
H ∩ S L ∞ ( F q ) = Stab S L ∞ ( F q ) ( x ) = { g ∈ S L ∞ ( F q ) : g ⋅ x = x } . (27) H \cap\mathrm{SL}_{\infty}(\mathbb{F}_q) = \operatorname{Stab}_{\mathrm{SL}_{\infty}(\mathbb{F}_q)}(x) = \{g \in\mathrm{SL}_{\infty}(\mathbb{F}_q) : g \cdot x = x\}. \tag*{(27)} H ∩ SL ∞ ( F q ) = Stab SL ∞ ( F q ) ( x ) = { g ∈ SL ∞ ( F q ) : g ⋅ x = x } . (27) As the property of regular linear independence (introduced in Theorem 7.5) is Haar k , ℓ \operatorname{Haar}^{k,\ell} Haar k , ℓ v v v φ \varphi φ H ∈ Sub ( G L ∞ ( F q ) ) H \in\operatorname{Sub}(\mathrm{GL}_{\infty}(\mathbb{F}_q)) H ∈ Sub ( GL ∞ ( F q ))
H = Stab G L ∞ ( F q ) K ( x ) = { A ∈ G L ∞ ( F q ) : A ⋅ x = x and det A ∈ K } . H = \operatorname{Stab}^{K}_{\mathrm{GL}_{\infty}(\mathbb{F}_q)}(x) = \{A \in\mathrm{GL}_{\infty}(\mathbb{F}_q) : A \cdot x = x \text{ and } \det A \in K\}. H = Stab GL ∞ ( F q ) K ( x ) = { A ∈ GL ∞ ( F q ) : A ⋅ x = x and det A ∈ K } . We show containment in one direction. Consider a matrix A ∈ H A \in H A ∈ H B ∈ H B \in H B ∈ H det B = det A \det B = \det A det B = det A B − 1 A ∈ H ∩ S L ∞ ( F q ) B^{-1}A \in H \cap\mathrm{SL}_{\infty}(\mathbb{F}_q) B − 1 A ∈ H ∩ SL ∞ ( F q ) B − 1 A B^{-1}A B − 1 A x x x
A ⋅ x = B ⋅ x . A \cdot x = B \cdot x. A ⋅ x = B ⋅ x . Applying this observation to the conjugate matrices B = g A g − 1 B = gAg^{-1} B = g A g − 1 g ∈ H ∩ S L ∞ ( F q ) g \in H \cap\mathrm{SL}_{\infty}(\mathbb{F}_q) g ∈ H ∩ SL ∞ ( F q )
A ⋅ x = g A g − 1 ⋅ x = g ⋅ ( A ⋅ x ) . A \cdot x = gAg^{-1} \cdot x = g \cdot(A \cdot x). A ⋅ x = g A g − 1 ⋅ x = g ⋅ ( A ⋅ x ) . In other words, the tuple A ⋅ x A \cdot x A ⋅ x H ∩ S L ∞ ( F q ) H \cap\mathrm{SL}_{\infty}(\mathbb{F}_q) H ∩ SL ∞ ( F q ) Theorem 8.9 we deduce that A v i ∈ span { v 1 , … , v k } Av^i \in\operatorname{span}\{v^1,\ldots,v^k\} A v i ∈ span { v 1 , … , v k } ( A t ) − 1 φ i ∈ span { φ 1 , … , φ ℓ } (A^t)^{-1}\varphi^i \in\operatorname{span}\{\varphi^1,\ldots,\varphi^\ell\} ( A t ) − 1 φ i ∈ span { φ 1 , … , φ ℓ } 1 ≤ i ≤ k 1 \le i \le k 1 ≤ i ≤ k A v i = ∑ j = 1 k α j v j A v^{i}=\sum_{j=1}^{k}\alpha_{j}v^{j} A v i = ∑ j = 1 k α j v j α j ∈ F q \alpha_{j}\in\mathbb{F}_{q} α j ∈ F q m ∈ N m\in\mathbb{N} m ∈ N A ∈ SL m ( F q ) A\in\operatorname{SL}_{m}(\mathbb{F}_{q}) A ∈ SL m ( F q )
v > m i = ( A v i ) > m = ∑ j = 1 k α j v > m j . v^{i}_{>m}=(Av^{i})_{>m}=\sum_{j=1}^{k}\alpha_{j}v^{j}_{>m}. v > m i = ( A v i ) > m = j = 1 ∑ k α j v > m j . As the tuple x x x ( v > m j ) j = 1 k (v^{j}_{>m})_{j=1}^{k} ( v > m j ) j = 1 k α j = δ i j \alpha_{j}=\delta_{ij} α j = δ ij A v i = v i Av^{i}=v^{i} A v i = v i φ i \varphi^{i} φ i A ⋅ x = x A\cdot x=x A ⋅ x = x
Lastly, certainly det A ∈ K \det A\in K det A ∈ K K K K
We show containment in the opposite direction. Let A ∈ GL ∞ ( F q ) A\in\operatorname{GL}_{\infty}(\mathbb{F}_{q}) A ∈ GL ∞ ( F q ) x x x det A ∈ K \det A\in K det A ∈ K B ∈ H B\in H B ∈ H det B = det A \det B=\det A det B = det A B B B det ( H ) = K \det(H)=K det ( H ) = K B B B x x x B − 1 A B^{-1}A B − 1 A x x x B − 1 A ∈ H ∩ SL ∞ ( F q ) B^{-1}A\in H\cap\operatorname{SL}_{\infty}(\mathbb{F}_{q}) B − 1 A ∈ H ∩ SL ∞ ( F q ) A = B ⋅ ( B − 1 A ) ∈ H A=B\cdot(B^{-1}A)\in H A = B ⋅ ( B − 1 A ) ∈ H
To conclude, consider the μ 0 \mu_{0} μ 0 Ψ : Sub ( SL ∞ ( F q ) ) → Sub ( GL ∞ ( F q ) ) \Psi:\operatorname{Sub}(\operatorname{SL}_{\infty}(\mathbb{F}_{q}))\to\operatorname{Sub}(\operatorname{GL}_{\infty}(\mathbb{F}_{q})) Ψ : Sub ( SL ∞ ( F q )) → Sub ( GL ∞ ( F q )) Stab SL ∞ ( F q ) ( x ) \operatorname{Stab}_{\operatorname{SL}_{\infty}(\mathbb{F}_{q})}(x) Stab SL ∞ ( F q ) ( x ) Stab GL ∞ ( F q ) K ( x ) \operatorname{Stab}^{K}_{\operatorname{GL}_{\infty}(\mathbb{F}_{q})}(x) Stab GL ∞ ( F q ) K ( x ) Ψ \Psi Ψ x = ( v , φ ) x=(v,\varphi) x = ( v , φ ) Stab SL ∞ ( F q ) ( x ) \operatorname{Stab}_{\operatorname{SL}_{\infty}(\mathbb{F}_{q})}(x) Stab SL ∞ ( F q ) ( x ) v v v φ \varphi φ Theorem 8.9 ). The above discussion shows that H = Ψ ( H ∩ SL ∞ ( F q ) ) H=\Psi(H\cap\operatorname{SL}_{\infty}(\mathbb{F}_{q})) H = Ψ ( H ∩ SL ∞ ( F q )) μ \mu μ H ∈ Sub ( GL ∞ ( F q ) ) H\in\operatorname{Sub}(\operatorname{GL}_{\infty}(\mathbb{F}_{q})) H ∈ Sub ( GL ∞ ( F q )) μ = Ψ ∗ μ 0 \mu=\Psi_{*}\mu_{0} μ = Ψ ∗ μ 0 μ 0 \mu_{0} μ 0
μ = ( Ψ ∘ Stab SL ∞ ( F q ) ) ∗ Haar k , ℓ = ( Stab GL ∞ ( F q ) K ) ∗ Haar k , ℓ = μ k , ℓ K \mu=(\Psi\circ\operatorname{Stab}_{\operatorname{SL}_{\infty}(\mathbb{F}_{q})})_{*}\operatorname{Haar}^{k,\ell}=(\operatorname{Stab}^{K}_{\operatorname{GL}_{\infty}(\mathbb{F}_{q})})_{*}\operatorname{Haar}^{k,\ell}=\mu^{K}_{k,\ell} μ = ( Ψ ∘ Stab SL ∞ ( F q ) ) ∗ Haar k , ℓ = ( Stab GL ∞ ( F q ) K ) ∗ Haar k , ℓ = μ k , ℓ K as required. □
Permutation stability of elementary enrichment
In this section we give an explicit realization of the diagonal product ⨂ i PSL n i ( F q ) \bigotimes_{i}\operatorname{PSL}_{n_i}(\mathbb{F}_{q}) ⨂ i PSL n i ( F q ) Theorem 1.7 saying this diagonal product is permutation stable.
Permutation stability
We recall the definition of permutation stability. For each n ∈ N n\in\mathbb{N} n ∈ N S n S_n S n n n n d n d_n d n
d n ( σ , τ ) = 1 − 1 n ∣ Fix ( σ − 1 τ ) ∣ ∀ σ , τ ∈ S n . d_n(\sigma,\tau)=1-\frac{1}{n}\left|\operatorname{Fix}(\sigma^{-1}\tau)\right|\qquad\forall\sigma,\tau\in S_n. d n ( σ , τ ) = 1 − n 1 Fix ( σ − 1 τ ) ∀ σ , τ ∈ S n . Let G G G almost-action of G G G f n : G → S n f_n:G\to S_n f n : G → S n
d n ( f n ( g ) f n ( h ) , f n ( g h ) ) → n → ∞ 0 ∀ g , h ∈ G . d_n(f_n(g)f_n(h),f_n(gh))\xrightarrow[n\to\infty]{}0\qquad\forall g,h\in G. d n ( f n ( g ) f n ( h ) , f n ( g h )) n → ∞ 0 ∀ g , h ∈ G . Such an almost-action f n f_n f n close to an action if there is a sequence of group homomorphisms ρ n : G → S n \rho_n:G\to S_n ρ n : G → S n
d n ( ρ n ( g ) , f n ( g ) ) → n → ∞ 0 ∀ g ∈ G . d_n(\rho_n(g),f_n(g))\xrightarrow[n\to\infty]{}0\qquad\forall g\in G. d n ( ρ n ( g ) , f n ( g )) n → ∞ 0 ∀ g ∈ G . Definition 9.1. The group G G G G G G
For finitely generated amenable groups, permutation stability can be understood in terms of properties of the space of invariant random subgroups; see Theorem 1.6 in the introduction by Becker, Lubotzky and Thom. This is our main motivation in this section.
Diagonal products
We give an explicit realization of the diagonal product group ⨂ i P S L n i ( F q ) \bigotimes_i \mathrm{PSL}_{n_i}(\mathbb{F}_q) ⨂ i PSL n i ( F q ) ∏ P S L n i ( F q ) \prod\mathrm{PSL}_{n_i}(\mathbb{F}_q) ∏ PSL n i ( F q ) [23 ] to make the parallel with the proof there as transparent as possible. In particular, these diagonal products will be denoted G ( P ) G(\mathcal{P}) G ( P )
Let S L Z ( F q ) \mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) SL Z ( F q ) F q \mathbb{F}_q F q Z \mathbb{Z} Z S L ∞ ( F q ) \mathrm{SL}_{\infty}(\mathbb{F}_q) SL ∞ ( F q )
For each subset I ⊆ Z I\subseteq\mathbb{Z} I ⊆ Z P S L I ( F q ) \mathrm{PSL}_{I}(\mathbb{F}_q) PSL I ( F q ) I I I r ∈ N r\in\mathbb{N} r ∈ N [ r ] = { − r , … , + r } ⊂ N [r]=\{-r,\ldots,+r\}\subset\mathbb{N} [ r ] = { − r , … , + r } ⊂ N r ∈ N r\in\mathbb{N} r ∈ N τ r + ( a ) , τ r − ( a ) ∈ P S L [ r ] ( F q ) \tau_r^+(a),\tau_r^-(a)\in\mathrm{PSL}_{[r]}(\mathbb{F}_q) τ r + ( a ) , τ r − ( a ) ∈ PSL [ r ] ( F q ) a ∈ F q a\in\mathbb{F}_q a ∈ F q ( 0 , 1 ) (0,1) ( 0 , 1 ) ( 1 , 0 ) (1,0) ( 1 , 0 ) r ∈ N r\in\mathbb{N} r ∈ N σ r ∈ P S L [ r ] ( F q ) \sigma_r\in\mathrm{PSL}_{[r]}(\mathbb{F}_q) σ r ∈ PSL [ r ] ( F q ) ( − r , … , r ) (-r,\ldots,r) ( − r , … , r )
Definition 9.2. Let P = ( n i ) \mathcal{P}=(n_i) P = ( n i ) n 1 ≥ 3 n_1\geq3 n 1 ≥ 3 i ∈ N i\in\mathbb{N} i ∈ N n i = 2 r i + 1 n_i=2r_i+1 n i = 2 r i + 1
P S L ( P ) = ∏ i ∈ N P S L [ r i ] ( F q ) . \mathrm{PSL}(\mathcal{P})=\prod_{i\in\mathbb{N}}\mathrm{PSL}_{[r_i]}(\mathbb{F}_q). PSL ( P ) = i ∈ N ∏ PSL [ r i ] ( F q ) . The group G ( P ) G(\mathcal{P}) G ( P ) P S L ( P ) \mathrm{PSL}(\mathcal{P}) PSL ( P ) σ \sigma σ τ ± ( a ) \tau^\pm(a) τ ± ( a ) a ∈ F q a\in\mathbb{F}_q a ∈ F q τ ± ( a ) i = τ r i ± ( a ) ∈ P S L [ r i ] ( F q ) \tau^\pm(a)_i=\tau_{r_i}^\pm(a)\in\mathrm{PSL}_{[r_i]}(\mathbb{F}_q) τ ± ( a ) i = τ r i ± ( a ) ∈ PSL [ r i ] ( F q ) σ i = σ r i ∈ P S L [ r i ] ( F q ) \sigma_i=\sigma_{r_i}\in\mathrm{PSL}_{[r_i]}(\mathbb{F}_q) σ i = σ r i ∈ PSL [ r i ] ( F q ) i ∈ N i\in\mathbb{N} i ∈ N
The group G ( P ) G(\mathcal{P}) G ( P ) r ∈ N r\in\mathbb{N} r ∈ N P S L ( [ r ] ) \mathrm{PSL}([r]) PSL ([ r ]) σ r \sigma_r σ r τ r + ( a ) \tau_r^+(a) τ r + ( a ) a ∈ F q a\in\mathbb{F}_q a ∈ F q G ( P ) G(\mathcal{P}) G ( P ) P S L ( P ) \mathrm{PSL}(\mathcal{P}) PSL ( P ) i ∈ N i\in\mathbb{N} i ∈ N L i ( P ) L_i(\mathcal{P}) L i ( P ) G ( P ) G(\mathcal{P}) G ( P )
L i ( P ) = ⟨ τ + ( a ) σ j , τ − ( a ) σ j | j ∈ [ r i − 1 ] , a ∈ F q × ⟩ . L_i(\mathcal{P})=\left\langle\tau^+(a)^{\sigma^j},\tau^-(a)^{\sigma^j}\ \middle|\ j\in[r_i-1],\,a\in\mathbb{F}_q^\times\right\rangle. L i ( P ) = ⟨ τ + ( a ) σ j , τ − ( a ) σ j j ∈ [ r i − 1 ] , a ∈ F q × ⟩ . The locally finite group N ( P ) N(\mathcal{P}) N ( P )
N ( P ) = ⋃ i ∈ N L i ( P ) N(\mathcal{P})=\bigcup_{i\in\mathbb{N}}L_i(\mathcal{P}) N ( P ) = i ∈ N ⋃ L i ( P ) is the commutator subgroup of G ( P ) G(\mathcal{P}) G ( P ) G ( P ) G(\mathcal{P}) G ( P ) τ ± ( a ) \tau^\pm(a) τ ± ( a ) a ∈ F q a\in\mathbb{F}_q a ∈ F q
Let σ ˉ \bar{\sigma} σ ˉ S L Z ( F q ) \mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) SL Z ( F q ) x ↦ x + 1 x\mapsto x+1 x ↦ x + 1 x ∈ Z x\in\mathbb{Z} x ∈ Z
V = Z ⋉ S L Z ( F q ) V=\mathbb{Z}\ltimes\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) V = Z ⋉ SL Z ( F q ) where Z ≅ ⟨ σ ˉ ⟩ \mathbb{Z}\cong\langle\bar{\sigma}\rangle Z ≅ ⟨ σ ˉ ⟩ S L Z ( F q ) \mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) SL Z ( F q ) σ ˉ \bar{\sigma} σ ˉ t ~ : G ( P ) → V \tilde{t}:G(\mathcal{P})\to V t ~ : G ( P ) → V σ \sigma σ σ ˉ \bar{\sigma} σ ˉ τ ± ( a ) \tau^\pm(a) τ ± ( a ) a ∈ F q a\in\mathbb{F}_q a ∈ F q S L Z ( F q ) \mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) SL Z ( F q ) U ( P ) = ker t ~ U(\mathcal{P})=\ker\tilde{t} U ( P ) = ker t ~ G ( P ) G(\mathcal{P}) G ( P )
U ( P ) ≅ ⨁ i ∈ N P S L [ r i ] ( F q ) ≤ N ( P ) . U(\mathcal{P})\cong\bigoplus_{i\in\mathbb{N}}\mathrm{PSL}_{[r_i]}(\mathbb{F}_q)\leq N(\mathcal{P}). U ( P ) ≅ i ∈ N ⨁ PSL [ r i ] ( F q ) ≤ N ( P ) . The subgroup N ( P ) N(\mathcal{P}) N ( P ) t ~ \tilde{t} t ~ Z \mathbb{Z} Z V V V G ( P ) / N ( P ) ≅ Z G(\mathcal{P})/N(\mathcal{P})\cong\mathbb{Z} G ( P ) / N ( P ) ≅ Z
Invariant random subgroups of V V V
Consider the standard action of the group S L Z ( F q ) \mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) SL Z ( F q ) V Z = F q Z V_{\mathbb{Z}}=\mathbb{F}_q^{\mathbb{Z}} V Z = F q Z V ^ Z = V Z \widehat{V}_{\mathbb{Z}}=V_{\mathbb{Z}} V Z = V Z § 2.2 ). For each pair of indices k , ℓ ∈ N 0 k,\ell\in\mathbb{N}_0 k , ℓ ∈ N 0 V Z k , ℓ = V Z k × V ^ Z ℓ V_{\mathbb{Z}}^{k,\ell}=V_{\mathbb{Z}}^k\times\widehat{V}_{\mathbb{Z}}^\ell V Z k , ℓ = V Z k × V Z ℓ S L Z ( F q ) \mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) SL Z ( F q ) Haar k , ℓ \operatorname{Haar}^{k,\ell} Haar k , ℓ Theorem 1.1 that any non-trivial invariant random subgroup of S L Z ( F q ) \mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) SL Z ( F q ) μ k , ℓ \mu_{k,\ell} μ k , ℓ Haar k , ℓ \operatorname{Haar}^{k,\ell} Haar k , ℓ Stab S L Z ( F q ) : V Z k , ℓ → Sub ( S L Z ( F q ) ) \operatorname{Stab}_{\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q)}:V_{\mathbb{Z}}^{k,\ell}\to\operatorname{Sub}(\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q)) Stab SL Z ( F q ) : V Z k , ℓ → Sub ( SL Z ( F q ))
The following lemma analyzes invariant random subgroups of S L Z ( F q ) \mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) SL Z ( F q ) [23 Corollary 4.5] and its argument invokes Theorem 1.1.
Lemma 9.3. Let μ \mu μ S L Z ( F q ) \mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) SL Z ( F q ) m ∈ N m\in\mathbb{N} m ∈ N μ \mu μ H H H σ ˉ m g \bar{\sigma}^m g σ ˉ m g g ∈ S L Z ( F q ) g\in\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) g ∈ SL Z ( F q ) H H H μ \mu μ
Proof. It suffices to verify that the invariant random subgroups μ k , ℓ ∈ IRS ( S L Z ( F q ) ) \mu_{k,\ell}\in\operatorname{IRS}(\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q)) μ k , ℓ ∈ IRS ( SL Z ( F q )) k + ℓ ≥ 1 k+\ell\geq1 k + ℓ ≥ 1 x = ( v , φ ) ∈ V Z k , ℓ x=(v,\varphi)\in V_{\mathbb{Z}}^{k,\ell} x = ( v , φ ) ∈ V Z k , ℓ H = Stab S L Z ( F q ) ( x ) H=\operatorname{Stab}_{\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q)}(x) H = Stab SL Z ( F q ) ( x ) m ∈ N m\in\mathbb{N} m ∈ N g ∈ S L Z ( F q ) g\in\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) g ∈ SL Z ( F q ) σ ˉ g m \bar{\sigma}^m_g σ ˉ g m H H H σ ˉ g m ⋅ x \bar{\sigma}^m_g\cdot x σ ˉ g m ⋅ x H H H Theorem 8.9 , this happens if and only if for all i = 1 , … , k i=1,\ldots,k i = 1 , … , k j = 1 , … , ℓ j=1,\ldots,\ell j = 1 , … , ℓ
σ ˉ m g v i ∈ span { v 1 , … , v k } and σ ˉ m ( g − 1 ) t φ j ∈ span { φ 1 , … , φ ℓ } . \bar{\sigma}^m g v^i\in\operatorname{span}\{v^1,\ldots,v^k\}
\quad\text{and}\quad
\bar{\sigma}^m(g^{-1})^t\varphi^j\in\operatorname{span}\{\varphi^1,\ldots,\varphi^\ell\}. σ ˉ m g v i ∈ span { v 1 , … , v k } and σ ˉ m ( g − 1 ) t φ j ∈ span { φ 1 , … , φ ℓ } . Given each g ∈ S L Z ( F q ) g\in\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) g ∈ SL Z ( F q ) α ∈ M k × k ( F q ) \alpha\in M_{k\times k}(\mathbb{F}_q) α ∈ M k × k ( F q ) β ∈ M ℓ × ℓ ( F q ) \beta\in M_{\ell\times\ell}(\mathbb{F}_q) β ∈ M ℓ × ℓ ( F q ) L ( m , g , α , β ) L(m,g,\alpha,\beta) L ( m , g , α , β ) x = ( v , φ ) ∈ V Z k , ℓ x=(v,\varphi)\in V_{\mathbb{Z}}^{k,\ell} x = ( v , φ ) ∈ V Z k , ℓ i = 1 , … , k i=1,\ldots,k i = 1 , … , k j = 1 , … , ℓ j=1,\ldots,\ell j = 1 , … , ℓ
σ ˉ m g v i = ∑ a = 1 k α i a v a and σ ˉ m ( g − 1 ) t φ j = ∑ b = 1 ℓ β j b φ b . \bar{\sigma}^m g v^i=\sum_{a=1}^{k}\alpha_{ia}v^a
\quad\text{and}\quad
\bar{\sigma}^m(g^{-1})^t\varphi^j=\sum_{b=1}^{\ell}\beta_{jb}\varphi^b. σ ˉ m g v i = a = 1 ∑ k α ia v a and σ ˉ m ( g − 1 ) t φ j = b = 1 ∑ ℓ β j b φ b . Clearly L ( m , g , α , β ) L(m,g,\alpha,\beta) L ( m , g , α , β ) V Z k , ℓ V_{\mathbb{Z}}^{k,\ell} V Z k , ℓ k + ℓ ≥ 1 k+\ell\geq1 k + ℓ ≥ 1 g ∈ S L Z ( F q ) g\in\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) g ∈ SL Z ( F q ) N ∈ N N\in\mathbb{N} N ∈ N g ∈ S L [ − N , N ] ( F q ) g\in\mathrm{SL}_{[-N,N]}(\mathbb{F}_q) g ∈ SL [ − N , N ] ( F q ) g g g N N N n ∈ Z n\in\mathbb{Z} n ∈ Z ∣ n ∣ > N + m \lvert n\rvert>N+m ∣ n ∣ > N + m n n n x ∈ V Z k , ℓ x\in V_{\mathbb{Z}}^{k,\ell} x ∈ V Z k , ℓ x x x L ( m , g , α , β ) L(m,g,\alpha,\beta) L ( m , g , α , β )
v n − m i = ∑ a = 1 k α i a v n a and φ n − m j = ∑ b = 1 ℓ β j b φ n b . v^i_{n-m}=\sum_{a=1}^{k}\alpha_{ia}v^a_n
\quad\text{and}\quad
\varphi^j_{n-m}=\sum_{b=1}^{\ell}\beta_{jb}\varphi^b_n. v n − m i = a = 1 ∑ k α ia v n a and φ n − m j = b = 1 ∑ ℓ β j b φ n b . As n n n n ∈ ( m + 1 ) Z n\in(m+1)\mathbb{Z} n ∈ ( m + 1 ) Z ∣ n ∣ > N + m \lvert n\rvert>N+m ∣ n ∣ > N + m n n n n − m n-m n − m n n n n n n L ( m , g , α , β ) L(m,g,\alpha,\beta) L ( m , g , α , β ) V Z k , ℓ V_{\mathbb{Z}}^{k,\ell} V Z k , ℓ Haar k , ℓ ( L ( m , g , α , β ) ) = 0 \operatorname{Haar}^{k,\ell}(L(m,g,\alpha,\beta))=0 Haar k , ℓ ( L ( m , g , α , β )) = 0
To conclude, note that by above paragraph every tuple x ∈ V Z k , ℓ x\in V_{\mathbb{Z}}^{k,\ell} x ∈ V Z k , ℓ σ ˉ g m \bar{\sigma}^m_g σ ˉ g m L ( m , g , α , β ) L(m,g,\alpha,\beta) L ( m , g , α , β ) α \alpha α β \beta β μ k , ℓ = ( Stab S L Z ( F q ) ) ∗ Haar k , ℓ \mu_{k,\ell}=(\operatorname{Stab}_{\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q)})_*\operatorname{Haar}^{k,\ell} μ k , ℓ = ( Stab SL Z ( F q ) ) ∗ Haar k , ℓ m ∈ N m\in\mathbb{N} m ∈ N
μ k , ℓ ( { H ∈ Sub ( S L Z ( F q ) ) : ∃ g ∈ S L Z ( F q ) such that σ ˉ g m normalizes H } ) ≤ ≤ ∑ g ∈ S L Z ( F q ) ∑ α ∈ M k × k ( F q ) ∑ β ∈ M ℓ × ℓ ( F q ) Haar k , ℓ ( L ( m , g , α , β ) ) = 0 , \begin{aligned}
\mu_{k,\ell}\bigl(\{H\in\operatorname{Sub}(\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q)):\exists g\in\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q)\text{ such that }\bar{\sigma}^m_g\text{ normalizes }H\}\bigr)
&\leq\\
&\leq\sum_{g\in\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q)}
\sum_{\alpha\in M_{k\times k}(\mathbb{F}_q)}
\sum_{\beta\in M_{\ell\times\ell}(\mathbb{F}_q)}
\operatorname{Haar}^{k,\ell}(L(m,g,\alpha,\beta))=0,
\end{aligned} μ k , ℓ ( { H ∈ Sub ( SL Z ( F q )) : ∃ g ∈ SL Z ( F q ) such that σ ˉ g m normalizes H } ) ≤ ≤ g ∈ SL Z ( F q ) ∑ α ∈ M k × k ( F q ) ∑ β ∈ M ℓ × ℓ ( F q ) ∑ Haar k , ℓ ( L ( m , g , α , β )) = 0 , as required.
Corollary 9.4. Let μ ~ \tilde{\mu} μ ~ V = Z ⋉ S L Z ( F q ) ≅ ⟨ σ ˉ ⟩ ⋉ S L Z ( F q ) V = \mathbb{Z} \ltimes\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) \cong\langle\bar{\sigma}\rangle\ltimes\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) V = Z ⋉ SL Z ( F q ) ≅ ⟨ σ ˉ ⟩ ⋉ SL Z ( F q )
Then exactly one of the following two statements is true:
μ ~ \tilde{\mu} μ ~ H H H V V V S L Z ( F q ) \mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) SL Z ( F q )
μ ~ \tilde{\mu} μ ~ ⟨ σ ˉ m ⟩ ⋉ S L Z ( F q ) \langle\bar{\sigma}^{m}\rangle\ltimes\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) ⟨ σ ˉ m ⟩ ⋉ SL Z ( F q ) m ∈ N m \in\mathbb{N} m ∈ N
Proof. The proof of the analogous fact in the case of B.H. Neumann groups is [23 Corollary 4.6] . The proof in the present case is exactly the same, where we make use of Lemma 9.3 in place of the analogous [23 Corollary 4.5] . □
Invariant random subgroups of G ( P ) G(\mathcal{P}) G ( P )
The diagonal product ⨂ i P S L n i ( F q ) \bigotimes_i \mathrm{PSL}_{n_i}(\mathbb{F}_q) ⨂ i PSL n i ( F q ) P = ( n i ) \mathcal{P}=(n_i) P = ( n i ) G ( P ) G(\mathcal{P}) G ( P ) Theorem 1.6 , to show that the diagonal product in question ⨂ i S L n i ( F q ) \bigotimes_i \mathrm{SL}_{n_i}(\mathbb{F}_q) ⨂ i SL n i ( F q )
Theorem 9.5. All invariant random subgroups of G ( P ) G(\mathcal{P}) G ( P )
Recall that an invariant random subgroup μ \mu μ μ \mu μ ∗ * ∗
Proof of Theorem 9.5. The proof is structurally identical to that of [23 Theorem 1.3] . Rather than providing a full derivation, we highlight the necessary modifications to the definitions and indicate where Theorem 1.1 is invoked. We make use of exactly the same notations as in [23 ] .
Fix a monotone sequence P = ( n i ) \mathcal{P}=(n_i) P = ( n i ) n 1 ≥ 3 n_1 \ge3 n 1 ≥ 3 G = G ( P ) G=G(\mathcal{P}) G = G ( P ) n i = 2 r i + 1 n_i=2r_i+1 n i = 2 r i + 1 i ∈ N i\in\mathbb{N} i ∈ N μ \mu μ G G G μ \mu μ [22 Corollary 2.6] .
Let μ ~ = t ~ ∗ μ \tilde{\mu}=\tilde{t}_*\mu μ ~ = t ~ ∗ μ μ \mu μ t ~ \tilde{t} t ~ μ ~ \tilde{\mu} μ ~ V = Z ⋉ S L Z ( F q ) V=\mathbb{Z}\ltimes\mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q) V = Z ⋉ SL Z ( F q ) μ ~ \tilde{\mu} μ ~
First, consider the case where μ ~ \tilde{\mu} μ ~ H H H S L Z ( F q ) ◃ V \mathrm{SL}_{\mathbb{Z}}(\mathbb{F}_q)\triangleleft V SL Z ( F q ) ◃ V μ \mu μ N = N ( P ) ◃ G N=N(\mathcal{P})\triangleleft G N = N ( P ) ◃ G [23 Section 5] . For each i ∈ N i\in\mathbb{N} i ∈ N
D i = D i ( P ) = ker ( G → Z / ( 2 r i − 1 ) Z ) D_i=D_i(\mathcal{P})=\ker\left(G\to\mathbb{Z}/(2r_i-1)\mathbb{Z}\right) D i = D i ( P ) = ker ( G → Z / ( 2 r i − 1 ) Z ) via the quotient map G / N ≅ Z G/N\cong\mathbb{Z} G / N ≅ Z
G i = G i ( P ) = ker ( G → ∏ j = 1 i P S L [ r j ] ( F q ) ) G_i=G_i(\mathcal{P})=\ker\left(G\to\prod_{j=1}^{i}\mathrm{PSL}_{[r_j]}(\mathbb{F}_q)\right) G i = G i ( P ) = ker ( G → j = 1 ∏ i PSL [ r j ] ( F q ) ) so that D i D_i D i G i G_i G i G G G i ∈ N i\in\mathbb{N} i ∈ N
F i = F i ( P ) = ⟨ σ i : i ∈ [ 2 r i − 1 ] ⟩ ⋅ L i F_i=F_i(\mathcal{P})=\langle\sigma^i:i\in[2r_i-1]\rangle\cdot L_i F i = F i ( P ) = ⟨ σ i : i ∈ [ 2 r i − 1 ]⟩ ⋅ L i so that F i F_i F i G G G μ \mu μ H H H i ∈ N i\in\mathbb{N} i ∈ N
K i = K i ( H ) = ( H ∩ L i ) ( G i ∩ D i ) K_i=K_i(H)=(H\cap L_i)(G_i\cap D_i) K i = K i ( H ) = ( H ∩ L i ) ( G i ∩ D i ) so that K i K_i K i G G G F i F_i F i N G ( K i ) N_G(K_i) N G ( K i ) ( K i , F i ) (K_i,F_i) ( K i , F i ) μ \mu μ H H H [22 Definition 1.7] . The arguments presented there carry over to our case without modification. The existence of such a Weiss approximation in a finitely generated residually finite amenable group implies that the invariant random subgroup μ \mu μ [22 Theorem 1.8] .
Lastly, consider the remaining case where μ ~ \tilde{\mu} μ ~ ⟨ σ ˉ n ⟩ ⋊ SL Z ( F q ) \langle\bar{\sigma}^{n}\rangle\rtimes\operatorname{SL}_{\mathbb{Z}}(F_q) ⟨ σ ˉ n ⟩ ⋊ SL Z ( F q ) n ∈ N n \in\mathbb{N} n ∈ N [23 Proposition 2.5] that μ \mu μ G G G [23 Proposition 2.5] is stated for B.H. Neumann groups, its parallel version in our setting is proved identically). Hence μ \mu μ □ \square □
