Introduction
Hierarchies and Morse accessibility
Suppose a finitely generated group Γ \Gamma Γ that splitting also split again, and so on. This type of iterated graphs of groups splitting leads to the notion of a hierarchy of splittings for the group Γ \Gamma Γ
In the development of various types of hierarchies, boundaries have proven to be a powerful tool. They frequently reveal splittings and hierarchies inherent in the group’s structure. For instance, topological properties of different types of boundaries can reveal splittings and even full JSJ decompositions for different classes of groups, including hyperbolic groups [4 ] , relatively hyperbolic groups [5 ] [29 ] [30 ] , and CAT(0) groups [38 ] . The most important example for this paper, however, is the Dunwoody hierarchy for hyperbolic groups: Every hyperbolic group admits a finite hierarchy over finite edge groups where the final vertex groups are hyperbolic and either one-ended or finite [23 ] . In terms of the Gromov boundary, this hierarchy can be described as one in which the edge groups have empty Gromov boundary and where every vertex group has connected (possible empty) Gromov boundary. The Gromov boundaries of the one-ended vertex groups embed in the Gromov boundary of the ambient group and yield its infinite connected components. This is the origin of the idea in [KMZ26] to investigate hierarchies of this kind not only for the Gromov boundary of a hyperbolic group but also for the Morse boundary for a general finitely generated group.
The Morse boundary was introduced by Charney–Sultan [CS15], and Cordes [17 ] with the intention of providing a tool analogous to the Gromov boundary of a hyperbolic group for groups that are not Gromov hyperbolic. It is a generalization in the sense that the two boundaries coincide for hyperbolic groups and both are a quasi-isometry invariant for the group.
When transferring the above from Gromov boundaries to Morse boundaries, the natural subgroups to split over are subgroups H H H relative Morse boundary ( ∂ ∗ H , Γ ) (\partial_*H,\Gamma) ( ∂ ∗ H , Γ ) H H H weak Morse hierarchy . One might hope that such a splitting process terminates at some point with final vertex groups that have connected, possibly empty (relative) Morse boundary. If it does, we call the tree a Morse hierarchy . In [KMZ26], the authors began to investigate which finitely generated groups admit such a hierarchy, and called such groups Morse accessible . In this paper, we formalize the notion of Morse accessibility in detail, using and extending the results in [24 ] . This allows us to verify that Morse hierarchies indeed generalize the Dunwoody hierarchy for hyperbolic groups in the following sense:
Theorem A. Let Γ \Gamma Γ H \mathcal{H} H Γ \Gamma Γ
L \mathcal{L} L L L L H \mathcal{H} H ( ∂ ∗ L , Γ ) ≠ ∅ (\partial_*L,\Gamma) \ne\varnothing ( ∂ ∗ L , Γ ) = ∅
S \mathcal{S} S ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ
Moreover, every infinite connected component C C C ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ
C = ( ∂ ∗ stab ( C ) , Γ ) . C=(\partial_*\operatorname{stab}(C),\Gamma). C = ( ∂ ∗ stab ( C ) , Γ ) . For example, suppose Γ \Gamma Γ Figure 1 . One Morse hierarchy of Γ \Gamma Γ Z \mathbb{Z} Z
Figure 1. The pictured tree yields a Morse hierarchy for the fundamental group of the amalgam at the top.
There are four groups associated to the leaves of this tree, the fundamental groups of the genus two surface, the torus, and two one-point spaces. Importantly, the set L \mathcal{L} L π 1 ( S ) \pi_1(S) π 1 ( S ) S S S Figure 1 . The theorem says that the set L \mathcal{L} L Γ \Gamma Γ L \mathcal{L} L § 6 .
A direct consequence of the theorem above is that we can characterize when a Morse-accessible group has totally disconnected Morse boundary. This extends and uses results from [24 ] .
Corollary B. Let Γ \Gamma Γ H \mathcal{H} H Γ \Gamma Γ L L L H \mathcal{H} H ( ∂ ∗ L , Γ ) = ∅ (\partial_* L,\Gamma)=\varnothing ( ∂ ∗ L , Γ ) = ∅
Just as with group splittings, it is natural to consider how a Morse hierarchy behaves with respect to quasi-isometry. For instance, the quasi-isometry invariance of accessibility follows from the characterization of accessibility in [43 ] . We apply the results from [KMZ26] to show the quasi-isometric rigidity statements in Theorem D and Theorem E below. In order to obtain these results, we show that the set of infinite Morse boundary components is point-convergent. Point-convergence is a notion that was introduced in [KMZ26], and generalizes the notion of a null-family of subsets. Null-families have been important in the study of boundaries. In particular, they were used to characterize quasi-convex subgroups of hyperbolic groups [GMRS98, Mj08].
Theorem C. The set of all infinite connected components of the Morse boundary of a Morse accessible group is point-convergent.
By Corollary 1.2, Corollary 1.3, and Theorem 1.5 in [KMZ26], Theorem C directly implies the following two statements.
Corollary D. Let Γ \Gamma Γ Γ ′ \Gamma' Γ ′ Γ \Gamma Γ
if C C C ∂ ∗ Γ ′ \partial_*\Gamma' ∂ ∗ Γ ′ H H H C C C H H H ∂ ∗ Γ ′ \partial_*\Gamma' ∂ ∗ Γ ′ ( ∂ ∗ H , Γ ′ ) (\partial_*H,\Gamma') ( ∂ ∗ H , Γ ′ )
an infinite component C C C ∂ ∗ Γ ′ \partial_*\Gamma' ∂ ∗ Γ ′
if C C C ∂ ∗ Γ ′ = ∂ ∗ Γ \partial_*\Gamma'=\partial_*\Gamma ∂ ∗ Γ ′ = ∂ ∗ Γ C C C Γ \Gamma Γ Γ ′ \Gamma' Γ ′
Applying the previous theorem to Morse accessible group whose Morse boundary contains a connected component homeomorphic to a circle yields:
Corollary E. Let Γ \Gamma Γ ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ C C C Γ \Gamma Γ
Note, there are two natural topologies one can put on the Morse boundary. In this paper, we use the direct limit topology introduced in [CS15] and [17 ] . A metrizable topology was introduced in [16 ] and we refer to this as the Cashen–Mackay topology. Almost all Morse hierarchies discussed in this paper remain valid when considered with respect to the Cashen–Mackay topology. This is discussed in the Appendix Section A.
(Weak) Morse hierarchies of Coxeter groups and applications.
We show that every Coxeter groups has a weak Morse hierarchy. If the Coxeter group does not contains a spherical nexus (a condition restricting the behavior of special spherical subgroups) this weak Morse hierarchy is a genuine Morse hierarchy. Many Coxeter groups do not contain a spherical nexus, as for instance all right-angled Coxeter groups (RACGs), and are therefore Morse accessible. We derive consequences for their Morse boundaries and quasi-isometry classification in § 5 . For definitions related to Coxeter groups, see § 4 .
To obtain a finite Morse hierarchies for Coxeter groups, we use ideas from [31 ] to develop a graph decomposition algorithm over subgraphs that correspond to special subgroups having empty relative Morse boundary in the ambient group. See Figure 4 for how the algorithm works. Then we apply the results from Cordes–Levcovitz [14 ] to show that the vertex groups we are left with one-ended and have connected relative Morse boundary (this is detected by the Coxeter graph being relatively wide-avoidant ). These vertex groups are special subgroups that are either finite or the stabilizer of an infinite connected component of the Morse boundary. Conversely, we can identify the stabilizer of each infinite connected component of the Morse boundary as a conjugate of one of these non-finite special subgroups that remain after applying the algorithm. This is a remarkable relation between the topology of the Morse boundary and the combinatorics of the graph defining the Coxeter group.
Figure 4. Construction of a weak Morse hierarchy for a Coxeter group W Δ W_{\Delta} W Δ 2 ∣ V ∣ 2^{|V|} 2 ∣ V ∣ Δ \Delta Δ
Theorem F. (Theorem 4.17) Every Coxeter group has a weak Morse hierarchy in which each leaf group L L L W Λ W_\Lambda W Λ
Λ \Lambda Λ Δ \Delta Δ W Λ W_\Lambda W Λ ( ∂ ∗ W Λ , W Δ ) ≠ ∅ (\partial_*W_\Lambda,W_\Delta) \ne\varnothing ( ∂ ∗ W Λ , W Δ ) = ∅
Λ \Lambda Λ Δ \Delta Δ ( ∂ ∗ W Λ , W Δ ) = ∅ (\partial_*W_\Lambda,W_\Delta) = \varnothing ( ∂ ∗ W Λ , W Δ ) = ∅
Moreover, if W Δ W_\Delta W Δ ( ∂ ∗ W Λ , W Δ ) (\partial_*W_\Lambda,W_\Delta) ( ∂ ∗ W Λ , W Δ )
We illustrate the process for obtaining a finite Morse hierarchy in the RACG case in Figure 2 . In this case, we iteratively split the defining graph along cliques and subgraphs contained in joins which represent a direct products of two infinite subgroups. For more details on this example, see § 6.4 .
Figure 2. The pictured tree yields a Morse hierarchy of the RACG with the defining graph at the top. At each level we split over the subgroup generated by the red vertices.
A direct consequence of Theorem A and Theorem F is the following.
Corollary G. Every infinite connected component C C C W Δ W_\Delta W Δ
C C C
C C C
has a stabilizer isomorphic to a one-ended special subgroup whose Coxeter graph is relatively wide-spherical-avoidant in Δ \Delta Δ
Again, we illustrate this corollary with the RACG example in Figure 2 . The leaf corresponding to the 6-cycle (on the bottom left of the figure) is the defining graph of a special subgroup H H H C C C ∂ H ≅ S 1 \partial H \cong S^1 ∂ H ≅ S 1 H H H ∂ H \partial H ∂ H Theorem 6.11 .
As a direct consequence of Theorem G (and implicitly Theorem B), we generalize the results in [31 ] for RACGs as follows:
Corollary H. If every leaf group of the weak Morse hierarchy of W Δ W_\Delta W Δ Theorem F (Theorem 4.17 ) is spherical or contained in a special subgroup of W Δ W_\Delta W Δ ∂ ∗ W Δ \partial_*W_\Delta ∂ ∗ W Δ W Δ W_\Delta W Δ
In the § 5 , we will apply Theorem H and the proof of Theorem F to show that there is an algorithm to determine when certain Coxeter groups are not quasi-isometric to any right-angled Artin group (RAAG). This enables us to differentiate many new RACGs groups from each other up to quasi-isometry, even in the non triangle-free case.
For instance, the graph on the left in Figure 3 corresponds to a RACG with totally disconnected Morse boundary [31 ] . We show in § 5 that the RACG corresponding to the graph on the right of Figure 3 has an infinite connected component in its Morse boundary. So, the pictured graphs correspond to two RACGs whose Morse boundaries are not homeomorphic. Hence, the corresponding RACGs are not quasi-isometric.
Figure 3. The two RACGs corresponding to the pictured graphs are not quasi-isometric to each other. The graph on the left was studied in [15 Fig 10] .
For more examples, see Figure 5 in § 5 .
Figure 5. Graph (1) is quasi-isometric to the 5-cycle RAAG; graph (2) arises from graph (1) through adding 6 edges; graph (3) was studied in [15 Fig 10] ; graph (4) was discussed in [11 Fig. 13] ; graphs (5)-(11) are variations of graphs (1)-(3).
Outline
The paper is organized as follows. § 2 contains the necessary definitions and properties of Morse boundaries, relative Morse boundaries, and graphs of groups. Additionally, the theorems of [24 ] and [KMZ26] are recalled, and we provide slight variants of Theorem 2.1 from the former and Lemma 5.8 from the latter, which are needed in this paper. (Theorem 2.18 ).
In § 3 we give the formal definition of Morse accessibility which is a modified version of the notion introduced in [KMZ26]. In order to formulate our definition, we introduce Morse hierarchies and methodically transfer all the results proven for group splittings along groups with empty relative Morse boundaries from [24 ] to Morse hierarchies using induction. This will give Theorem A and Theorem B, which provide structural consequences of a group being Morse accessible. Additionally, Theorem 3.7 shows that the relative Morse boundary of a finitely generated undistorted subgroup of a finitely generated group cannot consist of just one point. Finally, in § 3.3 , we prove Theorem C which is necessary for obtaining the quasi-isometric rigidity statements in Theorem D and Theorem E.
In § 4 we recall and adjust the relevant results from Cordes-Levcovitz [14 ] and use them to prove Theorem F, Theorem G, and Theorem H.
In § 5 we illustrate how to use Morse hierarchies to distinguish groups up to quasi-isometry, with a focus on right-angled Artin groups (RAAGs) and right-angled Coxeter groups (RACGs).
In § 6 , we discuss how to obtain Morse hierarchies for several classes of groups including RAAGs, 3-manifold groups, surface amalgams. We give several examples of Morse hierarchies for RACGs that illustrate properties and consequences of having a Morse hierarchy.
Finally, in § 7 we consider alternative definitions of Morse accessibility.
AI Declaration
Artificial intelligence tools were used solely for grammatical editing and, in one instance, to assist with nomenclature. No AI tools, including LLMs, were used to develop, verify, or write any of the mathematical arguments presented in this article.
Acknowledgments
The authors would like to thank Emily Stark and Motiejus Valiunas for fruitful discussions.
Preliminary definitions and results
In this section, we will give the necessary definitions and background on Morse boundaries and relative boundaries for metric spaces and groups. We will also set up our notation and terminology for graphs of groups and recall the work in [24 ] needed here. We provide variants of the results in that paper that will allow us to work with hierarchies rather than just graphs of groups.
Morse Boundaries and Relative Morse Boundaries of Metric Spaces
In this subsection, we give the definition of the Morse boundary. There are two natural topologies one can consider on the Morse boundary. The first is the direct-limit topology, which we use throughout this paper. An alternative topology is the metrizable topology introduced by Cashen and Mackay [16 ] . The main advantages of the direct limit topology is its ease of computation. However, it has some notable drawbacks: It is not metrizable, and does not interact well with subgroups. We address which results carry over to the Cashen–Mackay topology in Section A.
Definition 2.1. Let N : [ 1 , ∞ ) × [ 0 , ∞ ) → [ 0 , ∞ ) N : [1,\infty) \times[0,\infty) \to[0,\infty) N : [ 1 , ∞ ) × [ 0 , ∞ ) → [ 0 , ∞ ) γ : I → X \gamma: I \to X γ : I → X N N N [ s , t ] ⊆ I [s,t] \subseteq I [ s , t ] ⊆ I I I I ( k , c ) (k,c) ( k , c ) γ ( s ) \gamma(s) γ ( s ) γ ( t ) \gamma(t) γ ( t ) N ( k , c ) N(k,c) N ( k , c ) γ ∣ [ s , t ] \gamma|_{[s,t]} γ ∣ [ s , t ] N N N Morse gauge .
As a set, the Morse boundary of a proper geodesic metric space X X X X X X o ∈ X o \in X o ∈ X γ : [ 0 , ∞ ) → X \gamma: [0,\infty) \to X γ : [ 0 , ∞ ) → X [ α ] [\alpha] [ α ] Morse boundary of X X X ∂ ∗ X \partial_*X ∂ ∗ X ∂ ∗ X o \partial_*X_o ∂ ∗ X o o o o
To topologize the boundary, first fix a Morse gauge N N N X X X N N N
∂ ∗ N X o = { [ α ] ∣ ∃ β ∈ [ α ] that is an N -Morse geodesic ray with base point o } . \partial_*^N X_o = \{[\alpha] \mid\exists\beta\in[\alpha] \text{ that is an } N\text{-Morse geodesic ray with base point } o\}. ∂ ∗ N X o = {[ α ] ∣ ∃ β ∈ [ α ] that is an N -Morse geodesic ray with base point o } . One can topologize this set with the compact-open topology. This topology is equivalent to the one defined by a neighborhood basis: at each point [ α ] ∈ ∂ ∗ N X o [\alpha] \in\partial_*^N X_o [ α ] ∈ ∂ ∗ N X o n ∈ N n \in\mathbb{N} n ∈ N V n N ( α ) V_n^N(\alpha) V n N ( α ) N N N γ : [ 0 , ∞ ) → X \gamma: [0,\infty) \to X γ : [ 0 , ∞ ) → X γ ( 0 ) = o \gamma(0)=o γ ( 0 ) = o d ( α ( t ) , γ ( t ) ) < δ N d(\alpha(t),\gamma(t)) < \delta_N d ( α ( t ) , γ ( t )) < δ N t < n t<n t < n δ N \delta_N δ N N N N [ α ] [\alpha] [ α ] ∂ ∗ N X o \partial_*^N X_o ∂ ∗ N X o
{ V n N ( α ) ∣ n ∈ N } . \{V_n^N(\alpha) \mid n \in\mathbb{N}\}. { V n N ( α ) ∣ n ∈ N } . The constant δ N \delta_N δ N [17 , 19 ] and is chosen so that any ray in the equivalence class of α \alpha α V n N ( α ) V_n^N(\alpha) V n N ( α )
Let M \mathcal{M} M M \mathcal{M} M N , N ′ ∈ M N,N' \in\mathcal{M} N , N ′ ∈ M N ≤ N ′ N \leq N' N ≤ N ′ N ( k , c ) ≤ N ′ ( k , c ) N(k,c) \leq N'(k,c) N ( k , c ) ≤ N ′ ( k , c ) k ≥ 1 k \geq1 k ≥ 1 c ≥ 0 c \geq0 c ≥ 0 X X X
∂ ∗ X o = lim M ∂ ∗ N X o \partial_*X_o = \lim_{\mathcal{M}} \partial_*^N X_o ∂ ∗ X o = M lim ∂ ∗ N X o with the induced direct limit topology, i.e., a set U U U ∂ ∗ X o \partial_*X_o ∂ ∗ X o U ∩ ∂ ∗ N X o U \cap\partial_*^N X_o U ∩ ∂ ∗ N X o N ∈ M N \in\mathcal{M} N ∈ M [17 , 19 ] .
The first salient feature of the Morse boundary is that it is invariant under quasi-isometry regardless of the geometry of the space.
Theorem 2.2 ([17 ] ). Let X X X Y Y Y f : X → Y f : X \to Y f : X → Y ∂ f : ∂ ∗ X o → ∂ ∗ Y f ( o ) \partial f : \partial_*X_o \to\partial_*Y_{f(o)} ∂ f : ∂ ∗ X o → ∂ ∗ Y f ( o )
As a corollary of this theorem, a change in basepoint induces a homeomorphism on the boundary. This justifies the omission of the basepoint from the notation. More generally, we will usually assume the basepoint is fixed and omit it from the notation ∂ ∗ N X o \partial_*^N X_o ∂ ∗ N X o ∂ ∗ X \partial_*X ∂ ∗ X ∂ ∗ N X \partial_*^N X ∂ ∗ N X
Definition 2.3 (relative Morse boundary, see [KMZ26]). Let Y Y Y X X X relative Morse boundary of Y Y Y X X X ( ∂ ∗ Y , X ) (\partial_*Y,X) ( ∂ ∗ Y , X )
( ∂ ∗ Y , X ) = { [ γ ] ∈ ∂ ∗ X ∣ ∃ k so that γ ⊆ N k ( Y ) } (\partial_*Y,X)=\{[\gamma]\in\partial_*X\mid\exists\ k\text{ so that }\gamma\subseteq N_k(Y)\} ( ∂ ∗ Y , X ) = {[ γ ] ∈ ∂ ∗ X ∣ ∃ k so that γ ⊆ N k ( Y )} where N k ( Y ) N_k(Y) N k ( Y ) k k k Y Y Y ( ∂ ∗ Y , X ) ⊆ ∂ ∗ X (\partial_*Y,X)\subseteq\partial_*X ( ∂ ∗ Y , X ) ⊆ ∂ ∗ X
To understand how a subset Y Y Y X X X ∂ ∗ X \partial_*X ∂ ∗ X ∂ ∗ Y \partial_*Y ∂ ∗ Y ∂ ∗ X \partial_*X ∂ ∗ X ∂ ∗ Y \partial_*Y ∂ ∗ Y ∂ ∗ X \partial_*X ∂ ∗ X Y Y Y X X X ∂ ∗ Y \partial_*Y ∂ ∗ Y
Definition 2.4 (surviving Morse boundary). Let ι \iota ι Y Y Y X X X surviving Morse boundary of Y Y Y X X X ∂ ∗ Y \partial_*Y ∂ ∗ Y Y Y Y X X X
M ( Y , X ) : = { [ γ ] ∈ ∂ ∗ Y ∣ ι ( γ ) is Morse in X } ⊆ ∂ ∗ Y . M(Y,X):=\{[\gamma]\in\partial_*Y\mid\iota(\gamma)\text{ is Morse in }X\}\subseteq\partial_*Y. M ( Y , X ) := {[ γ ] ∈ ∂ ∗ Y ∣ ι ( γ ) is Morse in X } ⊆ ∂ ∗ Y . Remark 2.5. A warning to the reader, in [24 ] , the notation ( ∂ ∗ Y , X ) (\partial_*Y,X) ( ∂ ∗ Y , X ) M ( Y , X ) M(Y,X) M ( Y , X ) Y Y Y X X X
Remark 2.6 (Choice of topology on subspace). Note that these two definitions of topologies on the boundary of a subset Y Y Y X X X [24 ] [Example on page 341]. Resolving this discrepancy requires care and is where much of the technical difficulties lie in the proofs in this paper.
Definition 2.7. Let X X X Y Y Y X X X undistorted in X X X Y Y Y X X X
If [ γ ] ∈ M ( Y , X ) [\gamma]\in M(Y,X) [ γ ] ∈ M ( Y , X ) ι ( [ γ ] ) \iota([\gamma]) ι ([ γ ]) ι ∗ : M ( Y , X ) → ∂ ∗ X \iota_*:M(Y,X)\to\partial_*X ι ∗ : M ( Y , X ) → ∂ ∗ X ( ∂ ∗ ι ( Y ) , X ) (\partial_*\iota(Y),X) ( ∂ ∗ ι ( Y ) , X ) M ( Y , X ) M(Y,X) M ( Y , X ) ι ∗ \iota_* ι ∗
Lemma 2.8 (see [KMZ26]). Let ( X , d X ) (X,d_X) ( X , d X ) ( Y , d Y ) (Y,d_Y) ( Y , d Y ) ι : Y → X \iota:Y\to X ι : Y → X γ : [ 0 , ∞ ) → X \gamma:[0,\infty)\to X γ : [ 0 , ∞ ) → X
[ γ ] ∈ ( ∂ ∗ ι ( Y ) , X ) [\gamma]\in(\partial_*\iota(Y),X) [ γ ] ∈ ( ∂ ∗ ι ( Y ) , X )
there exists a geodesic ray λ : [ 0 , ∞ ) → Y \lambda:[0,\infty)\to Y λ : [ 0 , ∞ ) → Y Y Y Y ι ∘ λ \iota\circ\lambda ι ∘ λ γ \gamma γ
This combined with the observations in [KMZ26] yields:
Lemma 2.9 (Lemma 2.3 in [24 ] ). Let ( X , d X ) (X,d_X) ( X , d X ) ( Y , d Y ) (Y,d_Y) ( Y , d Y ) ι : Y → X \iota:Y\to X ι : Y → X ι \iota ι ι ∗ : M ( Y , X ) → ( ∂ ∗ ι ( Y ) , X ) \iota_*:\mathcal{M}(Y,X)\to(\partial_*\iota(Y),X) ι ∗ : M ( Y , X ) → ( ∂ ∗ ι ( Y ) , X )
We now discuss the case where X X X Γ \Gamma Γ Y Y Y H H H Γ \Gamma Γ ι ∗ \iota_* ι ∗ H H H H H H ι ∗ \iota_* ι ∗ ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ [41 Thm 1.4] .
Definition 2.10 (stable subgroup). Let H < Γ H<\Gamma H < Γ H H H H H H stable subgroup of G G G N N N H H H N N N G G G
Tran shows that an infinite subgroup of a finitely generated group H H H Γ \Gamma Γ H H H Γ \Gamma Γ [41 Thm 4.8] .
Definition 2.11 (Morse subgroup [41 ] , [26 ] ). Let Γ \Gamma Γ H H H Γ \Gamma Γ H H H Morse in Γ \Gamma Γ K ≥ 1 K\ge1 K ≥ 1 C ≥ 0 C\ge0 C ≥ 0 M = M ( K , C ) M=M(K,C) M = M ( K , C ) ( K , C ) (K,C) ( K , C ) Γ \Gamma Γ H H H M M M H H H
We end this subsection with three lemmas about how the surviving and relative Morse boundaries of subgroups are related to the Morse boundary of the ambient group. These will be used in order to extend the results on graphs of groups discussed in the next subsection to hierarchies in § 3 .
Lemma 2.12. Suppose Γ \Gamma Γ Γ ′ ′ ≤ Γ \Gamma''\le\Gamma Γ ′′ ≤ Γ ι ˉ : Γ ′ ′ ↪ Γ ′ \bar{\iota}:\Gamma''\hookrightarrow\Gamma' ι ˉ : Γ ′′ ↪ Γ ′ ι : Γ ′ ↪ Γ \iota:\Gamma'\hookrightarrow\Gamma ι : Γ ′ ↪ Γ ι ∘ ι ˉ : Γ ′ ′ ↪ Γ \iota\circ\bar{\iota}:\Gamma''\hookrightarrow\Gamma ι ∘ ι ˉ : Γ ′′ ↪ Γ
ι ˉ ∗ : M ( Γ ′ ′ , Γ ′ ) → ( ∂ ∗ Γ ′ ′ , Γ ′ ) ⊆ ∂ ∗ Γ ′ \bar{\iota}_*:\mathcal{M}(\Gamma'',\Gamma')\to(\partial_*\Gamma'',\Gamma')\subseteq\partial_*\Gamma' ι ˉ ∗ : M ( Γ ′′ , Γ ′ ) → ( ∂ ∗ Γ ′′ , Γ ′ ) ⊆ ∂ ∗ Γ ′ ι ∗ : M ( Γ ′ , Γ ) → ( ∂ ∗ Γ ′ , Γ ) ⊆ ∂ ∗ Γ \iota_*:\mathcal{M}(\Gamma',\Gamma)\to(\partial_*\Gamma',\Gamma)\subseteq\partial_*\Gamma ι ∗ : M ( Γ ′ , Γ ) → ( ∂ ∗ Γ ′ , Γ ) ⊆ ∂ ∗ Γ ( ι ∘ ι ˉ ) ∗ : M ( Γ ′ ′ , Γ ) → ( ∂ ∗ Γ ′ ′ , Γ ) ⊆ ∂ ∗ Γ . (\iota\circ\bar{\iota})_*:\mathcal{M}(\Gamma'',\Gamma)\to(\partial_*\Gamma'',\Gamma)\subseteq\partial_*\Gamma. ( ι ∘ ι ˉ ) ∗ : M ( Γ ′′ , Γ ) → ( ∂ ∗ Γ ′′ , Γ ) ⊆ ∂ ∗ Γ. Then ι ∗ ( ( ∂ ∗ Γ ′ , Γ ′ ) ∩ M ( Γ ′ , Γ ) ) = ( ∂ ∗ Γ ′ ′ , Γ ) \iota_*((\partial_*\Gamma',\Gamma')\cap\mathcal{M}(\Gamma',\Gamma))=(\partial_*\Gamma'',\Gamma) ι ∗ (( ∂ ∗ Γ ′ , Γ ′ ) ∩ M ( Γ ′ , Γ )) = ( ∂ ∗ Γ ′′ , Γ )
Proof. Let ξ ∈ ι ∗ ( ( ∂ ∗ Γ ′ , Γ ′ ) ∩ M ( Γ ′ , Γ ) ) \xi\in\iota_*((\partial_*\Gamma',\Gamma')\cap\mathcal{M}(\Gamma',\Gamma)) ξ ∈ ι ∗ (( ∂ ∗ Γ ′ , Γ ′ ) ∩ M ( Γ ′ , Γ )) [ α ] ∈ ( ∂ ∗ Γ ′ , Γ ′ ) ∩ M ( Γ ′ , Γ ) [\alpha]\in(\partial_*\Gamma',\Gamma')\cap\mathcal{M}(\Gamma',\Gamma) [ α ] ∈ ( ∂ ∗ Γ ′ , Γ ′ ) ∩ M ( Γ ′ , Γ ) ι ∗ ( [ α ] ) = ξ \iota_*([\alpha])=\xi ι ∗ ([ α ]) = ξ [ α ] ∈ M ( Γ ′ , Γ ) [\alpha]\in\mathcal{M}(\Gamma',\Gamma) [ α ] ∈ M ( Γ ′ , Γ ) [ α ] ∈ ∂ ∗ Γ ′ [\alpha]\in\partial_*\Gamma' [ α ] ∈ ∂ ∗ Γ ′ ι ( α ) \iota(\alpha) ι ( α ) Γ \Gamma Γ [ α ] ∈ ( ∂ ∗ Γ ′ , Γ ′ ) [\alpha]\in(\partial_*\Gamma',\Gamma') [ α ] ∈ ( ∂ ∗ Γ ′ , Γ ′ ) [ β ] ∈ ∂ ∗ Γ ′ ′ [\beta]\in\partial_*\Gamma'' [ β ] ∈ ∂ ∗ Γ ′′ ι ˉ ∗ ( β ) \bar{\iota}_*(\beta) ι ˉ ∗ ( β ) Γ ′ \Gamma' Γ ′ α \alpha α ι ( ι ˉ ( β ) ) \iota(\bar{\iota}(\beta)) ι ( ι ˉ ( β )) Γ \Gamma Γ ι ( α ) \iota(\alpha) ι ( α ) ξ = ι ∗ ( [ α ] ) ∈ ( ∂ ∗ Γ ′ ′ , Γ ) \xi=\iota_*([\alpha])\in(\partial_*\Gamma'',\Gamma) ξ = ι ∗ ([ α ]) ∈ ( ∂ ∗ Γ ′′ , Γ )
On the other hand, let [ γ ] ∈ ( ∂ ∗ Γ ′ ′ , Γ ) [\gamma]\in(\partial_*\Gamma'',\Gamma) [ γ ] ∈ ( ∂ ∗ Γ ′′ , Γ ) γ ′ \gamma' γ ′ ∂ ∗ Γ ′ \partial_*\Gamma' ∂ ∗ Γ ′ [ γ ′ ] ∈ ( ∂ ∗ Γ ′ , Γ ′ ) ∩ M ( Γ ′ , Γ ) [\gamma']\in(\partial_*\Gamma',\Gamma')\cap\mathcal{M}(\Gamma',\Gamma) [ γ ′ ] ∈ ( ∂ ∗ Γ ′ , Γ ′ ) ∩ M ( Γ ′ , Γ ) ι ∗ ( [ γ ′ ] ) = [ γ ] \iota_*([\gamma'])=[\gamma] ι ∗ ([ γ ′ ]) = [ γ ] [ γ ] ∈ ( ∂ ∗ Γ ′ ′ , Γ ) [\gamma]\in(\partial_*\Gamma'',\Gamma) [ γ ] ∈ ( ∂ ∗ Γ ′′ , Γ ) α \alpha α Γ ′ ′ \Gamma'' Γ ′′ ι ( ι ˉ ( α ) ) \iota(\bar{\iota}(\alpha)) ι ( ι ˉ ( α ))
:::</lemma>
is Morse in Γ \Gamma Γ γ \gamma γ ι \iota ι ι ˉ \bar{\iota} ι ˉ ι ˉ ( α ) \bar{\iota}(\alpha) ι ˉ ( α ) Γ ′ \Gamma' Γ ′ γ ′ \gamma' γ ′ Γ ′ \Gamma' Γ ′ ι ˉ ( α ) \bar{\iota}(\alpha) ι ˉ ( α ) [ γ ′ ] = ι ˉ ∗ ( [ α ] ) ∈ ( ∂ ∗ Γ ′ ′ , Γ ′ ) [\gamma'] = \bar{\iota}_*([\alpha]) \in(\partial_*\Gamma'',\Gamma') [ γ ′ ] = ι ˉ ∗ ([ α ]) ∈ ( ∂ ∗ Γ ′′ , Γ ′ ) ι \iota ι ι ( γ ′ ) \iota(\gamma') ι ( γ ′ ) Γ \Gamma Γ [ γ ′ ] ∈ M ( Γ ′ , Γ ) [\gamma'] \in M(\Gamma',\Gamma) [ γ ′ ] ∈ M ( Γ ′ , Γ ) ι ( γ ′ ) \iota(\gamma') ι ( γ ′ ) ι ( ι ˉ ( α ) ) \iota(\bar{\iota}(\alpha)) ι ( ι ˉ ( α )) ι ∗ ( [ γ ′ ] ) = [ γ ] \iota_*([\gamma']) = [\gamma] ι ∗ ([ γ ′ ]) = [ γ ]
Lemma 2.13. Suppose Γ \Gamma Γ Γ ′ ′ ≤ Γ ′ \Gamma'' \leq\Gamma' Γ ′′ ≤ Γ ′ Γ ′ ′ \Gamma'' Γ ′′ Γ ′ \Gamma' Γ ′ Γ ′ \Gamma' Γ ′ Γ \Gamma Γ Γ ′ ′ \Gamma'' Γ ′′ Γ \Gamma Γ ( ∂ ∗ Γ ′ ′ , Γ ) ⊆ ( ∂ ∗ Γ ′ , Γ ) (\partial_*\Gamma'',\Gamma) \subseteq(\partial_*\Gamma',\Gamma) ( ∂ ∗ Γ ′′ , Γ ) ⊆ ( ∂ ∗ Γ ′ , Γ ) ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ
Proof. The composition of quasi-isometric embeddings is a quasi-isometric embedding so Γ ′ ′ \Gamma'' Γ ′′ Γ \Gamma Γ ( ∂ ∗ Γ ′ ′ , Γ ) ⊆ ( ∂ ∗ Γ ′ , Γ ) (\partial_*\Gamma'',\Gamma) \subseteq(\partial_*\Gamma',\Gamma) ( ∂ ∗ Γ ′′ , Γ ) ⊆ ( ∂ ∗ Γ ′ , Γ )
To show this, we assume γ : [ 0 , ∞ ) → Γ \gamma:[0,\infty) \to\Gamma γ : [ 0 , ∞ ) → Γ Γ \Gamma Γ [ γ ] [\gamma] [ γ ] ( ∂ ∗ Γ ′ ′ , Γ ) (\partial_*\Gamma'',\Gamma) ( ∂ ∗ Γ ′′ , Γ ) Γ ′ ′ \Gamma'' Γ ′′ Γ \Gamma Γ λ : [ 0 , ∞ ) → Γ ′ ′ \lambda:[0,\infty) \to\Gamma'' λ : [ 0 , ∞ ) → Γ ′′ Γ ′ ′ \Gamma'' Γ ′′ Γ \Gamma Γ D D D γ \gamma γ
We must show [ γ ] [\gamma] [ γ ] ( ∂ ∗ Γ ′ , Γ ) (\partial_*\Gamma',\Gamma) ( ∂ ∗ Γ ′ , Γ ) γ \gamma γ Γ \Gamma Γ k k k γ ⊆ N k ( Γ ′ ) \gamma\subseteq N_k(\Gamma') γ ⊆ N k ( Γ ′ ) ( ∂ ∗ Γ ′ , Γ ) (\partial_*\Gamma',\Gamma) ( ∂ ∗ Γ ′ , Γ )
Each point of γ \gamma γ D D D Γ ′ ′ \Gamma'' Γ ′′ Γ ′ ′ ⊆ Γ ′ \Gamma'' \subseteq\Gamma' Γ ′′ ⊆ Γ ′ γ \gamma γ D D D Γ ′ \Gamma' Γ ′
Since Γ \Gamma Γ Γ \Gamma Γ H H H Γ \Gamma Γ H H H
Lemma 2.14. If H H H Γ \Gamma Γ g ∈ Γ g \in\Gamma g ∈ Γ
g ( ∂ ∗ H , Γ ) = ( ∂ ∗ g H , Γ ) = ( ∂ ∗ g H g − 1 , Γ ) . g(\partial_*H,\Gamma) = (\partial_*gH,\Gamma) = (\partial_*gHg^{-1},\Gamma). g ( ∂ ∗ H , Γ ) = ( ∂ ∗ g H , Γ ) = ( ∂ ∗ g H g − 1 , Γ ) . Preliminaries on graph of groups
Convention 1. We use the terminology and notation for graph of groups as in [24 ] . In particular, we use the following characterization of group splittings: A finitely generated group Γ \Gamma Γ Γ \Gamma Γ T T T G G G G G G T T T G G G T T T Bass–Serre tree for the splitting. As in [24 ] , we refer to the vertex stabilizers for the action G ↷ T G \curvearrowright T G ↷ T vertex groups . Analogously, we refer to the edge stabilizers for the action G ↷ T G \curvearrowright T G ↷ T edge groups .
The quotient of T T T [40 Sec 5.4] yields a finite graph of groups which we denote by G ( Γ \ T ) \mathcal{G}(\Gamma\backslash T) G ( Γ\ T ) T T T G ( Γ ) \mathcal{G}(\Gamma) G ( Γ ) Γ \Gamma Γ G ( Γ ) \mathcal{G}(\Gamma) G ( Γ )
Alternatively, given a finite graph of groups G ( Γ ) \mathcal{G}(\Gamma) G ( Γ ) T T T Γ \Gamma Γ [40 Sec 5.3] . We will use this viewpoint in Section 3 and assume the action of Γ \Gamma Γ
Modified versions of fioravanti2022connected and karrer2024subgroups
Theorem 2.15 (Theorem A in [24 ] ). Suppose the action of a finitely generated group Γ \Gamma Γ T T T
(a) Each edge group E E E Γ \Gamma Γ
(b) ( ∂ ∗ E , Γ ) = ∅ (\partial_*E,\Gamma)=\varnothing ( ∂ ∗ E , Γ ) = ∅ E E E
Let x x x y y y T T T V x V_x V x V y V_y V y ( ∂ ∗ V x , Γ ) ∩ ( ∂ ∗ V y , Γ ) = ∅ (\partial_*V_x,\Gamma)\cap(\partial_*V_y,\Gamma)=\varnothing ( ∂ ∗ V x , Γ ) ∩ ( ∂ ∗ V y , Γ ) = ∅ V x V_x V x V y V_y V y Γ \Gamma Γ Γ \Gamma Γ C ⊆ ∂ ∗ Γ C\subseteq\partial_*\Gamma C ⊆ ∂ ∗ Γ V V V C C C ( ∂ ∗ V , Γ ) (\partial_*V,\Gamma) ( ∂ ∗ V , Γ )
Corollary 2.16. Let Γ \Gamma Γ C ⊆ ∂ ∗ Γ C\subseteq\partial_*\Gamma C ⊆ ∂ ∗ Γ ∣ C ∣ > 2 |C|>2 ∣ C ∣ > 2 ( ∂ ∗ V , Γ ) (\partial_*V,\Gamma) ( ∂ ∗ V , Γ ) V ≤ Γ V\leq\Gamma V ≤ Γ stab ( C ) ⊆ V \operatorname{stab}(C)\subseteq V stab ( C ) ⊆ V stab ( C ) \operatorname{stab}(C) stab ( C ) C C C Γ ↷ ∂ ∗ Γ \Gamma\curvearrowright\partial_*\Gamma Γ ↷ ∂ ∗ Γ
Proof. Let g ∈ Γ g\in\Gamma g ∈ Γ g C = C gC=C g C = C V V V C ⊆ ( ∂ ∗ V , Γ ) C\subseteq(\partial_*V,\Gamma) C ⊆ ( ∂ ∗ V , Γ ) g C ⊆ g ( ∂ ∗ V , Γ ) gC\subseteq g(\partial_*V,\Gamma) g C ⊆ g ( ∂ ∗ V , Γ ) g ( ∂ ∗ V , Γ ) = ( ∂ ∗ g V g − 1 , Γ ) g(\partial_*V,\Gamma)=(\partial_*gVg^{-1},\Gamma) g ( ∂ ∗ V , Γ ) = ( ∂ ∗ g V g − 1 , Γ ) g C ⊆ ( ∂ ∗ g V g − 1 , Γ ) gC\subseteq(\partial_*gVg^{-1},\Gamma) g C ⊆ ( ∂ ∗ g V g − 1 , Γ ) C = g C C=gC C = g C C ⊆ ( ∂ ∗ V , Γ ) ∩ ( ∂ ∗ g V g − 1 , Γ ) C\subseteq(\partial_*V,\Gamma)\cap(\partial_*gVg^{-1},\Gamma) C ⊆ ( ∂ ∗ V , Γ ) ∩ ( ∂ ∗ g V g − 1 , Γ ) g V g − 1 gVg^{-1} g V g − 1 V = g V g − 1 V=gVg^{-1} V = g V g − 1 g ∈ V g\in V g ∈ V
Motivated by the Theorem 2.15 and Theorem 2.16 above, we give the following definition that will be generalized to hierarchies in Section 3.
Definition 2.17. Let Γ \Gamma Γ G ( Γ ) \mathcal{G}(\Gamma) G ( Γ )
(a) Each edge group E E E Γ \Gamma Γ
(b) ( ∂ ∗ E , Γ ) = ∅ (\partial_*E,\Gamma)=\varnothing ( ∂ ∗ E , Γ ) = ∅ E E E
Then we call the graph of groups G ( Γ ) \mathcal{G}(\Gamma) G ( Γ ) weak Morse . If in addition, we have that ( ∂ ∗ V , Γ ) (\partial_*V,\Gamma) ( ∂ ∗ V , Γ ) V V V Morse .
When generalizing to hierarchies in Section 3, we need a variant of Theorem 2.15. In Theorem 2.15, no edge group contains a Morse ray. In the variant presented here, we skip this condition and instead study subspaces of ∂ ∗ Γ \partial_* \Gamma ∂ ∗ Γ [24 ] with minor adjustments.
Theorem 2.18 (Variant of Theorem A in [24 ] ). Suppose the action of a finitely generated group Γ \Gamma Γ T T T Γ \Gamma Γ A \mathcal{A} A
∂ ∗ Γ − ( ⋃ E edge group ( ∂ ∗ E , Γ ) ) \partial_*\Gamma-\left(\bigcup_{E\ \text{edge group}}(\partial_*E,\Gamma)\right) ∂ ∗ Γ − E edge group ⋃ ( ∂ ∗ E , Γ ) Let x x x y y y T T T V x V_x V x V y V_y V y ( ( ∂ ∗ V x , Γ ) ∩ A ) ∩ ( ( ∂ ∗ V y , Γ ) ∩ A ) = ∅ ((\partial_*V_x,\Gamma)\cap\mathcal{A})\cap((\partial_*V_y,\Gamma)\cap\mathcal{A})=\varnothing (( ∂ ∗ V x , Γ ) ∩ A ) ∩ (( ∂ ∗ V y , Γ ) ∩ A ) = ∅ C ⊆ A C\subseteq\mathcal{A} C ⊆ A V V V C C C ( ∂ ∗ V , Γ ) ∩ A (\partial_*V,\Gamma)\cap\mathcal{A} ( ∂ ∗ V , Γ ) ∩ A
Proof. Adjust the proof in Section 3 in [24 ] as follows:
In Assumption 3.1, remove the condition that the relative Morse boundary of every edge group is empty;
Note that the proofs of Lemmas 3.2 and 3.3 do not need the assumption that the edge groups have empty relative Morse boundary;
After the proof of Lemma 3.3, whenever the Morse boundary of the considered group is mentioned, replace it with A \mathcal{A} A ∂ M G \partial_M G ∂ M G A \mathcal{A} A
By Theorem 2.8, a Morse geodesic ray is of finite distance to an edge group E E E [24 ] ) of E E E
Whenever the assumption that the relative Morse boundaries (in the sense of [24 ] ) of the edge groups are empty, we instead use that A \mathcal{A} A
Whenever a statement is made about the relative Morse boundary of a subgroup, delete from this set all equivalence classes of Morse geodesic rays that are at finite distance of an edge group. This is in particular important for Lemma 3.9 and Corollary 3.10.
After these adjustments, the proof in Section 3 in [24 ] can be copied word by word. The notation for the relative Morse boundary in [24 ] is different from the notation in this paper (see Theorem 2.5). Thus, Theorem 2.18 follows after transferring the notation accordingly. ∎
We will also need the following variant of Lemma 6.7 in [KMZ26]. If γ : R → X \gamma:\mathbb{R}\to X γ : R → X X X X γ [ 0 , ∞ ) \gamma[0,\infty) γ [ 0 , ∞ ) γ ( − ∞ , 0 ] \gamma(-\infty,0] γ ( − ∞ , 0 ] γ + \gamma^{+} γ + γ − \gamma^{-} γ − X X X ∂ ∗ X \partial_*X ∂ ∗ X [ γ − ] [\gamma^{-}] [ γ − ] [ γ + ] [\gamma^{+}] [ γ + ]
Lemma 2.19. Suppose a finitely generated group Γ \Gamma Γ Γ \Gamma Γ
Let A \mathcal{A} A ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ
∂ ∗ Γ − ( ⋃ E e d g e g r o u p ( ∂ ∗ E , Γ ) ) \partial_*\Gamma-\left(\bigcup_{E\ \mathrm{edge\ group}}(\partial_*E,\Gamma)\right) ∂ ∗ Γ − E edge group ⋃ ( ∂ ∗ E , Γ ) Let N N N v v v X X X Γ \Gamma Γ ( γ n ) (\gamma_n) ( γ n ) N N N X X X γ \gamma γ v v v [ γ + ] [\gamma^+] [ γ + ] [ γ − ] [\gamma^-] [ γ − ] A \mathcal{A} A [ γ n − ] [\gamma_n^-] [ γ n − ] [ γ n + ] [\gamma_n^+] [ γ n + ] C n ⊆ A C_n\subseteq\mathcal{A} C n ⊆ A n n n i 0 i_0 i 0 V V V Γ \Gamma Γ i ≥ i 0 i\geq i_0 i ≥ i 0 [ γ + ] [\gamma^+] [ γ + ] [ γ − ] [\gamma^-] [ γ − ] [ γ i + ] [\gamma_i^+] [ γ i + ] [ γ i − ] ∈ ( ∂ ∗ V , Γ ) ∩ A [\gamma_i^-]\in(\partial_*V,\Gamma)\cap\mathcal{A} [ γ i − ] ∈ ( ∂ ∗ V , Γ ) ∩ A
Proof. Follow the proof by contradiction of Lemma 6.7 in [KMZ26], which relies on Lemmas 3.7 and 3.2 from [24 ] . Instead of these original lemmas, use the corresponding variants that are obtained by changing the setup in [24 ] to the setup here. This will lead to the conclusion that [ γ − ] [\gamma^-] [ γ − ] [ γ + ] [\gamma^+] [ γ + ] [ γ + ] [\gamma^+] [ γ + ] [ γ − ] [\gamma^-] [ γ − ] A \mathcal{A} A
Morse accessibility and its consequences
A proposed definition of Morse accessibility was introduced in [KMZ26]. In this section, we provide a formal definition of Morse accessibility which is a slight variation of the definition given in that paper (see § 7 for further explanation of the difference). In order to formulate our definition, we introduce Morse hierarchies and then methodically transfer all the results proven for group splittings along finitely generated undistorted subgroups with empty relative Morse boundaries from [24 ] to Morse hierarchies. This will give Theorem 3.20 (Theorem A) and Theorem 3.21 (Theorem B) which provide structural consequences of a group being Morse accessible. Additionally, Theorem 3.7 shows that the relative Morse boundary of a finitely generated undistorted subgroup of a finitely generated group cannot consist of just one point—i.e., it will contain 0, 2, or infinitely many points.
Definition of Morse hierarchies
3.1. Definition of Morse hierarchies. Recall we will use Convention 1 throughout this section when discussing splittings; in particular, actions on Bass–Serre trees are assumed to be non-elliptic and minimal.
Definition 3.1 (Hierarchies). A hierarchy H \mathcal{H} H Γ \Gamma Γ Γ \Gamma Γ r r r Γ \Gamma Γ v ∈ V ( H ) v \in\mathcal{V}(\mathcal{H}) v ∈ V ( H ) Γ v \Gamma_v Γ v G ( Γ v ) \mathcal{G}(\Gamma_v) G ( Γ v ) v v v V ( H ) \mathcal{V}(\mathcal{H}) V ( H ) G ( Γ v ) \mathcal{G}(\Gamma_v) G ( Γ v ) finite if H \mathcal{H} H h h h
h = max { d ( r , v ) ∣ v ∈ V ( H ) } , h = \max\{d(r,v) \mid v \in\mathcal{V}(\mathcal{H})\}, h = max { d ( r , v ) ∣ v ∈ V ( H )} , with r r r H \mathcal{H} H V V V Γ \Gamma Γ Γ v \Gamma_v Γ v v v v vertex group of H \mathcal{H} H § 2.2 . If, in addition, v v v H \mathcal{H} H L L L Γ v \Gamma_v Γ v leaf group .
Definition 3.3 (Morse hierarchies). A hierarchy H \mathcal{H} H Γ \Gamma Γ weak Morse if it is finite and for each non-leaf vertex v ∈ V ( H ) v \in\mathcal{V}(\mathcal{H}) v ∈ V ( H ) G ( Γ v ) \mathcal{G}(\Gamma_v) G ( Γ v ) Γ v \Gamma_v Γ v Γ v \Gamma_v Γ v Γ \Gamma Γ Morse hierarchy if for each leaf v v v H \mathcal{H} H ( ∂ ∗ Γ v , Γ ) (\partial_*\Gamma_v,\Gamma) ( ∂ ∗ Γ v , Γ ) ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ
Definition 3.4. A finitely generated group Γ \Gamma Γ Morse accessible if there exists a Morse hierarchy of Γ \Gamma Γ
We end this subsection by ruling out the possibility of a singleton connected component for a leaf group in a Morse hierarchy before moving on to the main results of the section.
Lemma 3.6. Suppose H H H Γ \Gamma Γ H H H ∣ ( ∂ ∗ H , Γ ) ∣ ≠ 1 \lvert(\partial_*H,\Gamma)\rvert\ne1 ∣( ∂ ∗ H , Γ )∣ = 1
Proof. Assume H H H Γ \Gamma Γ ∣ ( ∂ ∗ H , Γ ) ∣ = { [ ξ ] } \lvert(\partial_*H,\Gamma)\rvert= \{[\xi]\} ∣( ∂ ∗ H , Γ )∣ = {[ ξ ]} ⟨ g ⟩ \langle g\rangle ⟨ g ⟩ H H H ∣ ( ∂ ∗ H , Γ ) ∣ \lvert(\partial_*H,\Gamma)\rvert ∣( ∂ ∗ H , Γ )∣ [ ξ ] [\xi] [ ξ ] g i [ ξ ] = [ ξ ] g^i[\xi] = [\xi] g i [ ξ ] = [ ξ ] i ∈ N i \in\mathbb{N} i ∈ N ξ \xi ξ N N N [ ξ ] [\xi] [ ξ ] e e e
We first claim that { g i e } Z ≥ 0 \{g^i e\}_{\mathbb{Z}_{\ge0}} { g i e } Z ≥ 0 N ′ N' N ′ [ { g i e } ] = [ ξ ] [\{g^i e\}] = [\xi] [{ g i e }] = [ ξ ] i ∈ N i \in\mathbb{N} i ∈ N ξ ∪ g i ξ ∪ [ e , g i ] \xi\cup g^i\xi\cup[e,g^i] ξ ∪ g i ξ ∪ [ e , g i ] [8 ] , since both ξ \xi ξ g i ξ g^i\xi g i ξ N N N [ e , g i ] [e,g^i] [ e , g i ] N ′ N' N ′ N ′ N' N ′ N N N ⟨ g ⟩ \langle g\rangle ⟨ g ⟩ d ( e , g i ) → ∞ d(e,g^i) \to\infty d ( e , g i ) → ∞ i → ∞ i \to\infty i → ∞ [ e , g i ] [e,g^i] [ e , g i ] N ′ N' N ′ ⟨ g ⟩ \langle g\rangle ⟨ g ⟩ { e , g , g 2 , … , g i } \{e,g,g^2,\ldots,g^i\} { e , g , g 2 , … , g i } [19 ] , we know that [ e , g i ] [e,g^i] [ e , g i ] N ′ N' N ′ α \alpha α g g g [ α ] ∈ ( ∂ ∗ H , Γ ) [\alpha] \in(\partial_*H,\Gamma) [ α ] ∈ ( ∂ ∗ H , Γ ) [ α ] = [ ξ ] [\alpha] = [\xi] [ α ] = [ ξ ]
We can repeat this argument with the non-positive powers of g g g N ′ N' N ′ β \beta β g g g ⟨ g ⟩ \langle g\rangle ⟨ g ⟩ [ β ] = [ ξ ] [\beta] = [\xi] [ β ] = [ ξ ]
Corollary 3.7. If H H H Γ \Gamma Γ H H H Γ \Gamma Γ ( ∂ ∗ H , Γ ) (\partial_*H,\Gamma) ( ∂ ∗ H , Γ )
Proof. The subgroup H H H Γ \Gamma Γ ι : H ↪ Γ \iota: H \hookrightarrow\Gamma ι : H ↪ Γ M ( H , Γ ) \mathcal{M}(H,\Gamma) M ( H , Γ ) ∂ ∗ H \partial_*H ∂ ∗ H H H H Γ \Gamma Γ ι \iota ι
ι ∗ : M ( H , Γ ) → ( ∂ ∗ H , Γ ) ⊆ ∂ ∗ Γ \iota_* : \mathcal{M}(H,\Gamma) \to(\partial_*H,\Gamma) \subseteq\partial_*\Gamma ι ∗ : M ( H , Γ ) → ( ∂ ∗ H , Γ ) ⊆ ∂ ∗ Γ By [35 Cor 5.8] , ∂ ∗ H \partial_*H ∂ ∗ H
If ∂ ∗ H = ∅ \partial_*H = \varnothing ∂ ∗ H = ∅ M ( H , Γ ) = ∅ \mathcal{M}(H,\Gamma) = \varnothing M ( H , Γ ) = ∅ ( ∂ ∗ H , Γ ) = ι ∗ ( M ( H , Γ ) ) = ∅ (\partial_*H,\Gamma) = \iota_*(\mathcal{M}(H,\Gamma)) = \varnothing ( ∂ ∗ H , Γ ) = ι ∗ ( M ( H , Γ )) = ∅
If ∣ ∂ ∗ H ∣ = 2 \lvert\partial_*H\rvert= 2 ∣ ∂ ∗ H ∣ = 2 H H H Γ \Gamma Γ ( ∂ ∗ H , Γ ) = ι ∗ ( M ( H , Γ ) ) (\partial_*H,\Gamma) = \iota_*(\mathcal{M}(H,\Gamma)) ( ∂ ∗ H , Γ ) = ι ∗ ( M ( H , Γ ))
Finally, assume that ∣ ∂ ∗ H ∣ = ∞ |\partial_* H| = \infty ∣ ∂ ∗ H ∣ = ∞ ( ∂ ∗ H , Γ ) ≠ ∅ (\partial_* H,\Gamma) \ne\varnothing ( ∂ ∗ H , Γ ) = ∅ H H H γ \gamma γ ι ∗ ( [ γ ] ) ∈ ( ∂ ∗ H , Γ ) \iota_*([\gamma]) \in(\partial_* H,\Gamma) ι ∗ ([ γ ]) ∈ ( ∂ ∗ H , Γ ) [35 Cor 6.2] , the action of H H H ∂ ∗ H \partial_* H ∂ ∗ H H [ γ ] H[\gamma] H [ γ ] ι ∗ \iota_* ι ∗ H H H H [ γ ] H[\gamma] H [ γ ] ( ∂ ∗ H , Γ ) (\partial_* H,\Gamma) ( ∂ ∗ H , Γ )
Thus the corollary is proven.
Corollary 3.8. If Γ \Gamma Γ
Proof. By Theorem 3.15 in the next subsection, every leaf group is finitely generated and undistorted. Thus, Theorem 3.7 applies. □
From graphs of groups to hierarchies: the proofs of thmMorseaccessiblegroupsIntro and corMorseaccessiblegroupsIntro
In this subsection, we will prove Theorem 3.20 and Theorem 3.21 which are Theorem A and Theorem B from the introduction. The strategy is to carefully generalize the results on splittings of groups along subgroups with empty relative Morse boundaries from [24 ] to hierarchies using induction.
Let Γ \Gamma Γ H \mathcal{H} H Γ \Gamma Γ H \mathcal{H} H H \mathcal{H} H H \mathcal{H} H Γ \Gamma Γ H \mathcal{H} H Γ \Gamma Γ [24 ] via induction, we introduce the rooted blow-up tree H ‾ \overline{\mathcal{H}} H H \mathcal{H} H Γ ↷ H ‾ \Gamma\curvearrowright\overline{\mathcal{H}} Γ ↷ H H \mathcal{H} H H ‾ \overline{\mathcal{H}} H H ‾ \overline{\mathcal{H}} H Γ v \Gamma_v Γ v v ∈ H v \in\mathcal{H} v ∈ H H \mathcal{H} H
Definition 3.9 (blow-up tree of H \mathcal{H} H Γ \Gamma Γ H \mathcal{H} H r r r H ‾ \overline{\mathcal{H}} H H \mathcal{H} H
V ( H ‾ ) = { w g Γ v ∣ g ∈ Γ and v ∈ V ( H ) } V(\overline{\mathcal{H}}) = \{w_g\Gamma_v \mid g \in\Gamma\text{ and } v \in V(\mathcal{H})\} V ( H ) = { w g Γ v ∣ g ∈ Γ and v ∈ V ( H )}
E ( H ‾ ) = { { w g Γ u , w h Γ v } | { u , v } ∈ E ( H ) and g Γ u ⊆ h Γ v or g Γ v ⊆ h Γ u } E(\overline{\mathcal{H}}) = \left\{\{w_g\Gamma_u,w_h\Gamma_v\} \,\middle|\, \{u,v\} \in E(\mathcal{H}) \text{ and } g\Gamma_u \subseteq h\Gamma_v \text{ or } g\Gamma_v \subseteq h\Gamma_u\right\} E ( H ) = { { w g Γ u , w h Γ v } ∣ { u , v } ∈ E ( H ) and g Γ u ⊆ h Γ v or g Γ v ⊆ h Γ u }
The root of H ‾ \overline{\mathcal{H}} H w Γ r w_{\Gamma_r} w Γ r
Remark 3.10. Let w g Γ v w_g\Gamma_v w g Γ v H ‾ \overline{\mathcal{H}} H g ∈ Γ g \in\Gamma g ∈ Γ Γ v \Gamma_v Γ v v ∈ H v \in\mathcal{H} v ∈ H u 1 , … , u n u_1,\ldots,u_n u 1 , … , u n v v v H \mathcal{H} H Γ u 1 , … , Γ u n \Gamma_{u_1},\ldots,\Gamma_{u_n} Γ u 1 , … , Γ u n D D D w g Γ v w_g\Gamma_v w g Γ v
D = { w g h Γ u i ∣ h ∈ Γ v , i ∈ { 1 , … , n } } . D = \{w_{gh}\Gamma_{u_i} \mid h \in\Gamma_v, i \in\{1,\ldots,n\}\}. D = { w g h Γ u i ∣ h ∈ Γ v , i ∈ { 1 , … , n }} . Let T v T_v T v G ( Γ v ) \mathcal{G}(\Gamma_v) G ( Γ v ) T v T_v T v G ( Γ v ) \mathcal{G}(\Gamma_v) G ( Γ v ) D D D g T v gT_v g T v T v T_v T v g g g T v T_v T v g Γ v g − 1 g\Gamma_vg^{-1} g Γ v g − 1 g T v gT_v g T v g g g
Recall that the i i i i i i
Lemma 3.11. Suppose H ‾ \overline{\mathcal H} H H \mathcal H H i i i H ‾ \overline{\mathcal H} H g Γ w g\Gamma_w g Γ w g ∈ Γ g \in\Gamma g ∈ Γ Γ w \Gamma_w Γ w i i i H \mathcal H H H \mathcal H H H ‾ \overline{\mathcal H} H
We observe:
Lemma 3.12 (The action of Γ \Gamma Γ H ‾ \overline{\mathcal H} H Γ \Gamma Γ H \mathcal H H H ‾ \overline{\mathcal H} H H \mathcal H H Γ \Gamma Γ H ‾ \overline{\mathcal H} H H ‾ \overline{\mathcal H} H w Γ u w\Gamma_u w Γ u H ‾ \overline{\mathcal H} H g Γ u g − 1 g\Gamma_ug^{-1} g Γ u g − 1
Corollary 3.13. Let Γ \Gamma Γ H \mathcal H H
{ g Γ v g − 1 ∣ g ∈ Γ , v ∈ V ( H ) } \{g\Gamma_vg^{-1} \mid g \in\Gamma, v \in V(\mathcal H)\} { g Γ v g − 1 ∣ g ∈ Γ , v ∈ V ( H )} of H \mathcal H H
{ stab ( w ) ∣ w ∈ V ( H ‾ ) } . \{\operatorname{stab}(w) \mid w \in V(\overline{\mathcal H})\}. { stab ( w ) ∣ w ∈ V ( H )} . And, the set of leaf groups
{ g Γ v g − 1 ∣ g ∈ Γ , v is a leaf of H } \{g\Gamma_vg^{-1} \mid g \in\Gamma, v\text{ is a leaf of }\mathcal H\} { g Γ v g − 1 ∣ g ∈ Γ , v is a leaf of H } coincides with the set of leaf stabilizers
{ stab ( w ) ∣ w is a leaf of H ‾ } . \{\operatorname{stab}(w) \mid w\text{ is a leaf of }\overline{\mathcal H}\}. { stab ( w ) ∣ w is a leaf of H } . If the height of H \mathcal H H Γ \Gamma Γ H \mathcal H H
The above enables us to generalize the results in [24 ] to weak Morse hierarchies via induction. Here, we use the following crucial property of weak Morse hierarchies:
restricts the action of Γ \Gamma Γ H ‾ n \overline{\mathcal{H}}_n H n H ‾ m \overline{\mathcal{H}}_m H m Γ \Gamma Γ H ‾ m \overline{\mathcal{H}}_m H m H m \mathcal{H}_m H m m ≤ n m \le n m ≤ n
As a first step, the described approach can be used to extend Lemma 2.15 from [24 ] which shows that in a graph of groups decomposition of Γ \Gamma Γ Γ \Gamma Γ
Lemma 3.15. Let H \mathcal{H} H Γ \Gamma Γ
If every edge group of H \mathcal{H} H H \mathcal{H} H Γ \Gamma Γ
Suppose a a a H \mathcal{H} H H \mathcal{H} H b b b H \mathcal{H} H B B B b b b A A A a a a H \mathcal{H} H ( ∂ ∗ A , Γ ) ⊆ ( ∂ ∗ B , Γ ) (\partial_* A,\Gamma) \subseteq(\partial_* B,\Gamma) ( ∂ ∗ A , Γ ) ⊆ ( ∂ ∗ B , Γ )
Theorem A in [24 ] (see Theorem 2.15) shows that if a finitely generated group Γ \Gamma Γ
If v v v w w w v v v w w w [ v , w ] [v,w] [ v , w ] V ( [ v , w ] ) \mathcal{V}([v,w]) V ([ v , w ]) x ∈ [ v , w ] x \in[v,w] x ∈ [ v , w ] x x x
Lemma 3.16. Let H \mathcal{H} H Γ \Gamma Γ H ‾ \overline{\mathcal{H}} H r r r A A A B B B H \mathcal{H} H ( ∂ ∗ A , Γ ) ∩ ( ∂ ∗ B , Γ ) ≠ ∅ (\partial_* A,\Gamma) \cap(\partial_* B,\Gamma) \ne\varnothing ( ∂ ∗ A , Γ ) ∩ ( ∂ ∗ B , Γ ) = ∅ A A A a a a H ‾ \overline{\mathcal{H}} H B B B b b b H ‾ \overline{\mathcal{H}} H [ r , a ] ⊆ [ r , b ] [r,a] \subseteq[r,b] [ r , a ] ⊆ [ r , b ] [ r , b ] ⊆ [ r , a ] [r,b] \subseteq[r,a] [ r , b ] ⊆ [ r , a ]
Proof. We use induction on the height of a weak Morse hierarchy.
Induction base: Let H 1 \mathcal{H}_1 H 1 Γ \Gamma Γ Γ \Gamma Γ H 1 \mathcal{H}_1 H 1 Γ \Gamma Γ [24 ] (see Theorem 2.15) directly implies the claim.
Induction step: Let H n + 1 \mathcal{H}_{n+1} H n + 1 Γ \Gamma Γ n + 1 n+1 n + 1 H ‾ n + 1 \overline{\mathcal{H}}_{n+1} H n + 1 A A A B B B H n + 1 \mathcal{H}_{n+1} H n + 1 ( ∂ ∗ A , Γ ) ∩ ( ∂ ∗ B , Γ ) ≠ ∅ (\partial_* A,\Gamma) \cap(\partial_* B,\Gamma) \ne\varnothing ( ∂ ∗ A , Γ ) ∩ ( ∂ ∗ B , Γ ) = ∅ A A A B B B H ‾ n + 1 \overline{\mathcal{H}}_{n+1} H n + 1
Let a , b a,b a , b H ‾ n + 1 \overline{\mathcal{H}}_{n+1} H n + 1 A = stab ( a ) A = \operatorname{stab}(a) A = stab ( a ) B = stab ( b ) B = \operatorname{stab}(b) B = stab ( b ) [ r , a ] ⊆ [ r , b ] [r,a] \subseteq[r,b] [ r , a ] ⊆ [ r , b ] [ r , b ] ⊆ [ r , a ] [r,b] \subseteq[r,a] [ r , b ] ⊆ [ r , a ]
Claim 1: Either [ r , a ] ⊆ [ r , b ] [r,a] \subseteq[r,b] [ r , a ] ⊆ [ r , b ] [ r , b ] ⊆ [ r , a ] [r,b] \subseteq[r,a] [ r , b ] ⊆ [ r , a ] a a a b b b H ‾ n + 1 \overline{\mathcal{H}}_{n+1} H n + 1 z ∈ V ( H ‾ n + 1 ) z \in V(\overline{\mathcal{H}}_{n+1}) z ∈ V ( H n + 1 )
Claim 2: If a a a b b b H ‾ n + 1 \overline{\mathcal{H}}_{n+1} H n + 1 a = b a=b a = b
We now prove the uniqueness of a a a b b b
Uniqueness of a a a We first prove uniqueness assuming Claim 1 and Claim 2. By symmetry it suffices to show that the vertex a a a a a a a ′ a' a ′ H ‾ n + 1 \overline{\mathcal{H}}_{n+1} H n + 1 stab ( a ) = stab ( a ′ ) = A \operatorname{stab}(a) = \operatorname{stab}(a') = A stab ( a ) = stab ( a ′ ) = A ( ∂ ∗ A , Γ ) ∩ ( ∂ ∗ B , Γ ) ≠ ∅ (\partial_* A,\Gamma) \cap(\partial_* B,\Gamma) \ne\varnothing ( ∂ ∗ A , Γ ) ∩ ( ∂ ∗ B , Γ ) = ∅ ( ∂ ∗ A , Γ ) ≠ ∅ (\partial_* A,\Gamma) \ne\varnothing ( ∂ ∗ A , Γ ) = ∅ A = B A=B A = B a a a a ′ a' a ′ r r r H ‾ n + 1 \overline{\mathcal{H}}_{n+1} H n + 1 H ‾ n + 1 \overline{\mathcal{H}}_{n+1} H n + 1 a = a ′ a=a' a = a ′
We will now prove Claim 1 and Claim 2.
Proof of Claim 1: Let H ‾ n \overline{\mathcal{H}}_n H n H ‾ n + 1 \overline{\mathcal{H}}_{n+1} H n + 1 n n n r r r H ‾ n + 1 \overline{\mathcal{H}}_{n+1} H n + 1 H ‾ n \overline{\mathcal{H}}_n H n H n \mathcal{H}_n H n Γ \Gamma Γ n n n a a a b b b a a a H ‾ n + 1 \overline{\mathcal{H}}_{n+1} H n + 1 a ˉ \bar{a} a ˉ a a a b ˉ \bar{b} b ˉ r r r b b b b = r b=r b = r b ˉ = r \bar{b}=r b ˉ = r b ˉ \bar{b} b ˉ b b b a ˉ \bar{a} a ˉ b ˉ \bar{b} b ˉ H ‾ n \overline{\mathcal{H}}_n H n A ˉ \bar{A} A ˉ B ˉ \bar{B} B ˉ a ˉ \bar{a} a ˉ b ˉ \bar{b} b ˉ ( ∂ ∗ B , Γ ) ⊆ ( ∂ ∗ B ˉ , Γ ) (\partial_* B,\Gamma) \subseteq(\partial_* \bar{B},\Gamma) ( ∂ ∗ B , Γ ) ⊆ ( ∂ ∗ B ˉ , Γ ) ( ∂ ∗ A , Γ ) ⊆ ( ∂ ∗ A ˉ , Γ ) (\partial_* A,\Gamma) \subseteq(\partial_* \bar{A},\Gamma) ( ∂ ∗ A , Γ ) ⊆ ( ∂ ∗ A ˉ , Γ ) ( ∂ ∗ A , Γ ) ∩ ( ∂ ∗ B , Γ ) ≠ ∅ (\partial_* A,\Gamma) \cap(\partial_* B,\Gamma) \ne\varnothing ( ∂ ∗ A , Γ ) ∩ ( ∂ ∗ B , Γ ) = ∅ ( ∂ ∗ A ˉ , Γ ) ∩ ( ∂ ∗ B ˉ , Γ ) ≠ ∅ (\partial_* \bar{A},\Gamma) \cap(\partial_* \bar{B},\Gamma) \ne\varnothing ( ∂ ∗ A ˉ , Γ ) ∩ ( ∂ ∗ B ˉ , Γ ) = ∅ a ˉ \bar{a} a ˉ H ‾ n \overline{\mathcal{H}}_n H n [ r , b ˉ ] ⊆ [ r , a ˉ ] [r,\bar{b}] \subseteq[r,\bar{a}] [ r , b ˉ ] ⊆ [ r , a ˉ ] a ˉ = b ˉ \bar{a}=\bar{b} a ˉ = b ˉ a a a b b b b ∈ [ r , a ˉ ] ⊆ H ‾ n b \in[r,\bar{a}] \subseteq\overline{\mathcal{H}}_n b ∈ [ r , a ˉ ] ⊆ H n [ r , b ] ⊆ [ r , a ˉ ] ⊆ [ r , a ] [r,b] \subseteq[r,\bar{a}] \subseteq[r,a] [ r , b ] ⊆ [ r , a ˉ ] ⊆ [ r , a ]
Proof of Claim 2: Let a a a b b b H ‾ n + 1 \overline{\mathcal{H}}_{n+1} H n + 1 z z z A = stab ( a ) A=\operatorname{stab}(a) A = stab ( a ) B = stab ( b ) B=\operatorname{stab}(b) B = stab ( b ) Z = stab ( z ) Z=\operatorname{stab}(z) Z = stab ( z ) H ‾ n + 1 \overline{\mathcal{H}}_{n+1} H n + 1 g ∈ Γ g \in\Gamma g ∈ Γ h a , h b ∈ Γ z h_a,h_b \in\Gamma_z h a , h b ∈ Γ z Γ a \Gamma_a Γ a Γ b \Gamma_b Γ b G ( Γ z ) \mathcal{G}(\Gamma_z) G ( Γ z ) z = w g Γ z z=w g\Gamma_z z = w g Γ z a = w g h a Γ a a=w g h_a\Gamma_a a = w g h a Γ a b = w g h b Γ b b=w g h_b\Gamma_b b = w g h b Γ b g h a Γ a = g h b Γ b g h_a\Gamma_a=g h_b\Gamma_b g h a Γ a = g h b Γ b
Let G ( Γ z ) \mathcal{G}(\Gamma_z) G ( Γ z ) Γ z \Gamma_z Γ z T z T_z T z G ( Γ z ) \mathcal{G}(\Gamma_z) G ( Γ z ) G ( Γ z ) \mathcal{G}(\Gamma_z) G ( Γ z ) g T z gT_z g T z T z T_z T z g g g Z = g Γ z g − 1 Z=g\Gamma_zg^{-1} Z = g Γ z g − 1 g G ( Γ ) g − 1 g\mathcal{G}(\Gamma)g^{-1} g G ( Γ ) g − 1 g T z gT_z g T z g T z gT_z g T z x x x g h a Γ a gh_a\Gamma_a g h a Γ a y y y g h b Γ b gh_b\Gamma_b g h b Γ b x = y x=y x = y g h a Γ a = g h b Γ b gh_a\Gamma_a=gh_b\Gamma_b g h a Γ a = g h b Γ b
To simplify notation, we assume that g = id g=\operatorname{id} g = id A A A B B B Γ z \Gamma_z Γ z G ( Γ z ) \mathcal{G}(\Gamma_z) G ( Γ z ) g ≠ id g\ne\operatorname{id} g = id
The stabilizers A = Γ a h a A=\Gamma_a^{h_a} A = Γ a h a B = Γ b h b B=\Gamma_b^{h_b} B = Γ b h b Γ a \Gamma_a Γ a Γ b \Gamma_b Γ b Γ z \Gamma_z Γ z Γ z \Gamma_z Γ z Γ z \Gamma_z Γ z Γ \Gamma Γ ι : Γ z ↪ Γ \iota:\Gamma_z\hookrightarrow\Gamma ι : Γ z ↪ Γ M ( Γ z , Γ ) \mathcal{M}(\Gamma_z,\Gamma) M ( Γ z , Γ ) ∂ ∗ Γ z \partial_*\Gamma_z ∂ ∗ Γ z Γ z \Gamma_z Γ z Γ \Gamma Γ ι \iota ι
ι ∗ : M ( Γ z , Γ ) → ( ∂ ∗ Γ z , Γ ) ⊆ ∂ ∗ Γ . \iota_*:\mathcal{M}(\Gamma_z,\Gamma)\to(\partial_*\Gamma_z,\Gamma)\subseteq\partial_*\Gamma. ι ∗ : M ( Γ z , Γ ) → ( ∂ ∗ Γ z , Γ ) ⊆ ∂ ∗ Γ. As A A A B B B Γ z \Gamma_z Γ z
( ∂ ∗ A , Γ ) ∩ ( ∂ ∗ B , Γ ) = ι ∗ ( M ( Γ z , Γ ) ∩ ( ∂ ∗ A , Γ z ) ) ∩ ι ∗ ( M ( Γ z , Γ ) ∩ ( ∂ ∗ B , Γ z ) ) . (\partial_*A,\Gamma)\cap(\partial_*B,\Gamma)=\iota_*(\mathcal{M}(\Gamma_z,\Gamma)\cap(\partial_*A,\Gamma_z))\cap\iota_*(\mathcal{M}(\Gamma_z,\Gamma)\cap(\partial_*B,\Gamma_z)). ( ∂ ∗ A , Γ ) ∩ ( ∂ ∗ B , Γ ) = ι ∗ ( M ( Γ z , Γ ) ∩ ( ∂ ∗ A , Γ z )) ∩ ι ∗ ( M ( Γ z , Γ ) ∩ ( ∂ ∗ B , Γ z )) . Injectivity of ι ∗ \iota_* ι ∗
( ∂ ∗ A , Γ ) ∩ ( ∂ ∗ B , Γ ) = ι ∗ ( ( M ( Γ z , Γ ) ∩ ( ∂ ∗ A , Γ z ) ) ∩ ( M ( Γ z , Γ ) ∩ ( ∂ ∗ B , Γ z ) ) ) = ι ∗ ( M ( Γ z , Γ ) ∩ ( ∂ ∗ A , Γ z ) ∩ ( ∂ ∗ B , Γ z ) ) . \begin{aligned}
(\partial_*A,\Gamma)\cap(\partial_*B,\Gamma)
&=\iota_*\big((\mathcal{M}(\Gamma_z,\Gamma)\cap(\partial_*A,\Gamma_z))\cap(\mathcal{M}(\Gamma_z,\Gamma)\cap(\partial_*B,\Gamma_z))\big)\\
&=\iota_*(\mathcal{M}(\Gamma_z,\Gamma)\cap(\partial_*A,\Gamma_z)\cap(\partial_*B,\Gamma_z)).
\end{aligned} ( ∂ ∗ A , Γ ) ∩ ( ∂ ∗ B , Γ ) = ι ∗ ( ( M ( Γ z , Γ ) ∩ ( ∂ ∗ A , Γ z )) ∩ ( M ( Γ z , Γ ) ∩ ( ∂ ∗ B , Γ z )) ) = ι ∗ ( M ( Γ z , Γ ) ∩ ( ∂ ∗ A , Γ z ) ∩ ( ∂ ∗ B , Γ z )) . Accordingly,
( ∂ ∗ A , Γ ) ∩ ( ∂ ∗ B , Γ ) ≠ ∅ ⇒ M ( Γ z , Γ ) ∩ ( ∂ ∗ A , Γ z ) ∩ ( ∂ ∗ B , Γ z ) ≠ ∅ . (\partial_*A,\Gamma)\cap(\partial_*B,\Gamma)\ne\varnothing\Rightarrow\mathcal{M}(\Gamma_z,\Gamma)\cap(\partial_*A,\Gamma_z)\cap(\partial_*B,\Gamma_z)\ne\varnothing. ( ∂ ∗ A , Γ ) ∩ ( ∂ ∗ B , Γ ) = ∅ ⇒ M ( Γ z , Γ ) ∩ ( ∂ ∗ A , Γ z ) ∩ ( ∂ ∗ B , Γ z ) = ∅ . Since every edge group of G z \mathcal{G}_z G z ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ
M ( Γ z , Γ ) ⊆ ∂ ∗ Γ z − ( ⋃ E edge group of G z ( ∂ ∗ E , Γ z ) ) . \mathcal{M}(\Gamma_z,\Gamma)\subseteq\partial_*\Gamma_z-\left(\bigcup_{E\ \text{edge group of }\mathcal{G}_z}(\partial_*E,\Gamma_z)\right). M ( Γ z , Γ ) ⊆ ∂ ∗ Γ z − E edge group of G z ⋃ ( ∂ ∗ E , Γ z ) . Theorem 2.18 implies that A A A B B B T z T_z T z x = y x=y x = y
Theorem A in [24 ] (see Theorem 2.15) and Theorem 2.16 shows that the infinite Morse boundary components and their stabilizers of a weak Morse splitting arise from the vertex groups. The following proposition generalizes this result to weak Morse hierarchies of arbitrary height. Recall that if C C C ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ stab ( C ) \operatorname{stab}(C) stab ( C ) C C C Γ ↷ ∂ ∗ Γ \Gamma\curvearrowright\partial_*\Gamma Γ ↷ ∂ ∗ Γ
Proposition 3.17. Let H ‾ \overline{\mathcal{H}} H H \mathcal{H} H Γ \Gamma Γ r r r C C C ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ v v v H ‾ \overline{\mathcal{H}} H
stabilizer W W W w w w [ r , v ] [r,v] [ r , v ]
C ⊆ ( ∂ ∗ W , Γ ) ( † ) stab ( C ) ⊆ W ( ‡ ) \begin{aligned}
C &\subseteq(\partial_*W,\Gamma) &&(\dagger)\\
\operatorname{stab}(C) &\subseteq W &&(\ddagger)
\end{aligned} C stab ( C ) ⊆ ( ∂ ∗ W , Γ ) ⊆ W ( † ) ( ‡ ) Proof. We show the claim by induction on the height of a weak Morse hierarchy.
Induction base: Let H 1 \mathcal{H}_1 H 1 Γ \Gamma Γ H 1 \mathcal{H}_1 H 1 ( † ) (\dagger) ( † ) ( ‡ ) (\ddagger) ( ‡ )
Induction step: Let H n + 1 \mathcal{H}_{n+1} H n + 1 n + 1 n+1 n + 1 Γ \Gamma Γ H ‾ n + 1 \overline{\mathcal{H}}_{n+1} H n + 1 r r r C C C ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ H n \mathcal{H}_n H n n n n Γ \Gamma Γ H ‾ n \overline{\mathcal{H}}_n H n z z z w ∈ V ( [ r , z ] ) w \in V([r,z]) w ∈ V ([ r , z ]) ( † ) (\dagger) ( † ) ( ‡ ) (\ddagger) ( ‡ ) [ r , z ] [r,z] [ r , z ] r r r v v v H ‾ n + 1 \overline{\mathcal{H}}_{n+1} H n + 1 [ r , v ] [r,v] [ r , v ] ( † ) (\dagger) ( † ) ( ‡ ) (\ddagger) ( ‡ )
Existence of a leaf v v v [ r , v ] [r,v] [ r , v ] ( † ) (\dagger) ( † ) For the existence, it suffices to show that there exists one descendant v v v z z z V V V C ⊆ ( ∂ ∗ V , Γ ) C \subseteq(\partial_*V,\Gamma) C ⊆ ( ∂ ∗ V , Γ ) H n + 1 \mathcal{H}_{n+1} H n + 1 z = g z Γ z z = g_z\Gamma_z z = g z Γ z g z ∈ Γ g_z \in\Gamma g z ∈ Γ Γ z \Gamma_z Γ z H \mathcal{H} H Z = g z Γ z g z − 1 Z = g_z\Gamma_zg_z^{-1} Z = g z Γ z g z − 1 z z z g z = id g_z = \operatorname{id} g z = id C ⊆ ( ∂ ∗ Γ z , Γ ) C \subseteq(\partial_*\Gamma_z,\Gamma) C ⊆ ( ∂ ∗ Γ z , Γ ) g z ≠ id g_z \ne\operatorname{id} g z = id
ι ∗ : M ( Γ z , Γ ) → ( ∂ ∗ Γ z , Γ ) \iota_*:\mathcal{M}(\Gamma_z,\Gamma)\to(\partial_*\Gamma_z,\Gamma) ι ∗ : M ( Γ z , Γ ) → ( ∂ ∗ Γ z , Γ ) By Lemma 2.3 in [FK22] (see Theorem 2.9), the map ι ∗ \iota_* ι ∗ ι ∗ \iota_* ι ∗ S ⊆ M ( Γ v , Γ ) S \subseteq\mathcal{M}(\Gamma_v,\Gamma) S ⊆ M ( Γ v , Γ ) ι ∗ ( S ) = C \iota_*(S) = C ι ∗ ( S ) = C C z C_z C z M ( Γ v , Γ ) \mathcal{M}(\Gamma_v,\Gamma) M ( Γ v , Γ ) S S S Γ z \Gamma_z Γ z G ( Γ z ) \mathcal{G}(\Gamma_z) G ( Γ z )
M ( Γ z , Γ ) ⊆ ∂ ∗ Γ z − ( ⋃ E edge group of G z ( ∂ ∗ E , Γ z ) ) \mathcal{M}(\Gamma_z,\Gamma)\subseteq\partial_*\Gamma_z-\left(\bigcup_{\substack{E\text{ edge group of }\mathcal{G}_z}}(\partial_*E,\Gamma_z)\right) M ( Γ z , Γ ) ⊆ ∂ ∗ Γ z − E edge group of G z ⋃ ( ∂ ∗ E , Γ z ) Thus, by Theorem 2.18 , there exists a vertex of G z \mathcal{G}_z G z Γ v \Gamma_v Γ v g v ∈ Γ g_v \in\Gamma g v ∈ Γ C z C_z C z M ( Γ z , Γ ) \mathcal{M}(\Gamma_z,\Gamma) M ( Γ z , Γ )
C z ⊆ ( ∂ ∗ g v Γ v g v − 1 , Γ z ) ∩ M ( Γ z , Γ ) . C_z\subseteq(\partial_*g_v\Gamma_vg_v^{-1},\Gamma_z)\cap\mathcal{M}(\Gamma_z,\Gamma). C z ⊆ ( ∂ ∗ g v Γ v g v − 1 , Γ z ) ∩ M ( Γ z , Γ ) . Using that Γ v \Gamma_v Γ v Γ z \Gamma_z Γ z Γ z \Gamma_z Γ z Γ \Gamma Γ
C = ι ∗ ( S ) ⊆ ι ∗ ( C z ) ⊆ ι ∗ ( ( ∂ ∗ g v Γ v g v − 1 , Γ z ) ∩ M ( Γ z , Γ ) ) = ( ∂ ∗ g v Γ v g v − 1 , Γ ) C=\iota_*(S)\subseteq\iota_*(C_z)\subseteq\iota_*\left((\partial_*g_v\Gamma_vg_v^{-1},\Gamma_z)\cap\mathcal{M}(\Gamma_z,\Gamma)\right)=(\partial_*g_v\Gamma_vg_v^{-1},\Gamma) C = ι ∗ ( S ) ⊆ ι ∗ ( C z ) ⊆ ι ∗ ( ( ∂ ∗ g v Γ v g v − 1 , Γ z ) ∩ M ( Γ z , Γ ) ) = ( ∂ ∗ g v Γ v g v − 1 , Γ ) As the vertex w g v Γ v w_{g_v\Gamma_v} w g v Γ v z z z V = g v Γ v g v − 1 V = g_v\Gamma_vg_v^{-1} V = g v Γ v g v − 1 v = w g v Γ v v = w_{g_v\Gamma_v} v = w g v Γ v H n + 1 \mathcal{H}_{n+1} H n + 1 V = g v Γ v g v − 1 V = g_v\Gamma_vg_v^{-1} V = g v Γ v g v − 1 C ⊆ ( ∂ ∗ V , Γ ) C \subseteq(\partial_*V,\Gamma) C ⊆ ( ∂ ∗ V , Γ ) [ v , r ] [v,r] [ v , r ]
Uniqueness of [ r , v ] [r,v] [ r , v ] It remains to show that the choice of the leaf v v v V V V L L L H n + 1 \mathcal{H}_{n+1} H n + 1 v v v ( ∂ ∗ L , Γ ) ∩ ( ∂ ∗ V , Γ ) = ∅ (\partial_*L,\Gamma) \cap(\partial_*V,\Gamma) = \varnothing ( ∂ ∗ L , Γ ) ∩ ( ∂ ∗ V , Γ ) = ∅ C ⊆ ( ∂ ∗ V , Γ ) C \subseteq(\partial_*V,\Gamma) C ⊆ ( ∂ ∗ V , Γ ) C ⊈ ( ∂ ∗ L , Γ ) C \nsubseteq(\partial_*L,\Gamma) C ⊈ ( ∂ ∗ L , Γ )
The path [ r , v ] [r,v] [ r , v ] In order to prove that [ r , v ] [r,v] [ r , v ] stab ( C ) ⊆ V \operatorname{stab}(C) \subseteq V stab ( C ) ⊆ V g ∈ Γ g \in\Gamma g ∈ Γ g C = C gC = C g C = C w ∈ [ r , z ] w \in[r,z] w ∈ [ r , z ] stab ( C ) ⊆ stab ( z ) = Γ z \operatorname{stab}(C) \subseteq\operatorname{stab}(z) = \Gamma_z stab ( C ) ⊆ stab ( z ) = Γ z g ∈ Γ z g \in\Gamma_z g ∈ Γ z
Recall from the proof of (†) that
C z ⊆ ( ∂ ∗ V , Γ z ) ∩ M ( Γ z , Γ ) . C_z \subseteq(\partial_*V,\Gamma_z) \cap\mathcal{M}(\Gamma_z,\Gamma). C z ⊆ ( ∂ ∗ V , Γ z ) ∩ M ( Γ z , Γ ) . Hence,
g C z ⊆ g ( ∂ ∗ V , Γ z ) ∩ g M ( Γ z , Γ ) . gC_z \subseteq g(\partial_*V,\Gamma_z) \cap g\mathcal{M}(\Gamma_z,\Gamma). g C z ⊆ g ( ∂ ∗ V , Γ z ) ∩ g M ( Γ z , Γ ) . As g ∈ Γ z g \in\Gamma_z g ∈ Γ z g ( ∂ ∗ V , Γ z ) = ( ∂ ∗ g V g − 1 , Γ z ) g(\partial_*V,\Gamma_z) = (\partial_*gVg^{-1},\Gamma_z) g ( ∂ ∗ V , Γ z ) = ( ∂ ∗ g V g − 1 , Γ z ) g ∈ Γ z g \in\Gamma_z g ∈ Γ z g M ( Γ z , Γ ) = M ( Γ z , Γ ) g\mathcal{M}(\Gamma_z,\Gamma) = \mathcal{M}(\Gamma_z,\Gamma) g M ( Γ z , Γ ) = M ( Γ z , Γ ) ι ∗ \iota_* ι ∗ Γ z \Gamma_z Γ z
g C = g ι ∗ ( S ) = ι ∗ ( g S ) ⊆ ι ∗ ( g C z ) ⊆ ι ∗ ( g ( ∂ ∗ V , Γ z ) ∩ g M ( Γ z , Γ ) ) = ι ∗ ( ( ∂ ∗ g V g − 1 , Γ z ) ∩ M ( Γ z , Γ ) ) = ( ∂ ∗ g V g − 1 , Γ ) . gC = g\iota_*(S) = \iota_*(gS) \subseteq\iota_*(gC_z) \subseteq\iota_*\bigl(g(\partial_*V,\Gamma_z) \cap g\mathcal{M}(\Gamma_z,\Gamma)\bigr)
= \iota_*\bigl((\partial_*gVg^{-1},\Gamma_z) \cap\mathcal{M}(\Gamma_z,\Gamma)\bigr) = (\partial_*gVg^{-1},\Gamma). g C = g ι ∗ ( S ) = ι ∗ ( g S ) ⊆ ι ∗ ( g C z ) ⊆ ι ∗ ( g ( ∂ ∗ V , Γ z ) ∩ g M ( Γ z , Γ ) ) = ι ∗ ( ( ∂ ∗ g V g − 1 , Γ z ) ∩ M ( Γ z , Γ ) ) = ( ∂ ∗ g V g − 1 , Γ ) . As [ r , v ] [r,v] [ r , v ] C ⊆ ( ∂ ∗ V , Γ ) C \subseteq(\partial_*V,\Gamma) C ⊆ ( ∂ ∗ V , Γ ) C = g C C = gC C = g C C ⊆ ( ∂ ∗ V , Γ ) ∩ ( ∂ ∗ g V g − 1 , Γ ) C \subseteq(\partial_*V,\Gamma) \cap(\partial_*gVg^{-1},\Gamma) C ⊆ ( ∂ ∗ V , Γ ) ∩ ( ∂ ∗ g V g − 1 , Γ ) g V g − 1 = V gVg^{-1} = V g V g − 1 = V g ∈ V g \in V g ∈ V
Theorem A will follow from the following two propositions.
Proposition 3.18. Let Γ \Gamma Γ H \mathcal{H} H C C C ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ L L L C = ( ∂ ∗ L , Γ ) C = (\partial_*L,\Gamma) C = ( ∂ ∗ L , Γ )
Proof. Let C C C ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ L L L H \mathcal{H} H C ⊆ ( ∂ ∗ L , Γ ) C \subseteq(\partial_*L,\Gamma) C ⊆ ( ∂ ∗ L , Γ ) stab ( C ) ⊆ L \operatorname{stab}(C) \subseteq L stab ( C ) ⊆ L ( ∂ ∗ L , Γ ) (\partial_*L,\Gamma) ( ∂ ∗ L , Γ ) ( ∂ ∗ L , Γ ) ⊆ C (\partial_*L,\Gamma) \subseteq C ( ∂ ∗ L , Γ ) ⊆ C C = ( ∂ ∗ L , Γ ) C = (\partial_*L,\Gamma) C = ( ∂ ∗ L , Γ )
It remains to show that L ⊆ stab ( C ) L \subseteq\operatorname{stab}(C) L ⊆ stab ( C ) g ′ ∈ L g' \in L g ′ ∈ L ξ ∈ C \xi\in C ξ ∈ C g ′ ξ ∈ C g'\xi\in C g ′ ξ ∈ C γ : R + → Γ \gamma: \mathbb{R}^{+} \to\Gamma γ : R + → Γ ξ = [ γ ] \xi= [\gamma] ξ = [ γ ] C = ( ∂ ∗ L , Γ ) C = (\partial_*L,\Gamma) C = ( ∂ ∗ L , Γ ) D D D L L L γ \gamma γ D > 0 D > 0 D > 0 t ≥ 0 t \geq0 t ≥ 0 h ∈ L h \in L h ∈ L d ( γ ( t ) , h ) ≤ D d(\gamma(t),h) \leq D d ( γ ( t ) , h ) ≤ D d ( g ′ ⋅ γ ( t ) , g ′ ⋅ h ) ≤ D d(g'\cdot\gamma(t),g'\cdot h) \leq D d ( g ′ ⋅ γ ( t ) , g ′ ⋅ h ) ≤ D g ′ ⋅ h ∈ L g'\cdot h \in L g ′ ⋅ h ∈ L g ′ ⋅ γ ( t ) g'\cdot\gamma(t) g ′ ⋅ γ ( t ) L L L t ≥ 0 t \geq0 t ≥ 0 ξ = g ′ [ γ ] = [ g ′ γ ] ∈ ( ∂ ∗ L , Γ ) = C \xi= g'[\gamma] = [g'\gamma] \in(\partial_*L,\Gamma) = C ξ = g ′ [ γ ] = [ g ′ γ ] ∈ ( ∂ ∗ L , Γ ) = C
:::
Proposition 3.19. Let Γ \Gamma Γ H \mathcal{H} H L L L ( ∂ ∗ L , Γ ) (\partial_*L,\Gamma) ( ∂ ∗ L , Γ ) ( ∂ ∗ L , Γ ) (\partial_*L,\Gamma) ( ∂ ∗ L , Γ ) ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ L L L Γ ↷ ∂ ∗ Γ \Gamma\curvearrowright\partial_*\Gamma Γ ↷ ∂ ∗ Γ
Proof. Let L L L ( ∂ ∗ L , Γ ) (\partial_*L,\Gamma) ( ∂ ∗ L , Γ ) C C C ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ ( ∂ ∗ L , Γ ) (\partial_*L,\Gamma) ( ∂ ∗ L , Γ ) L ′ L' L ′ C = ( ∂ ∗ L ′ , Γ ) C=(\partial_*L',\Gamma) C = ( ∂ ∗ L ′ , Γ ) ( ∂ ∗ L , Γ ) ∩ ( ∂ ∗ L ′ , Γ ) ≠ ∅ (\partial_*L,\Gamma)\cap(\partial_*L',\Gamma)\ne\emptyset ( ∂ ∗ L , Γ ) ∩ ( ∂ ∗ L ′ , Γ ) = ∅ L = L ′ L=L' L = L ′
Theorem 3.20 (Theorem A). Let Γ \Gamma Γ H \mathcal{H} H Γ \Gamma Γ
L \mathcal{L} L L L L H \mathcal{H} H ( ∂ ∗ L , Γ ) ≠ ∅ (\partial_*L,\Gamma)\ne\emptyset ( ∂ ∗ L , Γ ) = ∅
S \mathcal{S} S ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ
Moreover, every infinite connected component C C C ∂ ∗ Γ \partial_*\Gamma ∂ ∗ Γ
C = ( ∂ ∗ stab ( C ) , Γ ) . C=(\partial_*\operatorname{stab}(C),\Gamma). C = ( ∂ ∗ stab ( C ) , Γ ) . Proof. Follows directly from Theorem 3.18 and Theorem 3.19.
In Theorem 3.6 we prove that no leaf group of a Morse accessible group can have a singleton as relative Morse boundary, therefore we can conclude Theorem B:
Corollary 3.21 (Theorem B). Let Γ \Gamma Γ H \mathcal{H} H Γ \Gamma Γ L L L H \mathcal{H} H ( ∂ ∗ L , Γ ) = ∅ (\partial_*L,\Gamma)=\emptyset ( ∂ ∗ L , Γ ) = ∅
Point-convergence of the set of infinite Morse boundary components
In this section, we prove that the set of infinite Morse boundary components of a Morse accessible group is point-convergent. By Corollary 1.2, Corollary 1.3, and Theorem 1.5 in [KMZ26], this directly implies the quasi-isometry rigidity statements Theorem D and Theorem E in the introduction.
Point-convergence , a notion introduced in [KMZ26] which generalizes the notion of a null-family.
Definition 3.22. Let Y Y Y ( A n ) n ∈ N (A_n)_{n\in\mathbb{N}} ( A n ) n ∈ N Y Y Y
L = lim n → ∞ A n L=\lim_{n\to\infty} A_n L = n → ∞ lim A n is the set of points x ∈ Y x\in Y x ∈ Y x = lim n → ∞ x n x=\lim_{n\to\infty}x_n x = lim n → ∞ x n x n ∈ A n x_n\in A_n x n ∈ A n
Definition 3.23. Let Y Y Y { A i } i ∈ I \{A_i\}_{i\in I} { A i } i ∈ I Y Y Y point-convergent if for all sequences ( A i n ) n ∈ N (A_{i_n})_{n\in\mathbb{N}} ( A i n ) n ∈ N { A i } i ∈ I \{A_i\}_{i\in I} { A i } i ∈ I
∣ lim n → ∞ A i n ∣ ≤ 1. \left|\lim_{n\to\infty} A_{i_n}\right|\le1. n → ∞ lim A i n ≤ 1. We show the following stronger variant of Theorem C:
Theorem 3.24. Let V V V Γ \Gamma Γ ( ∂ ∗ V , Γ ) (\partial_* V,\Gamma) ( ∂ ∗ V , Γ ) Γ \Gamma Γ
In order to show Theorem 3.24 we need the following compatibility lemma. In this lemma one might consider the special case where Γ ′ \Gamma' Γ ′ Γ ′ \Gamma' Γ ′ ∂ ∗ Γ ′ \partial_*\Gamma' ∂ ∗ Γ ′ Γ \Gamma Γ
Lemma 3.25. Let Γ \Gamma Γ Γ ′ \Gamma' Γ ′ Γ \Gamma Γ Γ \Gamma Γ Γ ′ \Gamma' Γ ′ Γ \Gamma Γ Γ ′ \Gamma' Γ ′ C C C ( ∂ ∗ Γ ′ , Γ ) (\partial_*\Gamma',\Gamma) ( ∂ ∗ Γ ′ , Γ ) V V V Γ ′ \Gamma' Γ ′ C C C ( ∂ ∗ V , Γ ) (\partial_*V,\Gamma) ( ∂ ∗ V , Γ )
Proof. Let C C C ( ∂ ∗ Γ ′ , Γ ) (\partial_*\Gamma',\Gamma) ( ∂ ∗ Γ ′ , Γ ) ι : Γ ′ → Γ \iota:\Gamma' \to\Gamma ι : Γ ′ → Γ ι \iota ι
M ( Γ ′ , Γ ) : = { [ γ ] ∈ ∂ ∗ Γ ′ ∣ ι ( γ ) is Morse in Γ } ⊆ ∂ ∗ Γ ′ . \mathcal{M}(\Gamma',\Gamma):=\{[\gamma]\in\partial_*\Gamma'\mid\iota(\gamma)\text{ is Morse in }\Gamma\}\subseteq\partial_*\Gamma'. M ( Γ ′ , Γ ) := {[ γ ] ∈ ∂ ∗ Γ ′ ∣ ι ( γ ) is Morse in Γ } ⊆ ∂ ∗ Γ ′ . Let ι ∗ \iota_* ι ∗ M ( Γ ′ , Γ ) → ( ∂ ∗ Γ ′ , Γ ) ⊆ ∂ ∗ Γ \mathcal{M}(\Gamma',\Gamma)\to(\partial_*\Gamma',\Gamma)\subseteq\partial_*\Gamma M ( Γ ′ , Γ ) → ( ∂ ∗ Γ ′ , Γ ) ⊆ ∂ ∗ Γ ι \iota ι ι ∗ [ γ ] = [ ι ∗ ( γ ) ] \iota_*[\gamma]=[\iota_*(\gamma)] ι ∗ [ γ ] = [ ι ∗ ( γ )] ι ∗ \iota_* ι ∗ ι ∗ \iota_* ι ∗ ι ∗ − 1 ( C ) \iota_*^{-1}(C) ι ∗ − 1 ( C ) M ( Γ ′ , Γ ) \mathcal{M}(\Gamma',\Gamma) M ( Γ ′ , Γ )
By assumption, ( ∂ ∗ E , Γ ) = ∅ (\partial_*E,\Gamma)=\varnothing ( ∂ ∗ E , Γ ) = ∅ E E E Γ ′ \Gamma' Γ ′ C ∩ ( ∂ ∗ E , Γ ) = ∅ C\cap(\partial_*E,\Gamma)=\varnothing C ∩ ( ∂ ∗ E , Γ ) = ∅ E E E Γ ′ \Gamma' Γ ′
M ( Γ ′ , Γ ) ⊆ ∂ ∗ Γ ′ ∖ ( ⋃ E edge group of Γ ′ ( ∂ ∗ E , Γ ′ ) ) . \mathcal{M}(\Gamma',\Gamma)\subseteq\partial_*\Gamma'\setminus\left(\bigcup_{E\text{ edge group of }\Gamma'}(\partial_*E,\Gamma')\right). M ( Γ ′ , Γ ) ⊆ ∂ ∗ Γ ′ ∖ E edge group of Γ ′ ⋃ ( ∂ ∗ E , Γ ′ ) . By Theorem 2.18 , there exists exactly one vertex group V V V Γ ′ \Gamma' Γ ′ ι ∗ − 1 ( C ) \iota_*^{-1}(C) ι ∗ − 1 ( C ) ( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ ) (\partial_*V,\Gamma')\cap\mathcal{M}(\Gamma',\Gamma) ( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ ) ι ∗ \iota_* ι ∗ C C C ι ∗ ( ( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ ) ) \iota_*((\partial_*V,\Gamma')\cap\mathcal{M}(\Gamma',\Gamma)) ι ∗ (( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ )) ι ∗ ( ( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ ) ) ⊆ ( ∂ ∗ V , Γ ) \iota_*((\partial_*V,\Gamma')\cap\mathcal{M}(\Gamma',\Gamma))\subseteq(\partial_*V,\Gamma) ι ∗ (( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ )) ⊆ ( ∂ ∗ V , Γ ) Γ ′ \Gamma' Γ ′ Γ ′ \Gamma' Γ ′ V V V Γ ′ \Gamma' Γ ′ [24 ] . As Γ ′ \Gamma' Γ ′ Γ \Gamma Γ
The following lemma can be viewed as a variant of Lemma 6.7 in [KMZ26].
Lemma 3.26. Let Γ \Gamma Γ Γ ′ \Gamma' Γ ′ Γ \Gamma Γ Γ \Gamma Γ Γ ′ \Gamma' Γ ′ Γ \Gamma Γ Γ ′ \Gamma' Γ ′
Let ( ξ i ) i ∈ N (\xi_i)_{i\in\mathbb{N}} ( ξ i ) i ∈ N ( ζ i ) i ∈ N (\zeta_i)_{i\in\mathbb{N}} ( ζ i ) i ∈ N ( ∂ ∗ Γ ′ , Γ ) (\partial_{*}\Gamma',\Gamma) ( ∂ ∗ Γ ′ , Γ ) ξ = lim i → ∞ ξ i \xi=\lim_{i\to\infty}\xi_i ξ = lim i → ∞ ξ i ζ = lim i → ∞ ζ i \zeta=\lim_{i\to\infty}\zeta_i ζ = lim i → ∞ ζ i ( ∂ ∗ Γ ′ , Γ ) (\partial_{*}\Gamma',\Gamma) ( ∂ ∗ Γ ′ , Γ ) ξ i \xi_i ξ i ζ i \zeta_i ζ i ( ∂ ∗ Γ ′ , Γ ) (\partial_{*}\Gamma',\Gamma) ( ∂ ∗ Γ ′ , Γ ) i i i i 0 i_0 i 0 V V V Γ ′ \Gamma' Γ ′ i ≥ i 0 i\ge i_0 i ≥ i 0 ξ i ∈ ( ∂ ∗ V , Γ ) \xi_i\in(\partial_{*}V,\Gamma) ξ i ∈ ( ∂ ∗ V , Γ ) ζ i ∈ ( ∂ ∗ V , Γ ) \zeta_i\in(\partial_{*}V,\Gamma) ζ i ∈ ( ∂ ∗ V , Γ ) ζ , ξ ∈ ( ∂ ∗ V , Γ ) \zeta,\xi\in(\partial_{*}V,\Gamma) ζ , ξ ∈ ( ∂ ∗ V , Γ )
Proof. By assumption, ( ∂ ∗ E , Γ ) = ∅ (\partial_{*}E,\Gamma)=\varnothing ( ∂ ∗ E , Γ ) = ∅ E E E Γ ′ \Gamma' Γ ′
M ( Γ ′ , Γ ) ⊆ ∂ ∗ Γ ′ − ( ⋃ E edge group of Γ ′ ( ∂ ∗ E , Γ ′ ) ) . \mathcal{M}(\Gamma',\Gamma)\subseteq\partial_{*}\Gamma'-\left(\bigcup_{E\ \text{edge group of }\Gamma'}(\partial_{*}E,\Gamma')\right). M ( Γ ′ , Γ ) ⊆ ∂ ∗ Γ ′ − E edge group of Γ ′ ⋃ ( ∂ ∗ E , Γ ′ ) . Let ξ ˉ i : = ι ∗ − 1 ( ξ i ) \bar{\xi}_i:=\iota_{*}^{-1}(\xi_i) ξ ˉ i := ι ∗ − 1 ( ξ i ) ζ ˉ i : = ι ∗ − 1 ( ζ i ) \bar{\zeta}_i:=\iota_{*}^{-1}(\zeta_i) ζ ˉ i := ι ∗ − 1 ( ζ i ) ξ ˉ : = ι ∗ − 1 ( ξ ) \bar{\xi}:=\iota_{*}^{-1}(\xi) ξ ˉ := ι ∗ − 1 ( ξ ) ζ ˉ : = ι ∗ − 1 ( ζ ) \bar{\zeta}:=\iota_{*}^{-1}(\zeta) ζ ˉ := ι ∗ − 1 ( ζ ) ι ∗ \iota_{*} ι ∗ ( ξ ˉ i ) i (\bar{\xi}_i)_i ( ξ ˉ i ) i ξ ˉ ∈ M ( Γ ′ , Γ ) \bar{\xi}\in\mathcal{M}(\Gamma',\Gamma) ξ ˉ ∈ M ( Γ ′ , Γ ) ( ζ ˉ i ) i (\bar{\zeta}_i)_i ( ζ ˉ i ) i ζ ˉ ∈ M ( Γ ′ , Γ ) \bar{\zeta}\in\mathcal{M}(\Gamma',\Gamma) ζ ˉ ∈ M ( Γ ′ , Γ ) ξ ˉ i \bar{\xi}_i ξ ˉ i ζ ˉ i \bar{\zeta}_i ζ ˉ i M ( Γ ′ , Γ ) \mathcal{M}(\Gamma',\Gamma) M ( Γ ′ , Γ ) ξ i \xi_i ξ i ζ i \zeta_i ζ i ( ∂ ∗ Γ ′ , Γ ) (\partial_{*}\Gamma',\Gamma) ( ∂ ∗ Γ ′ , Γ ) ι ∗ \iota_{*} ι ∗ ι ∗ \iota_{*} ι ∗ ξ ˉ \bar{\xi} ξ ˉ ζ ˉ \bar{\zeta} ζ ˉ ξ ˉ i \bar{\xi}_i ξ ˉ i ζ ˉ i \bar{\zeta}_i ζ ˉ i i i i
For each i i i γ i \gamma_i γ i X ′ X' X ′ Γ ′ \Gamma' Γ ′ γ i − = ξ ˉ i \gamma_i^{-}=\bar{\xi}_i γ i − = ξ ˉ i γ i + = ζ ˉ i \gamma_i^{+}=\bar{\zeta}_i γ i + = ζ ˉ i [17 ] ). Following the proof of Lemma 3.14 in [KMZ26] one can show that there exists a Morse gauge N N N γ i \gamma_i γ i N N N X ′ X' X ′ γ i \gamma_i γ i x ∈ X ′ x\in X' x ∈ X ′ ( γ i ) i (\gamma_i)_i ( γ i ) i N N N γ \gamma γ Γ ′ \Gamma' Γ ′ γ − = lim i → ∞ γ i − = ξ ˉ \gamma^{-}=\lim_{i\to\infty}\gamma_i^{-}=\bar{\xi} γ − = lim i → ∞ γ i − = ξ ˉ γ + = lim i → ∞ γ i + = ζ ˉ \gamma^{+}=\lim_{i\to\infty}\gamma_i^{+}=\bar{\zeta} γ + = lim i → ∞ γ i + = ζ ˉ i 0 i_0 i 0 V V V Γ ′ \Gamma' Γ ′ i ≥ i 0 i\ge i_0 i ≥ i 0 [ ξ ˉ i ] = [ γ i + ] ∈ ( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ ) [\bar{\xi}_i]=[\gamma_i^{+}]\in(\partial_{*}V,\Gamma')\cap\mathcal{M}(\Gamma',\Gamma) [ ξ ˉ i ] = [ γ i + ] ∈ ( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ ) [ ζ ˉ i ] = [ γ i − ] ∈ ( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ ) [\bar{\zeta}_i]=[\gamma_i^{-}]\in(\partial_{*}V,\Gamma')\cap\mathcal{M}(\Gamma',\Gamma) [ ζ ˉ i ] = [ γ i − ] ∈ ( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ ) ξ ˉ , ζ ˉ ∈ ( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ ) \bar{\xi},\bar{\zeta}\in(\partial_{*}V,\Gamma')\cap\mathcal{M}(\Gamma',\Gamma) ξ ˉ , ζ ˉ ∈ ( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ ) ι ∗ \iota_{*} ι ∗ i ≥ i 0 i\ge i_0 i ≥ i 0 [ ξ i ] ∈ ι ∗ ( ( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ ) ) [\xi_i]\in\iota_{*}((\partial_{*}V,\Gamma')\cap\mathcal{M}(\Gamma',\Gamma)) [ ξ i ] ∈ ι ∗ (( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ )) [ ζ i ] ∈ ι ∗ ( ( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ ) ) [\zeta_i]\in\iota_{*}((\partial_{*}V,\Gamma')\cap\mathcal{M}(\Gamma',\Gamma)) [ ζ i ] ∈ ι ∗ (( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ )) ξ , ζ ∈ ι ∗ ( ( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ ) ) \xi,\zeta\in\iota_{*}((\partial_{*}V,\Gamma')\cap\mathcal{M}(\Gamma',\Gamma)) ξ , ζ ∈ ι ∗ (( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ ))
It remains to show that ι ∗ ( ( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ ) ) ⊆ ( ∂ ∗ V , Γ ) \iota_{*}((\partial_{*}V,\Gamma')\cap\mathcal{M}(\Gamma',\Gamma))\subseteq(\partial_{*}V,\Gamma) ι ∗ (( ∂ ∗ V , Γ ′ ) ∩ M ( Γ ′ , Γ )) ⊆ ( ∂ ∗ V , Γ ) Γ ′ \Gamma' Γ ′ Γ ′ \Gamma' Γ ′ V V V Γ ′ \Gamma' Γ ′ [24 ] . As Γ ′ \Gamma' Γ ′ Γ \Gamma Γ □ \square □
Proof of Theorem 3.24. Let V V V H \mathcal{H} H Γ \Gamma Γ V = g Γ v g − 1 V=g\Gamma_vg^{-1} V = g Γ v g − 1 g ∈ Γ g\in\Gamma g ∈ Γ Γ v \Gamma_v Γ v v v v H \mathcal{H} H ( ∂ ∗ V , Γ ) = ( ∂ ∗ g Γ v g − 1 , Γ ) = g ( ∂ ∗ Γ v , Γ ) (\partial_{*}V,\Gamma)=(\partial_{*}g\Gamma_vg^{-1},\Gamma)=g(\partial_{*}\Gamma_v,\Gamma) ( ∂ ∗ V , Γ ) = ( ∂ ∗ g Γ v g − 1 , Γ ) = g ( ∂ ∗ Γ v , Γ ) Γ \Gamma Γ ( ∂ ∗ V , Γ ) (\partial_{*}V,\Gamma) ( ∂ ∗ V , Γ ) ( ∂ ∗ Γ v , Γ ) (\partial_{*}\Gamma_v,\Gamma) ( ∂ ∗ Γ v , Γ ) Γ v \Gamma_v Γ v v v v H \mathcal{H} H ( ∂ ∗ Γ v , Γ ) (\partial_{*}\Gamma_v,\Gamma) ( ∂ ∗ Γ v , Γ )
Let H \mathcal{H} H v v v H \mathcal{H} H v v v H \mathcal{H} H v v v v v v ( ∂ ∗ Γ v , Γ ) (\partial_* \Gamma_v,\Gamma) ( ∂ ∗ Γ v , Γ ) ( ∂ ∗ Γ v , Γ ) (\partial_* \Gamma_v,\Gamma) ( ∂ ∗ Γ v , Γ )
Let v v v H \mathcal{H} H n + 1 n+1 n + 1 H \mathcal{H} H v v v Γ v \Gamma_v Γ v v v v ( ∂ ∗ Γ v , Γ ) (\partial_* \Gamma_v,\Gamma) ( ∂ ∗ Γ v , Γ ) ( C i ) i ∈ N (C_i)_{i\in\mathbb{N}} ( C i ) i ∈ N C i C_i C i ( ∂ ∗ Γ v , Γ ) (\partial_* \Gamma_v,\Gamma) ( ∂ ∗ Γ v , Γ ) ( ξ i ) i ∈ N (\xi_i)_{i\in\mathbb{N}} ( ξ i ) i ∈ N ( ζ i ) i ∈ N (\zeta_i)_{i\in\mathbb{N}} ( ζ i ) i ∈ N ( ∂ ∗ Γ v , Γ ) (\partial_* \Gamma_v,\Gamma) ( ∂ ∗ Γ v , Γ ) ξ i , ζ i ∈ C i \xi_i,\zeta_i\in C_i ξ i , ζ i ∈ C i ξ = lim i → ∞ ξ i \xi=\lim_{i\to\infty}\xi_i ξ = lim i → ∞ ξ i ζ = lim i → ∞ ζ i \zeta=\lim_{i\to\infty}\zeta_i ζ = lim i → ∞ ζ i ( ∂ ∗ Γ v , Γ ) (\partial_* \Gamma_v,\Gamma) ( ∂ ∗ Γ v , Γ ) Theorem 3.26 , there exists i 0 i_0 i 0 V V V Γ v \Gamma_v Γ v i ≥ i 0 i\ge i_0 i ≥ i 0 ξ i ∈ ( ∂ ∗ V , Γ ) \xi_i\in(\partial_* V,\Gamma) ξ i ∈ ( ∂ ∗ V , Γ ) ζ i ∈ ( ∂ ∗ V , Γ ) \zeta_i\in(\partial_* V,\Gamma) ζ i ∈ ( ∂ ∗ V , Γ ) ξ , ζ ∈ ( ∂ ∗ V , Γ ) \xi,\zeta\in(\partial_* V,\Gamma) ξ , ζ ∈ ( ∂ ∗ V , Γ ) Theorem 3.25 implies that the connected components C i C_i C i ( ∂ ∗ V , Γ ) (\partial_* V,\Gamma) ( ∂ ∗ V , Γ ) i ≥ n 0 i\ge n_0 i ≥ n 0 C i C_i C i ( ∂ ∗ V , Γ ) (\partial_* V,\Gamma) ( ∂ ∗ V , Γ ) ( C i ) i (C_i)_i ( C i ) i ( ∂ ∗ V , Γ ) (\partial_* V,\Gamma) ( ∂ ∗ V , Γ ) ξ \xi ξ ζ \zeta ζ ( ∂ ∗ V , Γ ) (\partial_* V,\Gamma) ( ∂ ∗ V , Γ ) V V V Γ u \Gamma_u Γ u u u u v v v ( ∂ ∗ V , Γ ) (\partial_* V,\Gamma) ( ∂ ∗ V , Γ ) ( ∂ ∗ Γ u , Γ ) (\partial_* \Gamma_u,\Gamma) ( ∂ ∗ Γ u , Γ ) ( ∂ ∗ Γ u , Γ ) (\partial_* \Gamma_u,\Gamma) ( ∂ ∗ Γ u , Γ ) u u u u u u n < n + 1 n<n+1 n < n + 1
Remark 3.27. The above shows that the set of infinite connected components of ( ∂ ∗ V , Γ ) (\partial_* V,\Gamma) ( ∂ ∗ V , Γ ) V V V ι ∗ \iota_* ι ∗ M ( V , Γ ) \mathcal{M}(V,\Gamma) M ( V , Γ ) ∂ ∗ V \partial_* V ∂ ∗ V ι ∗ \iota_* ι ∗ M ( V , Γ ) \mathcal{M}(V,\Gamma) M ( V , Γ ) ι ∗ \iota_* ι ∗ ( ∂ ∗ V , Γ ) (\partial_* V,\Gamma) ( ∂ ∗ V , Γ )
(Weak) Morse hierarchies of Coxeter groups
In this section we prove all Coxeter groups have a weak Morse hierarchy, which we prove in the spherical nexus free (Theorem 4.15) case to be a genuine Morse hierarchy (Theorem F ). This shows that every Coxeter group without a spherical nexus is Morse accessible. We will begin this section with preliminaries on Coxeter groups, followed by results that “relativize” the necessary results from [14 ] , and then prove the theorem in § 4.3 . For a discussion of consequences of this theorem and examples see § 5 and § 6 .
Preliminaries on Coxeter groups
A Coxeter graph is a simplicial graph Δ \Delta Δ { 2 , 3 , 4 , … } \{2,3,4,\ldots\} { 2 , 3 , 4 , … } s , t ∈ Δ s,t\in\Delta s , t ∈ Δ m s t m_{st} m s t V ( Δ ) V(\Delta) V ( Δ ) Δ \Delta Δ
The Coxeter group W Δ W_{\Delta} W Δ Δ \Delta Δ
⟨ V ( Δ ) ∣ s 2 = 1 for all s ∈ V ( Δ ) , ( s t ) m s t = 1 for each ( s , t ) ∈ E ( Δ ) ⟩ \langle V(\Delta) \mid s^{2}=1\ \text{ for all }s\in V(\Delta),\ (st)^{m_{st}}=1\ \text{ for each }(s,t)\in E(\Delta)\rangle ⟨ V ( Δ ) ∣ s 2 = 1 for all s ∈ V ( Δ ) , ( s t ) m s t = 1 for each ( s , t ) ∈ E ( Δ )⟩ A Coxeter group is called a right-angled Coxeter group (RACG) if all edges of its Coxeter graph are labeled with 2. In this case, the usual labeling becomes unnecessary for extracting the group’s presentation. Consequently, if Δ \Delta Δ defining graph of the RACG.
Remark 4.1. Note that there is another common convention for Coxeter graphs where there are unlabeled edges and edges labeled by { 4 , 5 , 6 , … } ∪ { ∞ } \{4,5,6,\ldots\}\cup\{\infty\} { 4 , 5 , 6 , … } ∪ { ∞ } s s s t t t m s t = 3 m_{st}=3 m s t = 3 s s s t t t m s t = 2 m_{st}=2 m s t = 2 [3 Définition 4] for instance). In this paper we do not use this convention.
Given a subgraph Λ ⊆ Δ \Lambda\subseteq\Delta Λ ⊆ Δ Λ \Lambda Λ induced if every edge in Δ \Delta Δ Λ \Lambda Λ Λ \Lambda Λ induced by Λ \Lambda Λ Δ \Delta Δ V ( Λ ) V(\Lambda) V ( Λ ) [20 ] for instance), we let W Λ W_{\Lambda} W Λ W Δ W_{\Delta} W Δ Λ \Lambda Λ special subgroups and conjugates of special subgroups are called parabolic subgroups .
Let W Δ W_{\Delta} W Δ W Δ W_{\Delta} W Δ irreducible if it does not split as a product of groups, and W Δ W_{\Delta} W Δ affine if it is irreducible and a Euclidean reflection group. The classification of affine Coxeter groups is well-known (see [20 ] ). The rank of a Coxeter group W Δ W_{\Delta} W Δ Δ \Delta Δ W Δ W_{\Delta} W Δ affine-free if for any subgraph Λ ⊆ Δ \Lambda\subseteq\Delta Λ ⊆ Δ W Λ W_{\Lambda} W Λ W Δ W_{\Delta} W Δ [6 ] .
Given a Coxeter graph Δ \Delta Δ Λ \Lambda Λ spherical subgraph if W Λ W_{\Lambda} W Λ Δ − Λ \Delta-\Lambda Δ − Λ Λ \Lambda Λ separating spherical subgraph . In the proof of Theorem F and the two corollaries, we use the following convention:
Convention 2. If Λ \Lambda Λ Λ \Lambda Λ empty . If Λ \Lambda Λ W Λ W_{\Lambda} W Λ
Remark 4.2. Given a Coxeter graph Δ \Delta Δ W Δ W_{\Delta} W Δ Δ \Delta Δ does not have a separating spherical subgraph [20 Theorem 8.7.2] . In view of Convention 2, if Δ \Delta Δ ∅ \emptyset ∅ W Δ W_\Delta W Δ
The main ingredient of the proof are the results of Cordes–Levcovitz discussed in § 4.2 . Using these results, we will prove Morse-accessibility of affine-free Coxeter groups in § 4.3 . Afterwards, we will discuss consequences and examples in § 6 .
Relativizing results of Cordes–Levcovitz
In this subsection, we review and “relativize” the results on Morse boundaries of affine free Coxeter groups of Cordes–Levcovitz which we later use to give a Morse-hierarchy for certain affine-free Coxeter groups, which includes all RACGs.
Definition 4.3. Let W Δ W_\Delta W Δ Λ ⊆ Δ \Lambda\subseteq\Delta Λ ⊆ Δ wide if there is a decomposition V ( Λ ) = P ⊔ Q V(\Lambda)=P\sqcup Q V ( Λ ) = P ⊔ Q Q Q Q P P P Q Q Q
(1) both W P W_P W P W Q W_Q W Q
(2) W P W_P W P
We call ( P , Q ) (P,Q) ( P , Q ) wide decomposition of Λ \Lambda Λ Q Q Q W Λ ≅ W P × W Q W_\Lambda\cong W_P \times W_Q W Λ ≅ W P × W Q W Λ W_\Lambda W Λ Λ \Lambda Λ
The following lemma describes which special subgroups have empty relative Morse boundaries in Coxeter groups.
Lemma 4.4. Let Δ \Delta Δ Λ \Lambda Λ Δ \Delta Δ W Λ W_\Lambda W Λ W Δ W_\Delta W Δ ( ∂ ∗ W Λ , W Δ ) = ∅ (\partial_*W_\Lambda,W_\Delta)=\emptyset ( ∂ ∗ W Λ , W Δ ) = ∅ Λ \Lambda Λ
Proof. As wide subgroups have no Morse rays [14 Lemma 3.2] , if Λ \Lambda Λ Λ ‾ \overline{\Lambda} Λ W Λ ‾ W_{\overline{\Lambda}} W Λ ( ∂ ∗ W Λ , W Δ ) (\partial_*W_\Lambda,W_\Delta) ( ∂ ∗ W Λ , W Δ ) Λ \Lambda Λ W Λ W_\Lambda W Λ ( ∂ ∗ W Λ , W Δ ) = ∅ (\partial_*W_\Lambda,W_\Delta)=\emptyset ( ∂ ∗ W Λ , W Δ ) = ∅ Λ \Lambda Λ v 1 , … , v n v_1,\ldots,v_n v 1 , … , v n Λ \Lambda Λ v 1 v 2 ⋯ v n v_1v_2\cdots v_n v 1 v 2 ⋯ v n W Δ W_\Delta W Δ ( ∂ ∗ W Λ , W Δ ) ≠ ∅ (\partial_*W_\Lambda,W_\Delta)\ne\emptyset ( ∂ ∗ W Λ , W Δ ) = ∅ [12 Proposition 4.5] . □ \square □
Definition 4.5. We say that the Coxeter graph Δ \Delta Δ W Δ W_\Delta W Δ wide-avoidant if given any wide subgraph Λ \Lambda Λ s , t ∈ Δ s,t\in\Delta s , t ∈ Δ α ⊆ Δ \alpha\subseteq\Delta α ⊆ Δ s s s t t t α ∩ Λ ⊆ { s , t } \alpha\cap\Lambda\subseteq\{s,t\} α ∩ Λ ⊆ { s , t } α ∩ E ( Λ ) = ∅ \alpha\cap E(\Lambda)=\emptyset α ∩ E ( Λ ) = ∅
Definition 4.6 (Special Join). Given a Coxeter graph Δ \Delta Δ special join is a triplet ( P , Q , K ) (P,Q,K) ( P , Q , K ) P P P Q Q Q K K K Δ \Delta Δ
(1) ( P , Q ) (P,Q) ( P , Q ) P ∪ Q P\cup Q P ∪ Q
(2) K K K W K W_K W K
(3) For all k ∈ K k \in K k ∈ K s ∈ P s \in P s ∈ P Δ \Delta Δ k k k s s s
Remark 4.7. If W Δ W_{\Delta} W Δ ( P , Q , K ) (P,Q,K) ( P , Q , K ) Δ \Delta Δ W P ∪ Q ∪ K = W P × W Q ∪ K W_{P\cup Q\cup K}=W_P\times W_{Q\cup K} W P ∪ Q ∪ K = W P × W Q ∪ K W Λ W_{\Lambda} W Λ Λ \Lambda Λ ( P , Q ) (P,Q) ( P , Q ) ( P , Q , ∅ ) (P,Q,\varnothing) ( P , Q , ∅ )
Definition 4.8 (wide-spherical-avoidant). We say that the Coxeter graph Δ \Delta Δ W Δ W_{\Delta} W Δ wide-spherical-avoidant if given any special join ( P , Q , K ) (P,Q,K) ( P , Q , K ) s , t ∈ Δ ∖ K s,t\in\Delta\setminus K s , t ∈ Δ ∖ K α \alpha α Δ \Delta Δ s s s t t t α ∩ ( K ∪ P ∪ Q ) ⊆ { s , t } \alpha\cap(K\cup P\cup Q)\subseteq\{s,t\} α ∩ ( K ∪ P ∪ Q ) ⊆ { s , t } α ∩ E ( K ∪ P ∪ Q ) = ∅ \alpha\cap E(K\cup P\cup Q)=\varnothing α ∩ E ( K ∪ P ∪ Q ) = ∅
Theorem 4.9 ([14 ] ). If W Δ W_{\Delta} W Δ W Δ W_{\Delta} W Δ
In the case when W Δ W_{\Delta} W Δ Δ \Delta Δ
(1) The Morse boundary ∂ ∗ W Δ \partial_*W_{\Delta} ∂ ∗ W Δ
(2) Δ \Delta Δ
Recently Cashen, Chu and Tao proved a full characterization of Morse geodesics [10 ] ; importantly in Corollary 1.5 they give a criterion in the affine case. This corollary allows us to extend Theorem 4.9 to the affine free case:
Corollary 4.10. If W Δ W_{\Delta} W Δ W Δ W_{\Delta} W Δ
Proof. In Corollary 1.5 of [10 ] , they prove that a geodesic is Morse if any only if there is an upper bound on the lengths of subsegments γ ′ \gamma' γ ′ γ \gamma γ Λ ⊂ Δ \Lambda\subset\Delta Λ ⊂ Δ W Λ W_{\Lambda} W Λ γ ′ \gamma' γ ′ Λ \Lambda Λ [14 ] , this is exactly the characterization they use to ensure the “filter construction” embeds stably in the Cayley graph ([14 Section 6.2] ). Thus, we can drop the affine free condition from the Cordes–Levcovitz result. □
This theorem can be generalized to a statement about relative Morse boundaries of special subgroups of W Δ W_{\Delta} W Δ
Definition 4.11. Let Δ \Delta Δ Λ \Lambda Λ Δ \Delta Δ Λ \Lambda Λ relatively wide-avoidant in Δ \Delta Δ J ⊆ Δ J\subseteq\Delta J ⊆ Δ s , t ∈ Λ s,t\in\Lambda s , t ∈ Λ α \alpha α Λ \Lambda Λ s s s t t t α ∩ J ⊆ { s , t } \alpha\cap J\subseteq\{s,t\} α ∩ J ⊆ { s , t }
We also say the subgraph Λ \Lambda Λ relatively wide-spherical-avoidant in Δ \Delta Δ ( P , Q , K ) (P,Q,K) ( P , Q , K ) Δ \Delta Δ s , t ∈ Λ ∖ K s,t\in\Lambda\setminus K s , t ∈ Λ ∖ K α \alpha α Λ \Lambda Λ s s s t t t α ∩ ( K ∪ P ∪ Q ) ⊆ { s , t } \alpha\cap(K\cup P\cup Q)\subseteq\{s,t\} α ∩ ( K ∪ P ∪ Q ) ⊆ { s , t }
Corollary 4.13 (Relative version of Theorem 4.10). Let W Δ W_\Delta W Δ Δ \Delta Δ Λ \Lambda Λ Δ \Delta Δ W Λ W_\Lambda W Λ Λ \Lambda Λ Δ \Delta Δ ( ∂ ∗ W Λ , W Δ ) (\partial_*W_\Lambda,W_\Delta) ( ∂ ∗ W Λ , W Δ )
Proof. In this proof we will be using the notation and constructions (e.g., fans/filters) from [14 ] . Let α , β ∈ ( ∂ ∗ W Λ , W Δ ) \alpha,\beta\in(\partial_*W_\Lambda,W_\Delta) α , β ∈ ( ∂ ∗ W Λ , W Δ ) W Λ W_\Lambda W Λ W Δ W_\Delta W Δ α \alpha α β \beta β W Δ W_\Delta W Δ Λ \Lambda Λ Λ \Lambda Λ F \mathcal{F} F W Δ W_\Delta W Δ Λ \Lambda Λ ∂ ∗ F \partial_*\mathcal{F} ∂ ∗ F ( ∂ ∗ W Λ , W Δ ) (\partial_*W_\Lambda,W_\Delta) ( ∂ ∗ W Λ , W Δ ) [14 ] yields the claim. □
The following lemma characterizes when an induced subgraph of a Coxeter group is one-ended and relatively wide-avoidant or contained in a spherical or wide subgroup. If such a relatively wide-avoidant subgraph of Δ \Delta Δ
Lemma 4.14. Let Λ \Lambda Λ Δ \Delta Δ
Λ \Lambda Λ S S S
There do not exist two proper induced subgraphs Δ 1 \Delta_1 Δ 1 Δ 2 \Delta_2 Δ 2 Λ \Lambda Λ
Λ \Lambda Λ
Proof. (1) ⇒ \Rightarrow ⇒ Λ \Lambda Λ Λ \Lambda Λ J J J Δ \Delta Δ s , t ∈ V ( Λ ) s,t \in V(\Lambda) s , t ∈ V ( Λ ) Λ \Lambda Λ Λ ∩ J ≠ Λ \Lambda\cap J \ne\Lambda Λ ∩ J = Λ Λ − J \Lambda-J Λ − J α \alpha α s s s t t t Λ − J \Lambda-J Λ − J α ∩ J ⊆ { s , t } \alpha\cap J \subseteq\{s,t\} α ∩ J ⊆ { s , t } Λ ∩ J = Λ \Lambda\cap J = \Lambda Λ ∩ J = Λ Λ \Lambda Λ Λ \Lambda Λ J J J Δ \Delta Δ J J J Λ \Lambda Λ Λ \Lambda Λ
(3) ⇒ \Rightarrow ⇒ Λ \Lambda Λ Λ \Lambda Λ S S S Λ − S \Lambda-S Λ − S Λ \Lambda Λ Λ \Lambda Λ S S S S S S J J J Δ \Delta Δ Λ − S \Lambda-S Λ − S v v v w w w Λ − S \Lambda-S Λ − S Λ − S \Lambda-S Λ − S v v v w w w Λ \Lambda Λ J J J Λ \Lambda Λ Δ \Delta Δ
(2) ⇒ \Rightarrow ⇒ ¬ ( 1 ) ⇒ ¬ ( 2 ) \neg(1) \Rightarrow\neg(2) ¬ ( 1 ) ⇒ ¬ ( 2 ) S S S Λ \Lambda Λ Δ \Delta Δ Λ − S \Lambda-S Λ − S Δ 1 \Delta_1 Δ 1 S S S Λ − S \Lambda-S Λ − S Δ 2 \Delta_2 Δ 2 S S S Λ − Δ 1 \Lambda-\Delta_1 Λ − Δ 1 Λ − S \Lambda-S Λ − S Δ 1 ≠ Λ \Delta_1 \ne\Lambda Δ 1 = Λ Δ 1 ≠ Λ \Delta_1 \ne\Lambda Δ 1 = Λ Λ = Δ 1 ∪ Δ 2 \Lambda=\Delta_1\cup\Delta_2 Λ = Δ 1 ∪ Δ 2 Δ 1 ∩ Δ 2 = S \Delta_1\cap\Delta_2=S Δ 1 ∩ Δ 2 = S S S S Δ \Delta Δ ¬ ( 2 ) \neg(2) ¬ ( 2 )
(1) ⇒ \Rightarrow ⇒ ¬ ( 2 ) ⇒ ¬ ( 1 ) \neg(2) \Rightarrow\neg(1) ¬ ( 2 ) ⇒ ¬ ( 1 ) Λ \Lambda Λ Δ 1 \Delta_1 Δ 1 Δ 2 \Delta_2 Δ 2 Λ \Lambda Λ Δ 1 ∩ Δ 2 \Delta_1\cap\Delta_2 Δ 1 ∩ Δ 2 Δ \Delta Δ Δ 1 \Delta_1 Δ 1 Δ 2 \Delta_2 Δ 2 Λ \Lambda Λ Δ 1 ∩ Δ 2 = ∅ \Delta_1\cap\Delta_2=\varnothing Δ 1 ∩ Δ 2 = ∅ Δ 1 ∩ Δ 2 \Delta_1\cap\Delta_2 Δ 1 ∩ Δ 2 Λ \Lambda Λ S = Δ 1 ∩ Δ 2 S=\Delta_1\cap\Delta_2 S = Δ 1 ∩ Δ 2 Δ \Delta Δ ¬ ( 1 ) \neg(1) ¬ ( 1 )
In fact, the lemma above gives us a way to check if a induced subgraph corresponding to a one-ended subgroup is relatively wide-avoidant. Indeed, if Λ \Lambda Λ Δ \Delta Δ Λ \Lambda Λ Δ \Delta Δ
The most general result characterizing connectivity of the (relative) Morse boundary, Theorem 4.10, requires the stronger wide-spherical-avoidant condition on graphs. There are Coxeter graphs, though, that are wide-avoidant, not wide-spherical-avoidant, but surprisingly(!) have connected Morse boundary. Since we do not know how to detect such graphs, and we are interested in the case when these groups have connected (relative) Morse boundary, we give a condition on a Coxeter graph (spherical-nexus free) that will allow us to not worry about the case when a graph is wide-avoidant but not wide-spherical-avoidant.
Definition 4.15 (Spherical nexus). We call an induced subgraph K K K Δ \Delta Δ W K W_K W K spherical nexus if Δ \Delta Δ ( P , Q , K ) (P,Q,K) ( P , Q , K ) Λ \Lambda Λ Δ \Delta Δ s , t ∈ Δ − K s,t \in\Delta-K s , t ∈ Δ − K α ⊆ Λ \alpha\subseteq\Lambda α ⊆ Λ s s s t t t K K K
The existence of a spherical nexus means that a graph cannot be wide-spherical-avoidant. The following lemma follows from the definitions:
Lemma 4.16. Suppose a Coxeter graph Δ \Delta Δ Λ \Lambda Λ Δ \Delta Δ
As pointed out in Theorem 4.7, if Δ \Delta Δ
Proof of weak Morse hierarchies for Coxeter groups
In this subsection, we give the proof of Theorem F , which we restate here:
Theorem 4.17 (Theorem F ). Every Coxeter group has a weak Morse hierarchy in which each leaf group L L L W Λ W_\Lambda W Λ
Λ \Lambda Λ Δ \Delta Δ W Λ W_\Lambda W Λ ( ∂ ∗ W Λ , W Δ ) ≠ ∅ (\partial_* W_\Lambda, W_\Delta) \ne\varnothing ( ∂ ∗ W Λ , W Δ ) = ∅
Λ \Lambda Λ Δ \Delta Δ ( ∂ ∗ W Λ , W Δ ) = ∅ (\partial_* W_\Lambda, W_\Delta) = \varnothing ( ∂ ∗ W Λ , W Δ ) = ∅
Moreover, if W Δ W_\Delta W Δ ( ∂ ∗ W Λ , W Δ ) (\partial_* W_\Lambda, W_\Delta) ( ∂ ∗ W Λ , W Δ )
The diagram in Figure 4 illustrates an algorithm on a Coxeter graph Δ \Delta Δ W Δ W_\Delta W Δ
Proof of Theorem F (Theorem 4.17 ). Let Δ \Delta Δ W Δ W_{\Delta} W Δ
I : = { E ⊆ Δ ∣ E is an induced subgraph of Δ } ∪ { ∅ } \mathcal{I}:=\{E\subseteq\Delta\mid E\text{ is an induced subgraph of }\Delta\}\cup\{\emptyset\} I := { E ⊆ Δ ∣ E is an induced subgraph of Δ } ∪ { ∅ } E : = { E ∈ I ∣ ∂ ∗ W E = ∅ } ∪ { ∅ } = { E ∈ I ∣ E is a spherical subgraph or a wide subgraph } ∪ { ∅ } . \begin{aligned}
\mathcal{E}:=\{E\in\mathcal{I}\mid\partial_{*}W_{E}=\emptyset\}\cup\{\emptyset\} \\
&=\{E\in\mathcal{I}\mid E\text{ is a spherical subgraph or a wide subgraph}\}\cup\{\emptyset\}.
\end{aligned} E := { E ∈ I ∣ ∂ ∗ W E = ∅ } ∪ { ∅ } = { E ∈ I ∣ E is a spherical subgraph or a wide subgraph } ∪ { ∅ } . We build the weak Morse hierarchy H \mathcal{H} H Figure 4 . We will start with a hierarchy H 0 \mathcal{H}_{0} H 0 H i \mathcal{H}_{i} H i H i − 1 \mathcal{H}_{i-1} H i − 1 H i − 1 \mathcal{H}_{i-1} H i − 1 H \mathcal{H} H
Base case: Let H 0 \mathcal{H}_{0} H 0 W Δ W_{\Delta} W Δ
The i + 1 i+1 i + 1 Suppose we constructed the hierarchy H i \mathcal{H}_{i} H i i i i v v v H i \mathcal{H}_{i} H i v v v W Λ W_{\Lambda} W Λ W Δ W_{\Delta} W Δ
∃ Λ 1 , Λ 2 ∈ I ∖ { Λ } and E ∈ E with Λ = Λ 1 ∪ Λ 2 and Λ 1 ∩ Λ 2 ⊆ E . (1) \exists\Lambda_1,\ \Lambda_2 \in\mathcal{I} \setminus\{\Lambda\}\ \text{ and } E \in\mathcal{E}\ \text{ with } \Lambda= \Lambda_1 \cup\Lambda_2\ \text{ and }\Lambda_1 \cap\Lambda_2 \subseteq E. \tag*{(1)} ∃ Λ 1 , Λ 2 ∈ I ∖ { Λ } and E ∈ E with Λ = Λ 1 ∪ Λ 2 and Λ 1 ∩ Λ 2 ⊆ E . (1) Yes: If there exist such Λ 1 \Lambda_1 Λ 1 Λ 2 \Lambda_2 Λ 2 W Λ = W Δ 1 ∗ W Λ 1 ∩ Λ 2 W Λ 2 W_\Lambda= W_{\Delta_1} *_{W_{\Lambda_1 \cap\Lambda_2}} W_{\Lambda_2} W Λ = W Δ 1 ∗ W Λ 1 ∩ Λ 2 W Λ 2 { v , v 1 } \{v,v_1\} { v , v 1 } { v , v 2 } \{v,v_2\} { v , v 2 } H i \mathcal{H}_i H i v 1 v_1 v 1 W Λ 1 W_{\Lambda_1} W Λ 1 v 2 v_2 v 2 W Λ 2 W_{\Lambda_2} W Λ 2
No: If no such Λ 1 \Lambda_1 Λ 1 Λ 2 \Lambda_2 Λ 2 Λ \Lambda Λ Δ \Delta Δ Λ \Lambda Λ Δ \Delta Δ v v v T i \mathcal{T}_i T i v v v H i + m \mathcal{H}_{i+m} H i + m m ≥ 1 m \ge1 m ≥ 1
As Δ \Delta Δ H \mathcal{H} H n n n (1) together with Theorem 4.4 ensures that every H i \mathcal{H}_i H i i ≤ n i \le n i ≤ n Λ \Lambda Λ H \mathcal{H} H Λ \Lambda Λ Δ \Delta Δ Λ \Lambda Λ Λ \Lambda Λ [20 Thm 8.7.2] . If Λ \Lambda Λ ( ∂ ∗ W Λ ) = ∅ (\partial_*W_\Lambda)=\varnothing ( ∂ ∗ W Λ ) = ∅ W Λ W_\Lambda W Λ Λ \Lambda Λ Δ \Delta Δ Λ \Lambda Λ ( ∂ ∗ W Λ , W Δ ) ≠ ∅ (\partial_*W_\Lambda,W_\Delta)\ne\varnothing ( ∂ ∗ W Λ , W Δ ) = ∅
If in addition, Δ \Delta Δ Λ \Lambda Λ Λ \Lambda Λ
In view of Theorem 4.14, the algorithm in the proof decides whether a Coxeter graph Δ \Delta Δ W Δ W_\Delta W Δ
Lemma 4.19. Let Δ \Delta Δ
The algorithm in Figure 4 terminates in at least one graph that is not spherical and is not contained in a wide subgraph of Δ \Delta Δ
Δ \Delta Δ W Δ W_\Delta W Δ
If, in addition, Δ \Delta Δ
Proof. The claim follows from Theorem 4.14 and Theorem 4.13. □ \square □
Corollary 4.20 (Theorem H). If every leaf group of the weak Morse hierarchy of W Δ W_\Delta W Δ Theorem 4.17 ) is spherical
or contained in a special subgroup of W Δ W_{\Delta} W Δ ∂ ∗ W Δ \partial_* W_{\Delta} ∂ ∗ W Δ W Δ W_{\Delta} W Δ
Application: Distinguishing Coxeter groups and RAAGs up to quasi-isometry
In this section we illustrate how to use Morse hierarchies to distinguish groups up to quasi-isometry, with a focus on right-angled Artin groups (RAAGs) and right-angled Coxeter groups (RACGs). Note that RACGs are examples of Coxeter groups that do not contain any spherical nexus.
In [31 ] , a procedure for constructing graphs that define RACGs with totally disconnected Morse boundary was given. Theorem H shows that all RACGs with totally disconnected boundary arise in this way (see Theorem 4.21).
Charney–Cordes–Sisto [9 ] proved that the Morse boundary of every right-angled Artin group (RAAG) is totally disconnected. Accordingly, if the Morse boundary of a given group G G G
Corollary 5.1. Let Δ \Delta Δ Figure 4 decides that Δ \Delta Δ ∂ ∗ W Δ \partial_* W_{\Delta} ∂ ∗ W Δ W Δ W_{\Delta} W Δ
Another method for finding infinite connected components in Morse boundaries of RACGs is to look for stably embedded virtually closed surface RACG-subgroups [1 , 11 ] . Such RACG-subgroups arise, for example, from stable cycles in the Coxeter graph, i.e., cycles of length at least 5 in which no non-adjacent vertices are contained in an induced 4-cycle. Our approach via Morse hierarchies, however, offers a more robust alternative, because the presence of a topologically embedded circle in the Morse boundary does not imply the existence of a stable cycle in the Coxeter graph [27 ] .
Besides Morse boundaries, there are existing tools that are known to distinguish RACGs from RAAGs up to quasi-isometry such as divergence, maximal product region graphs, and JSJ-decompositions. In [11 ] , definitions and an updated outline on what is known about RAAGedy RACGS is given.
We now apply our tools for Morse hierarchies we have developed to distinguish groups using connected components of the Morse boundary. If W Δ W_{\Delta} W Δ W Δ ′ W_{\Delta'} W Δ ′ C C C ∂ ∗ W Δ \partial_*W_{\Delta} ∂ ∗ W Δ C ′ C' C ′ ∂ ∗ W Δ ′ \partial_*W_{\Delta'} ∂ ∗ W Δ ′ C C C
If W Δ W_{\Delta} W Δ Theorem F (Theorem 4.17 ) yields a corresponding Morse hierarchy. In particular, we can determine all infinite connected components of ∂ ∗ W Δ \partial_*W_{\Delta} ∂ ∗ W Δ Δ \Delta Δ Figure 4 . The algorithm has as input the defining graph Δ \Delta Δ Λ \Lambda Λ Δ \Delta Δ Λ \Lambda Λ W Λ W_{\Lambda} W Λ C C C C = ( ∂ ∗ W Λ , W Δ ) C=(\partial_*W_{\Lambda},W_{\Delta}) C = ( ∂ ∗ W Λ , W Δ ) ∂ ∗ W Δ \partial_*W_{\Delta} ∂ ∗ W Δ W Δ W_{\Delta} W Δ
Moreover, Theorem D implies that an infinite Morse boundary component of W Δ W_{\Delta} W Δ [41 Thm 1.11] , and [26 Prop.4.9] and later reproved in [39 Thm 7.5] : A special subgroup W Λ W_{\Lambda} W Λ W Δ W_{\Delta} W Δ Λ \Lambda Λ square-complete . A subgraph Λ \Lambda Λ Δ \Delta Δ square-complete if each induced 4-cycle of Δ \Delta Δ Λ \Lambda Λ Λ \Lambda Λ [10 ] .
Thus, we obtain the following in the right-angled case. One could formulate the analogous statement for general Coxeter groups without a spherical nexus using the conditions in [10 ] .
Corollary 5.2. Let ∂ ∗ W Δ \partial_*W_{\Delta} ∂ ∗ W Δ ∂ ∗ W Δ ′ \partial_*W_{\Delta'} ∂ ∗ W Δ ′ C C C C C C W Δ W_{\Delta} W Δ W Δ ′ W_{\Delta'} W Δ ′ Δ \Delta Δ Δ ′ \Delta' Δ ′
The following examples in Figure 5 illustrate how the new quasi-isometry invariants discussed in this section are helpful for distinguishing RACGs from other RACGs or from RAAGs up to quasi-isometry.
We briefly describe the information contained in the table. Column 2: The Morse boundary column indicates whether there are none, one, or infinitely many infinite non-singleton connected components along with whether those non-trivial components are compact or not.
Column 3: The stable cycle indicates whether the defining graph contains a stable cycle or not.
Column 4: In [14 ] , the authors give combinatorial conditions on the defining that exactly characterize when the Morse boundary of the RACG is connected and locally connected (LC). These same conditions appear in the earlier paper [37 ] , where the authors investigate when the visual boundaries for a given RACG are locally connected (LC) versus not locally connected (NLC). We record this in the visual boundary column. They show that if one of these conditions is satisfied, then any visual boundary for the corresponding RACG is not locally connected (NLC). In [15 ] , Mihalik and Camp construct graph (3) to show that the combinatorial conditions in Theorem 4.9 are sufficient, but not necessary for local connectedness of the visual boundary. Specifically, the RACG group with graph (3) as defining graph has LC visual boundary but totally disconnected Morse boundary.
We now focus on the last three columns. For RACGs, quadratic divergence is equivalent to a combinatorial condition on the defining graph called CFS [21 ] ,[34 ] . Since quadratic divergence is a sufficient but not necessary condition for a RACG to be quasi-isometric to a RAAG, there has been other work developing finer invariants. In the triangle free case, the definition of CFS was strengthened to strongly CFS as a necessary condition for a RACG to be RAAGedy [11 ] .
Table 1.
We list the new insights through the study of Morse hierarchies that, according to our knowledge, cannot be obtained by other methods here:
(2) is not quasi-isometric to (7), (4), (10)
(3) is not quasi-isometric to (5), (6), (8)
(4) is not quasi-isometric to (6), (11)
(5) is not quasi-isometric to to (6)
(6) is not quasi-isometric to (8), (9)
Note that we can also apply Theorem D and Theorem E to the graphs above, meaning that in all groups quasi-isometric to the examples above, the infinite Morse boundary components arise from their stabilizers. Moreover, all compact components in such groups have stable stabilizers. In the case of circle components in the boundary, as for instance in graph (6), every group quasi-isometric to that group contains a stable subgroup that is virtually a hyperbolic surface group (see § 6.4 ).
For many more examples that illustrate Morse accessibility, including an illustration of the mentioned algorithm in Figure 4 for RACGs, see § 6 .
Examples
Morse accessible groups
In this section, we list examples of Morse-accessible groups and outline how to construct Morse hierarchies for them. We begin with the most basic example of finitely generated free group so that we can illustrate how we will describe Morse hierarchies with notation and figures.
Example 6.1 (Morse hierarchies of free groups). For F n F_n F n n ≥ 1 n \ge1 n ≥ 1 F n F_n F n n n n Figure 6 . In particular, it will be useful to view Z = { i d } { i d } \mathbb{Z}=\{id\}_{\{id\}} Z = { i d } { i d } Z \mathbb{Z} Z
Figure 6. Two Morse splittings of F 5 F_5 F 5 Z \mathbb{Z} Z
When illustrating Morse hierarchies, we will often label the vertices of the hierarchy tree with spaces whose fundamental groups yield the corresponding vertex groups, rather than the groups themselves. The associated graph of groups decomposition can then be indicated by coloring the edge spaces in red. Decomposing the space along these red subspaces yields pieces whose fundamental groups are precisely the vertex groups of the corresponding graph of groups decomposition. For instance, we can illustrate the Morse hierarchy on the left of Figure 6 alternatively as in Figure 7 .
Figure 7. A Morse hierarchy of height 2 of F 5 F_5 F 5
In Figure 8 , two Morse hierarchies for F 5 F_5 F 5 F 5 F_5 F 5 Z \mathbb{Z} Z Z = { i d } { i d } \mathbb{Z}=\{id\}_{\{id\}} Z = { i d } { i d } F 5 F_5 F 5 Z \mathbb{Z} Z
Figure 8. Two Morse hierarchies of F 5 F_5 F 5
Example 6.2 (Hyperbolic groups). If Γ \Gamma Γ Γ \Gamma Γ Γ \Gamma Γ
Example 6.3 (Non-hyperbolic group with height 0 Morse hierarchy). If G G G ω \omega ω [9 ] .
Example 6.4 (Free products). Suppose a finitely generated group G = A 1 ⋆ ⋯ ⋆ A n G = A_1 \star\cdots\star A_n G = A 1 ⋆ ⋯ ⋆ A n ∂ ∗ G \partial_*G ∂ ∗ G A i A_i A i G G G G G G A i A_i A i G G G
Example 6.5 (Morse Splittings and hierarchies of CAT(0) groups). Many Morse splittings and hierarchies of CAT(0) groups can be obtained by constructing the group iteratively via HNN-extensions and amalgamated free products, checking that the amalgams used satisfy the assumptions of [2 II.Thm 11.18, II.Thm 11.21] . Hierarchies constructed this way are:
Weak Morse, if one ensures that every edge group has empty relative Morse boundary. The easiest way to ensure this is to construct splittings along groups that have empty Morse boundary such as finite groups or direct products of infinite finitely generated groups.
(2) Morse, if in addition to (1), every leaf group has connected relative Morse boundary. One way to ensure this is to build Morse hierarchies in which the leaf groups have connected Morse boundaries that topologically embeds into the ambient group. The latter is satisfied for leaf groups that are stable (see Theorem 2.10) or Morse (see Theorem 2.11).
For a specific example, we can glue two hyperbolic 3-manifolds M 1 M_1 M 1 M 2 M_2 M 2 T T T T T T π 1 ( M 1 ) \pi_1(M_1) π 1 ( M 1 ) π 1 ( M 2 ) \pi_1(M_2) π 1 ( M 2 ) G G G [9 ] , the Morse boundaries of π 1 ( M 1 ) \pi_1(M_1) π 1 ( M 1 ) π 1 ( M 2 ) \pi_1(M_2) π 1 ( M 2 ) ω \omega ω
However, (1) and (2) can also be satisfied without the leaf groups being stable or Morse and the edge groups having empty Morse boundary. We illustrate this in § 6.3 and § 6.5 .
Example 6.6 (3-manifold groups). In [Zbi24], Zbinden calculates the Morse boundaries of closed, orientable 3-manifold groups. In the proof of Theorem 7 of that paper, Zbinden implicitly outlines how to obtain a Morse hierarchy for these groups. Briefly, every closed, orientable 3-manifold M M M ω \omega ω [9 ] ).
Example 6.7. (Right-angled Artin Groups) A right-angled Artin group (RAAG) A Δ A_\Delta A Δ W Δ W_\Delta W Δ [9 ] showed that the Morse boundary of every RAAG is totally disconnected. An alternative proof was given in [31 Thm 7.6] via implicitly constructing a weak Morse hierarchy. This weak Morse hierarchy is obtained by successively deleting vertices from the defining graph until one obtains a graph consisting of disconnected cliques. The weak Morse hierarchy can be extended to a Morse hierarchy by writing ∂ ∗ Z \partial_*\mathbb{Z} ∂ ∗ Z ∂ ∗ Z = i d ∗ i d \partial_*\mathbb{Z} = \mathrm{id}*\mathrm{id} ∂ ∗ Z = id ∗ id Figure 9 .
Figure 9. Three Morse hierarchies of a RAAG A Δ = Z ∗ ( F 2 × F 2 ) A_{\Delta} = \mathbb{Z} * (\mathbb{F}_{2} \times\mathbb{F}_{2}) A Δ = Z ∗ ( F 2 × F 2 ) Δ \Delta Δ
Example 6.8 (Coxeter groups). In § 4 , we show that every Coxeter group has a weak Morse hierarchy, which is a genuine Morse hierarchy for Coxeter groups that do not contain a spherical nexus (as for instance in the right-angled case). The proof is based on an algorithm on the defining graph illustrated in Figure 4 . In § 5 we explain how to use this algorithm to identify all infinite Morse boundary components of a Coxeter group that does not contain a spherical nexus, and how to use it in order to distinguish Coxeter groups up to quasi-isometry.
Non-uniqueness of Morse hierarchies
We have already illustrated the non-uniqueness of Morse hierarchies using F n \mathbb{F}_{n} F n
We study the disjoint union of a 4-cycle and a vertex. This RAAG is isomorphic to Z ∗ ( F 2 × F 2 ) \mathbb{Z} * (\mathbb{F}_{2} \times\mathbb{F}_{2}) Z ∗ ( F 2 × F 2 ) Figure 9 yield three different Morse hierarchies for the corresponding RAAG: Whenever a vertex labeled by a graph Δ \Delta Δ Δ 1 \Delta_{1} Δ 1 Δ 2 \Delta_{2} Δ 2 A Δ A_{\Delta} A Δ A Δ = A Δ 1 ∗ A Δ 1 ∩ Δ 2 A Δ 2 A_{\Delta} = A_{\Delta_{1}} *_{A_{\Delta_{1} \cap\Delta_{2}}} A_{\Delta_{2}} A Δ = A Δ 1 ∗ A Δ 1 ∩ Δ 2 A Δ 2 Z \mathbb{Z} Z Z = i d ∗ i d \mathbb{Z} = id_{*}id Z = i d ∗ i d Z \mathbb{Z} Z F 2 \mathbb{F}_{2} F 2 i d id i d Z \mathbb{Z} Z F 2 \mathbb{F}_{2} F 2
Comparing Morse hierarchies of two specific surface amalgams
In this subsection compare the Morse hierarchies of two specific surface amalgams in order to illustrate some features and technicalities of Morse hierarchies.
First, we study the space X X X S S S T T T α \alpha α Figure 10 . In order to create a Morse hierarchy for π 1 ( X ) \pi_1(X) π 1 ( X ) S S S T T T α \alpha α π 1 ( X ) \pi_1(X) π 1 ( X ) Z \mathbb{Z} Z Z \mathbb{Z} Z ∂ ∗ π 1 ( X ) \partial_*\pi_1(X) ∂ ∗ π 1 ( X ) α \alpha α T T T α \alpha α X ~ \widetilde{X} X X ~ \widetilde{X} X
Figure 10. The pictured tree yields a Morse hierarchy for the fundamental group of the space at the top of the picture.
Observe that although ∂ ∗ π 1 ( S ) \partial_*\pi_1(S) ∂ ∗ π 1 ( S ) relative Morse boundary in π 1 ( X ) \pi_1(X) π 1 ( X ) S S S S S S α \alpha α β \beta β α \alpha α π 1 ( S ) \pi_1(S) π 1 ( S ) α \alpha α α \alpha α β \beta β X ~ \widetilde{X} X S S S F 3 F_3 F 3 X X X π 1 ( X ) \pi_1(X) π 1 ( X ) ∂ ∗ π 1 ( X ) \partial_*\pi_1(X) ∂ ∗ π 1 ( X )
Next, we consider a space Y Y Y S 1 S_1 S 1 S 2 S_2 S 2 T T T Figure 11 . We can decompose Y Y Y π 1 ( Y ) \pi_1(Y) π 1 ( Y ) T T T Z 2 = Z ∗ Z \mathbb{Z}^2=\mathbb{Z}*\mathbb{Z} Z 2 = Z ∗ Z π 1 ( S 1 ) \pi_1(S_1) π 1 ( S 1 ) π 1 ( S 1 ) \pi_1(S_1) π 1 ( S 1 ) π 1 ( S 1 ) \pi_1(S_1) π 1 ( S 1 ) π 1 ( Y ) \pi_1(Y) π 1 ( Y ) π 1 ( S 1 ) \pi_1(S_1) π 1 ( S 1 ) Y Y Y π 1 ( S 1 ) \pi_1(S_1) π 1 ( S 1 ) ∂ ∗ π 1 ( Y ) \partial_*\pi_1(Y) ∂ ∗ π 1 ( Y ) Theorem 3.20 . Hence, π 1 ( Y ) \pi_1(Y) π 1 ( Y ) π 1 ( X ) \pi_1(X) π 1 ( X ) π 1 ( Y ) \pi_1(Y) π 1 ( Y )
Figure 11. The pictured tree yields a Morse hierarchy for the fundamental group of the space at the top of the picture.
Remark 6.9. The discussed hierarchies of π 1 ( X ) \pi_1(X) π 1 ( X ) π 1 ( Y ) \pi_1(Y) π 1 ( Y ) non-empty Morse boundary but empty relative Morse boundary in the ambient group. Moreover, neither π 1 ( X ) \pi_1(X) π 1 ( X ) π 1 ( Y ) \pi_1(Y) π 1 ( Y ) Morse , i.e. in which every vertex group has connected relative Morse boundary and in which every edge group is undistorted and has empty relative Morse boundary. boundary. Accordingly, the study of hierarchies rather than graph of groups is necessary in order to understand these examples.
Comparing Morse hierarchies of two specific RACGs
In this subsection we show explicitly how the algorithm outlined in Figure 4 in the proof of Theorem F produces Morse hierarchies from the defining graph of a RACG and how it can be used to distinguish RACGs up to quasi-isometry. In order to do so, we apply the algorithm in different ways to two RACGs and compare the Morse hierarchies we obtain this way. We highlight the different features and technicalities that might occur along the way.
We apply the algorithm to two toy examples, Δ 1 \Delta_1 Δ 1 Δ 2 \Delta_2 Δ 2 Figure 12 . Note that these groups are analogues of the groups in Section 6.3. The algorithm will show that ∂ ∗ W Δ 1 \partial_*W_{\Delta_1} ∂ ∗ W Δ 1 ∂ ∗ W Δ 2 \partial_*W_{\Delta_2} ∂ ∗ W Δ 2 W Δ 1 W_{\Delta_1} W Δ 1 W Δ 2 W_{\Delta_2} W Δ 2 W Δ 1 W_{\Delta_1} W Δ 1 W Δ 2 W_{\Delta_2} W Δ 2
Figure 12. Two examples inspired from Section 6 of [CS15]. The groups W Δ 1 W_{\Delta_1} W Δ 1 W Δ 2 W_{\Delta_2} W Δ 2 Δ 1 \Delta_1 Δ 1 Δ 2 \Delta_2 Δ 2
The three trees in Figure 13 , give three different Morse hierarchies of W Δ 1 W_{\Delta_1} W Δ 1 Δ 1 \Delta_1 Δ 1
Figure 13. Three trees corresponding to two Morse hierarchies of W Δ 1 W_{\Delta_1} W Δ 1
We now compare the three Morse hierarchies of W Δ 1 W_{\Delta_1} W Δ 1 C 6 C_6 C 6 C 4 C_4 C 4
However, if we follow the algorithm until the end, we are forced to decompose the 4-cycle C 4 C_4 C 4 C 4 C_4 C 4
We now consider the 6-cycle C 6 C_6 C 6 Δ 1 \Delta_1 Δ 1 Δ 1 \Delta_1 Δ 1 W Δ 1 W_{\Delta_1} W Δ 1
Note: Each edge of Δ 1 \Delta_1 Δ 1 Δ 1 \Delta_1 Δ 1 Δ 1 \Delta_1 Δ 1
All three hierarchies share the following crucial properties: Every edge of Δ 1 \Delta_1 Δ 1 W Δ 1 W_{\Delta_1} W Δ 1
Next, we examine W Δ 2 W_{\Delta_2} W Δ 2 Figure 14 shows three trees that correspond to three different Morse hierarchies. In contrast to Δ 1 \Delta_1 Δ 1 C 6 C_6 C 6 W C 6 W_{C_6} W C 6 W C 6 W_{C_6} W C 6 ∂ ∗ W Δ 2 \partial_* W_{\Delta_2} ∂ ∗ W Δ 2 W Δ 2 W_{\Delta_2} W Δ 2 Figure 11 , the cycle topologically embeds in the Morse boundary of the ambient group because W C 6 W_{C_6} W C 6 W Δ 2 W_{\Delta_2} W Δ 2
Figure 14. Three trees that yield Morse hierarchies of W Δ 2 W_{\Delta_2} W Δ 2
Indeed, by Theorem 5.2 , if W Δ ′ W_{\Delta'} W Δ ′ W Δ 2 W_{\Delta_2} W Δ 2 ∂ ∗ W Δ ′ \partial_* W_{\Delta'} ∂ ∗ W Δ ′ W Δ ′ W_{\Delta'} W Δ ′ Δ \Delta Δ Δ ′ \Delta' Δ ′ Theorem 5.2 also implies that Δ \Delta Δ W Δ W_\Delta W Δ W C 6 W_{C_6} W C 6 W Δ W_\Delta W Δ [42 , 13 , 25 ] W Δ W_\Delta W Δ [22 ] , Δ \Delta Δ
We observe that the Morse boundaries of W Δ 1 W_{\Delta_1} W Δ 1 W Δ 2 W_{\Delta_2} W Δ 2
Non-compact connected components
Recall that by Theorem D, an infinite Morse boundary component is compact if and only if its stabilizer is stable. It is naturally to ask if a similar characterization can be given for infinite Morse boundary components that are not compact. In order to give an intuition for this question, we first consider an easy example of a RACG with infinite non-compact Morse boundary components.
Example 6.10 (A RACG with non-compact Morse boundary components). The RACG whose defining graph is the one-skeleton of a 3-dimensional cube is a virtual hyperbolic 3-manifold group with torus cusp. Thus we can obtain a RACG example analogous to the 3-manifold example in Theorem 6.5 by identifying two copies of this along of one of its 4-cycle faces. Thus the Morse boundary of this amalgam consists of single-point components and components homeomorphic to ω \omega ω
In general, whenever a hierarchy for a RACG produced by the algorithm in Figure 4 has a leaf group whose corresponding induced subgraph contains an induced 4-cycle and is square-complete, the corresponding infinite Morse boundary component is not compact and arises from a special subgroup that is Morse, but not stable. This can be concluded from Tran’s characterization of stability and Morseness of special subgroups [41 ] .
In view of this example one might hope that every non-compact infinite connected component arises from a group that is Morse but not stable. However, this turns out not be the case. The next example shows that stabilizers of infinite Morse boundary components can behave less nicely than one might expect. In this example, an infinite non-compact Morse boundary component C C C H H H ∂ ∗ H \partial_* H ∂ ∗ H C C C
Example 6.11 (Hyperbolic RACG leaf group that is not Morse). Let Δ \Delta Δ Figure 15 and Λ \Lambda Λ Λ \Lambda Λ Δ \Delta Δ ( ∂ ∗ W Λ , W Δ ) (\partial_* W_\Lambda, W_\Delta) ( ∂ ∗ W Λ , W Δ ) ( ∂ ∗ W Λ , W Δ ) (\partial_* W_\Lambda, W_\Delta) ( ∂ ∗ W Λ , W Δ ) ∂ ∗ W Δ \partial_* W_\Delta ∂ ∗ W Δ W Λ W_\Lambda W Λ Λ \Lambda Λ W Λ W_\Lambda W Λ
Figure 15. The pictured tree yields a Morse hierarchy.
This also gives an example where ι \iota ι ι ∗ : M ( Y , X ) → ( ∂ ∗ ι ( Y ) , X ) \iota_* : \mathcal{M}(Y, X) \to(\partial_*\iota(Y), X) ι ∗ : M ( Y , X ) → ( ∂ ∗ ι ( Y ) , X )
To see this, let Υ \Upsilon Υ Λ \Lambda Λ { a , b , c , d } \{a, b, c, d\} { a , b , c , d } W Υ W_\Upsilon W Υ W Δ W_\Delta W Δ W Υ W_\Upsilon W Υ W Δ W_\Delta W Δ
Now consider the geodesic rays γ i = ( c d ) i ( a b ) i ( c d ) ∞ \gamma_i=(cd)^i(ab)^i(cd)^\infty γ i = ( c d ) i ( ab ) i ( c d ) ∞ γ i \gamma_i γ i a , b a,b a , b W Δ W_\Delta W Δ ( a b ) i (ab)^i ( ab ) i γ i \gamma_i γ i M i M_i M i M i M_i M i i i i M i < M i + 1 M_i<M_{i+1} M i < M i + 1 W Λ W_\Lambda W Λ W Δ W_\Delta W Δ γ i \gamma_i γ i ∂ ∗ W Δ \partial_*\!W_\Delta ∂ ∗ W Δ ( ∂ ∗ W Λ , W Δ ) (\partial_*\!W_\Lambda,W_\Delta) ( ∂ ∗ W Λ , W Δ ) W Λ W_\Lambda W Λ γ i \gamma_i γ i ( c d ) i (cd)^i ( c d ) i ∂ ∗ W Δ \partial_*\!W_\Delta ∂ ∗ W Δ W Δ W_\Delta W Δ M ( ∂ ∗ W Λ , W Δ ) \mathcal{M}(\partial_*\!W_\Lambda,W_\Delta) M ( ∂ ∗ W Λ , W Δ ) [27 ] in which the defining graph studied there does not contain any stable cycle.
Alternative definitions of Morse-accessibility
The concept of Morse accessibility was informally introduced in [KMZ26]. Our definition here is just one way to formalize the notion that worked for our purposes. We offer two alternative ways one could have defined Morse accessibility that might provide other interesting results.
Recall that in our definition of Morse hierarchy all edge groups and vertex groups are undistorted, and all leaf groups have connected relative Morse boundary. Alternatively keeping the undistorted condition, one could assume
all leaf groups have connected Morse boundary, or
all leaf groups have both, connected Morse boundary and connected relative Morse boundary.
Let L L L L L L
As a simple example for right-angled Coxeter groups, let Δ \Delta Δ K 3 , 2 K_{3,2} K 3 , 2 a , b a,b a , b x , y , z x,y,z x , y , z W Δ = A ∗ C B W_\Delta=A*_{C}B W Δ = A ∗ C B A = ⟨ a , b , x , y ⟩ A=\langle a,b,x,y\rangle A = ⟨ a , b , x , y ⟩ B = ⟨ a , b , z ⟩ B=\langle a,b,z\rangle B = ⟨ a , b , z ⟩ C = ⟨ a , b ⟩ C=\langle a,b\rangle C = ⟨ a , b ⟩ C C C C C C a , b a,b a , b a , b , x , y a,b,x,y a , b , x , y Z 2 \mathbb{Z}^2 Z 2 W Δ W_\Delta W Δ Δ ′ \Delta' Δ ′ Δ \Delta Δ ⟨ a , b ⟩ \langle a,b\rangle ⟨ a , b ⟩
However, in all examples we know, such hierarchies can be extended to hierarchies in which each leaf group with empty relative Morse boundary has empty Morse boundary as well. Thus L L L
If the Morse boundary of the leaf group L L L Figure 12 . However, in all examples we know, the other implication is satisfied. In all weak Morse hierarchies we know, every leaf group with connected non-empty relative Morse boundary has Morse boundary that is connected and non-empty as well.
These observations suggest that the answer to the following question might be positive:
Question 1. Suppose every leaf group of a weak Morse hierarchy has connected relative Morse boundary. Can this hierarchy be extended to one in which every leaf group has both, a connected relative Morse boundary and a connected Morse boundary?
A positive answer of this question would suggest that the last condition is the most natural choice for the notion of Morse accessibility. Nevertheless, for the purposes of the proofs and the underlying theory, it is necessary and sufficient only to assume that the leaf groups have connected relative Morse boundary. Therefore, we have chosen to require only this assumption.
One could also loosen the restrictions on the undistortedness of the leaf and/or edge groups in a Morse hierarchy, but there be dragons. Indeed, there would need to be many new results on splittings over undistorted distorted groups and its relationship to the Morse boundary. Indeed, there is an open question that would need to be answered first:
Question 2. Does there exist infinite groups H < G H < G H < G ∂ ∗ H = ∅ \partial_* H = \varnothing ∂ ∗ H = ∅ ( ∂ ∗ H , G ) (\partial_* H,G) ( ∂ ∗ H , G )
Such an H H H [7 Question 1.5] .
Question 3. Recall that Cashen and Mackay introduced another topology on the Morse boundary that is metrizable and thus coarser than the direct limit topology [16 ] . It is natural to ask: Do Morse hierarchies exist for the Cashen–Mackay topology?
In Section A of this paper, the second author and He address this question and give a partial positive answer.
Appendix A. Cashen–Mackay Morse hierarchies
Annette Karrer, Vivian He
As explained in § 7 , there are several possible variations of definitions of Morse hierarchies and their consequences. One variant arises by equipping the Morse boundary not with the direct limit topology introduced by Charney and Cordes [CS15, Cor17], but with the metrizable topology introduced by Cashen–Mackay instead [16 ] . In this appendix, we show that, with an additional quasi-convexity condition on the edge groups (see Equation (2) in Section A.3), we can transfer the machinery of Morse hierarchies to the Cashen–Mackay topology (see Section A.4); we will call these Cashen–Mackay Morse hierarchies .
The quasi-convexity condition we are imposing might feel very restrictive. However, many natural Morse hierarchies of groups satisfy this quasiconvexity condition. Indeed, it holds for almost all hierarchies considered in § 6.1 : hierarchies of RAAGs and Coxeter groups along special subgroups (in fact, they are convex [20 Proposition 4.8.1] ), prime- and JSJ-decompositions of closed oriented non-positively curved 3-manifolds, and surface amalgams (and more generally hierarchies of certain CAT(0) groups). See Theorem A.15 .
An important fact we are using is that the direct-limit topology is finer than the Cashen–Mackay topology; so if a set in connected in the direct-limit topology, it is connected in the Cashen–Mackay topology. As a result, if the leaf groups of a Morse hierarchy have connected relative Morse boundaries in the direct-limit topology, they also have connected relative Morse boundaries in the Cashen–Mackay topology. Hence, if the edge groups of a Morse hierarchy satisfy the quasi-convexity condition, then it is also a Cashen–Mackay Morse hierarchy.
Most of the arguments concerning Morse hierarchies in § 2 , § 3.1 and Section 3.2 transfer naturally to the Cashen–Mackay topology. In fact, there is only one key lemma [24 Lemma 3.5] that needs to be adapted and for which we need to impose the additional quasi-convexity condition on the edge groups. This lemma says that a splitting along undistorted groups with empty relative Morse boundary yields a corresponding decomposition of the Morse boundary into clopen sets. This is the Key Lemma of this appendix (Theorem A.5).
Acknowledgments
We thank Matthew Cordes and Kim Ruane for their helpful comments.
Setting and notation
Let X X X p ∈ X p \in X p ∈ X [16 ] . Cashen–Mackay work with equivalence classes of quasi-geodesic rays instead of equivalence classes of geodesic rays. As every Morse quasi-geodesic ray is at finite Hausdorff distance to a Morse geodesic ray, every such equivalence class contains a Morse geodesic ray as representative. We will rewrite the definition of the open neighborhood basis.
Definition A.1. Let [ γ ] ∈ ∂ ∗ X [\gamma] \in\partial_*X [ γ ] ∈ ∂ ∗ X γ \gamma γ r ≥ 1 r \ge1 r ≥ 1 U ( [ γ ] , r ) U([\gamma], r) U ([ γ ] , r ) [ η ] ∈ ∂ ∗ X [\eta] \in\partial_*X [ η ] ∈ ∂ ∗ X q ≥ 1 q \ge1 q ≥ 1 Q ≥ 0 Q \ge0 Q ≥ 0 ( q , Q ) (q,Q) ( q , Q ) η ′ ∈ [ η ] \eta' \in[\eta] η ′ ∈ [ η ]
d ( η ′ , γ ∩ ( N r ( p ) ) c ) ≤ κ ( ρ [ γ ] , q , Q ) . d(\eta', \gamma\cap(N_r(p))^c) \le\kappa(\rho_{[\gamma]}, q, Q). d ( η ′ , γ ∩ ( N r ( p ) ) c ) ≤ κ ( ρ [ γ ] , q , Q ) . The function ρ [ γ ] \rho_{[\gamma]} ρ [ γ ] κ \kappa κ [16 Definition 4.1] . For more information on the metrizable topology, see [16 Section 4] .
Form now on, we equip the Morse boundary with the Cashen–Mackay topology. In particular, relative Morse boundaries are now subspaces in the Morse boundary equipped with the Cashen–Mackay topology. Recall from Theorem 2.3 that, if X X X X X X ( ∂ ∗ Y , X ) (\partial_*Y, X) ( ∂ ∗ Y , X ) Y Y Y X X X ∂ ∗ X \partial_*X ∂ ∗ X M ( Y , X ) \mathcal{M}(Y, X) M ( Y , X ) Y Y Y X X X ∂ ∗ Y \partial_*Y ∂ ∗ Y Y Y Y X X X Y Y Y X X X
Theorem A.2 (Corollary 6.2 in [16 ] ). Let ( X , d X ) (X, d_X) ( X , d X ) ( Y , d Y ) (Y, d_Y) ( Y , d Y ) ι : Y → X \iota: Y \to X ι : Y → X ι \iota ι ι ∗ : M ( Y , X ) → ( ∂ ∗ ι ( Y ) , X ) \iota_*: \mathcal{M}(Y, X) \to(\partial_*\iota(Y), X) ι ∗ : M ( Y , X ) → ( ∂ ∗ ι ( Y ) , X )
Note that this is a stronger variant of Lemma 2.3 in [24 ] (see Theorem 2.9) for the direct-limit topology that is used in § 2 , § 3.1 and Section 3.2.
We now recall the setting and the notation in [24 ] : Let G G G G G G T T T E ( T ) \mathcal{E}(T) E ( T ) edge groups and the vertex stabilizers vertex groups . A half-space of T T T T T T e e e h h h h ∗ h^* h ∗ H ( T ) \mathcal{H}(T) H ( T )
Let X X X G G G o o o T T T f o : G → T f_o: G \to T f o : G → T f o ( g ) = f_o(g) = f o ( g ) = h ∈ H ( T ) \mathfrak{h} \in\mathcal{H}(T) h ∈ H ( T ) G = f o − 1 ( h ) ⊔ f o − 1 ( h ∗ ) G = f_o^{-1}(\mathfrak{h}) \sqcup f_o^{-1}(\mathfrak{h}^*) G = f o − 1 ( h ) ⊔ f o − 1 ( h ∗ ) e e e T T T G e G_e G e
The following lemma shows that, if every edge group is finitely generated and undistorted, we can give the Cayley graph a “tree-like” structure. More specifically, we can associate to every edge of T T T X ( e ) X(e) X ( e ) X X X T T T X ( e ) X(e) X ( e ) X X X
Lemma A.3 ([24 Lemma 3.2] ). Let G G G G G G e ∈ E ( T ) e \in\mathcal{E}(T) e ∈ E ( T ) X ( e ) ⊆ X X(e) \subseteq X X ( e ) ⊆ X
X ( e ) X(e) X ( e ) G e G_e G e G e G_e G e
X ( g e ) = g X ( e ) X(ge) = gX(e) X ( g e ) = g X ( e ) g ∈ G g \in G g ∈ G e ∈ E ( T ) e \in\mathcal{E}(T) e ∈ E ( T )
if h , h ∗ \mathfrak{h},\mathfrak{h}^* h , h ∗ e e e X ( e ) X(e) X ( e ) X X X f o − 1 ( h ) f_o^{-1}(\mathfrak{h}) f o − 1 ( h ) f o − 1 ( h ∗ ) f_o^{-1}(\mathfrak{h}^*) f o − 1 ( h ∗ )
for every r ≥ 0 r \ge0 r ≥ 0 N r ( f o − 1 ( h ) ) ∩ N r ( f o − 1 ( h ∗ ) ) ⊆ N r ( X ( e ) ) N_r(f_o^{-1}(\mathfrak{h})) \cap N_r(f_o^{-1}(\mathfrak{h}^*)) \subseteq N_r(X(e)) N r ( f o − 1 ( h )) ∩ N r ( f o − 1 ( h ∗ )) ⊆ N r ( X ( e )) X X X
each edge e ⊆ X e \subseteq X e ⊆ X X ( e ) X(e) X ( e ) e ∈ E ( T ) e \in\mathcal{E}(T) e ∈ E ( T )
If h \mathfrak{h} h e e e X ( h ) ⊆ X X(\mathfrak{h}) \subseteq X X ( h ) ⊆ X f o − 1 ( h ) ∖ X ( e ) ⊆ G f_o^{-1}(\mathfrak{h}) \setminus X(e) \subseteq G f o − 1 ( h ) ∖ X ( e ) ⊆ G X ∖ X ( e ) X \setminus X(e) X ∖ X ( e ) G e G_e G e X X X
X = X ( h ) ⊔ X ( e ) ⊔ X ( h ∗ ) . X = X(\mathfrak{h}) \sqcup X(e) \sqcup X(\mathfrak{h}^*). X = X ( h ) ⊔ X ( e ) ⊔ X ( h ∗ ) . Proof of Key Lemma
We now proceed to prove the key lemma. The version of the lemma for the direct limit topology is [24 Lemma 3.5] . In order to transfer it to the Cashen–Mackay topology, we impose the following additional quasi-convexity condition on the edge groups (see the third condition in Theorem A.5 below).
∀ e ∈ E ( T ) , X ( e ) in Lemma A.3 is quasi-convex in X . (2) \forall e \in\mathcal{E}(T),\ X(e)\text{ in Lemma A.3 is quasi-convex in }X. \tag*{(2)} ∀ e ∈ E ( T ) , X ( e ) in Lemma A.3 is quasi-convex in X . (2) Here, we refer to the usual definition of quasi-convexity of metric spaces:
Definition A.4. A subset Y Y Y X X X quasi-convex if there exists a constant D D D Y Y Y D D D N D ( Y ) N_D(Y) N D ( Y ) Y Y Y
It is important to note that the quasi-convexity condition we impose does depend on the choice of the generating set of G G G X X X G G G
The lemma considers the following (well-defined) subset of ∂ ∗ G \partial_*G ∂ ∗ G h ∈ H ( T ) \mathfrak{h} \in\mathcal{H}(T) h ∈ H ( T )
M ( h ) = { [ γ ] ∈ ∂ ∗ G ∣ γ is eventually contained in X ( h ) } . M(\mathfrak{h}) = \{[\gamma] \in\partial_*G \mid\gamma\text{ is eventually contained in }X(\mathfrak{h})\}. M ( h ) = {[ γ ] ∈ ∂ ∗ G ∣ γ is eventually contained in X ( h )} . Lemma A.5 (key lemma for the Cashen–Mackay topology). Let G G G X X X ∂ ∗ G \partial_*G ∂ ∗ G G G G T T T e ∈ E ( T ) e \in\mathcal{E}(T) e ∈ E ( T )
the edge group G e G_e G e G G G
the relative Morse boundary ( ∂ ∗ G e , G ) (\partial_*G_e,G) ( ∂ ∗ G e , G )
the graph X ( e ) X(e) X ( e ) Lemma A.3 is quasi-convex in X X X
Then for every h ∈ H ( T ) \mathfrak h \in\mathcal{H}(T) h ∈ H ( T ) ∂ M G = M ( h ) ⊔ M ( h ∗ ) \partial_M G = M(\mathfrak h) \sqcup M(\mathfrak h^*) ∂ M G = M ( h ) ⊔ M ( h ∗ )
Proof. Let h ∈ H ( T ) \mathfrak h \in\mathcal{H}(T) h ∈ H ( T ) e ∈ E ( T ) e \in\mathcal{E}(T) e ∈ E ( T ) p ∈ X ( h ) p \in X(\mathfrak h) p ∈ X ( h ) [24 ] that M ( h ) M(\mathfrak h) M ( h ) M ( h ∗ ) M(\mathfrak h^*) M ( h ∗ ) ∂ ∗ G \partial_*G ∂ ∗ G M ( h ) M(\mathfrak h) M ( h ) ( [ γ i ] ) i ([\gamma_i])_i ([ γ i ] ) i γ i \gamma_i γ i γ i ( 0 ) = p \gamma_i(0)=p γ i ( 0 ) = p i i i
lim i → ∞ ( [ γ i ] ) i = [ γ ] \lim_{i \to\infty}([\gamma_i])_i = [\gamma] i → ∞ lim ([ γ i ] ) i = [ γ ] for some Morse geodesic ray γ \gamma γ γ ( 0 ) = p \gamma(0)=p γ ( 0 ) = p [ γ i ] ∈ M ( h ) [\gamma_i] \in M(\mathfrak h) [ γ i ] ∈ M ( h ) i ∈ N i \in\mathbb{N} i ∈ N γ i \gamma_i γ i X ( h ) X(\mathfrak h) X ( h ) [ γ ] ∈ M ( h ) [\gamma] \in M(\mathfrak h) [ γ ] ∈ M ( h )
For each geodesic γ i \gamma_i γ i γ i ∩ X ( h ∗ ) \gamma_i \cap X(\mathfrak h^*) γ i ∩ X ( h ∗ ) γ i ( 0 ) = p ∈ X ( h ) \gamma_i(0)=p \in X(\mathfrak h) γ i ( 0 ) = p ∈ X ( h ) γ i ∈ M ( h ) \gamma_i \in M(\mathfrak h) γ i ∈ M ( h ) X ( h ) X(\mathfrak h) X ( h ) X ( h ∗ ) X(\mathfrak h^*) X ( h ∗ ) X ( e ) X(e) X ( e ) γ i ∩ X ( h ∗ ) \gamma_i \cap X(\mathfrak h^*) γ i ∩ X ( h ∗ ) X ( e ) X(e) X ( e ) X ( e ) X(e) X ( e ) D > 0 D>0 D > 0
γ i ⊆ X ( h ) ∪ N D ( X ( e ) ) for all i ∈ N . \gamma_i \subseteq X(\mathfrak h) \cup N_D(X(e)) \qquad\text{for all } i \in\mathbb{N}. γ i ⊆ X ( h ) ∪ N D ( X ( e )) for all i ∈ N . As [ γ i ] [\gamma_i] [ γ i ] [ γ ] [\gamma] [ γ ] R > 0 R>0 R > 0 N R > 0 N_R>0 N R > 0 i ≥ N R i \geq N_R i ≥ N R
d ( γ i , γ ( R , ∞ ) ) ≤ κ ( ρ [ γ ] , 1 , 0 ) . d(\gamma_i,\gamma(R,\infty)) \leq\kappa(\rho_{[\gamma]},1,0). d ( γ i , γ ( R , ∞ )) ≤ κ ( ρ [ γ ] , 1 , 0 ) . Note that κ ( ρ [ γ ] , 1 , 0 ) \kappa(\rho_{[\gamma]},1,0) κ ( ρ [ γ ] , 1 , 0 ) i ≥ N R i \geq N_R i ≥ N R t R i > R t_R^i>R t R i > R
γ ( t R i ) ∈ X ( h ) ∪ N D ′ ( X ( e ) ) where D ′ = D + κ ( ρ [ γ ] , 1 , 0 ) . (3) \gamma(t_R^i) \in X(\mathfrak h) \cup N_{D'}(X(e))
\qquad\text{where } D'=D+\kappa(\rho_{[\gamma]},1,0). \tag*{(3)} γ ( t R i ) ∈ X ( h ) ∪ N D ′ ( X ( e )) where D ′ = D + κ ( ρ [ γ ] , 1 , 0 ) . (3) We will show that there exists R 0 > 0 R_0>0 R 0 > 0 γ ( t R i ) ∈ X ( h ) \gamma(t_R^i) \in X(\mathfrak h) γ ( t R i ) ∈ X ( h ) R ≥ R 0 R \geq R_0 R ≥ R 0 i ≥ N R i \geq N_R i ≥ N R ( s j ) j (s_j)_j ( s j ) j ( t R 0 + i N R 0 + i ) i ∈ N \left(t_{R_0+i}^{N_{R_0+i}}\right)_{i\in\mathbb{N}} ( t R 0 + i N R 0 + i ) i ∈ N γ ( s j ) ∈ X ( h ) \gamma(s_j) \in X(\mathfrak h) γ ( s j ) ∈ X ( h ) γ \gamma γ X ( h ∗ ) X(\mathfrak h^*) X ( h ∗ ) ∂ ∗ G = M ( h ) ⊔ M ( h ∗ ) \partial_*G=M(\mathfrak h)\sqcup M(\mathfrak h^*) ∂ ∗ G = M ( h ) ⊔ M ( h ∗ ) [ γ ] ∈ M ( h ) [\gamma]\in M(\mathfrak h) [ γ ] ∈ M ( h )
It remains to show that there exists R 0 > 0 R_0>0 R 0 > 0
γ ( t R i ) ∈ X ( h ) for all R ≥ R 0 and i ≥ N R . \gamma(t_R^i) \in X(\mathfrak h)
\qquad\text{for all } R \geq R_0 \text{ and } i \geq N_R. γ ( t R i ) ∈ X ( h ) for all R ≥ R 0 and i ≥ N R . Suppose not, then there exists an increasing sequence ( t i ) i (t_i)_i ( t i ) i γ ( t i ) ∉ X ( h ) \gamma(t_i) \notin X(\mathfrak{h}) γ ( t i ) ∈ / X ( h ) γ ( t i ) ∈ N D ( X ( e ) ) \gamma(t_i) \in\mathcal{N}_{D}(X(\mathfrak{e})) γ ( t i ) ∈ N D ( X ( e )) x x x y y y X ( e ) X(\mathfrak{e}) X ( e )
d ( x , γ ( t i ) ) ≤ D ′ and ( y , γ ( t i ) ) ≤ D ′ . d(x,\gamma(t_i)) \le D' \qquad\text{and} \qquad(y,\gamma(t_i)) \le D'. d ( x , γ ( t i )) ≤ D ′ and ( y , γ ( t i )) ≤ D ′ . As in Lemma 3.3 in [24 ] , one argues that there is a quasi-geodesic α \alpha α x x x y y y X ( e ) X(\mathfrak{e}) X ( e ) X X X [24 Lemma 2.1.(1)] , the Hausdorff distance of α \alpha α γ ( [ t i , t i + 1 ] ) \gamma([t_i,t_i+1]) γ ([ t i , t i + 1 ]) M M M γ \gamma γ α \alpha α D ′ D' D ′ γ \gamma γ X ( e ) X(\mathfrak{e}) X ( e ) X ( e ) X(\mathfrak{e}) X ( e )
Remark A.6 (Direct limit versus Cashen–Mackay topology). If one works with the direct limit topology, it is not necessary to assume quasi-convexity of X ( e ) X(\mathfrak{e}) X ( e ) [ γ i ] [\gamma_i] [ γ i ] γ i \gamma_i γ i X ( e ) X(\mathfrak{e}) X ( e ) γ i ∩ X ( h ∗ ) \gamma_i \cap X(\mathfrak{h}^{*}) γ i ∩ X ( h ∗ )
Remark A.7 (Quasi-convexity condition for graph of spaces). In the proof above, we work with the setup and notation in [24 ] , to make the difference of Theorem A.5 and [24 Lemma 3.5] clear. However, it is possible to transfer § 3 in [24 ] to a setting where G G G X X X G \mathcal{G} G [24 ] holds for the Cashen–Mackay topology under the additional assumption that the edge spaces are quasi-convex within the total space X X X X X X
Cashen–Mackay Morse hierarchies
In this subsection, we explain how to transfer § 2 , § 3.1 and Section 3.2 to the Cashen–Mackay topology, and show that almost all Morse hierarchies in § 6 are Cashen–Mackay Morse hierarchies.
Using Theorem A.5, we conclude the following variant of [24 Theorem A, Corollary B] for the Cashen–Mackay topology (see Theorem 2.15) by adding the quasi-convexity condition 2 as condition (c).
Theorem A.8. Let G G G X X X ∂ ∗ G \partial_*G ∂ ∗ G G G G T T T e ∈ E ( T ) e \in E(T) e ∈ E ( T )
(a) the edge group G e G_e G e G G G
(b) the relative Morse boundary ( ∂ ∗ G e , G ) (\partial_*G_e,G) ( ∂ ∗ G e , G )
(c) the graph X ( e ) X(e) X ( e ) Lemma A.3 is quasi-convex in X X X
Let x x x y y y T T T V x V_x V x V y V_y V y ( ∂ ∗ V x , G ) ∩ ( ∂ ∗ V y , G ) = ∅ (\partial_*V_x,G) \cap(\partial_*V_y,G) = \varnothing ( ∂ ∗ V x , G ) ∩ ( ∂ ∗ V y , G ) = ∅ C ⊆ ∂ ∗ G C \subseteq\partial_*G C ⊆ ∂ ∗ G V V V C C C ( ∂ ∗ V , G ) (\partial_*V,G) ( ∂ ∗ V , G )
Proof of Theorem A.8. Follow section 3 of [24 ] , replacing Lemma 3.5 with Theorem A.5. □
Corollary A.9. Under the assumptions of Theorem A.8 , suppose additionally that ( ∂ ∗ V , G ) (\partial_*V,G) ( ∂ ∗ V , G ) V ≤ G V \leq G V ≤ G ∂ ∗ G \partial_*G ∂ ∗ G
The remaining statements in § 2 can be adjusted in the same manner as Theorem A.8 , and lead to the following definition of Cashen–Mackay Morse hierarchies via adding the quasi-convexity condition 2 to Theorem 3.3:
Definition A.11 (Cashen–Mackay Morse hierarchies). Let G G G ∂ ∗ G \partial_*G ∂ ∗ G H \mathcal{H} H G G G weak Cashen–Mackay Morse if it is finite and for each non-leaf vertex v ∈ V ( H ) v \in\mathcal{V}(\mathcal{H}) v ∈ V ( H ) G v G_v G v X v X_v X v T T T e ∈ E ( T ) e \in\mathcal{E}(T) e ∈ E ( T )
(a) the edge group G e G_e G e G v G_v G v
(b) the relative Morse boundary ( ∂ ∗ G e , G ) (\partial_*G_e,G) ( ∂ ∗ G e , G )
(c) the graph X ( e ) X(e) X ( e ) Lemma A.3 is quasi-convex in X v X_v X v
A weak Cashen–Mackay Morse hierarchy is called a Cashen–Mackay Morse hierarchy if for each leaf v v v H \mathcal{H} H ( ∂ ∗ G v , G ) (\partial_*G_v,G) ( ∂ ∗ G v , G ) ∂ ∗ G \partial_*G ∂ ∗ G
Using the transferred results of § 2 for the Cashen–Mackay topology, the statements and proofs in § 3.1 and Section 3.2 transfer to the Cashen–Mackay topology by replacing [24 Lem 2.3] (see Theorem 2.9) with [16 Cor 6.2] (see Theorem A.2 ). We obtain:
Theorem A.12 (Theorem A for Cashen–Mackay Morse hierarchies). Let G G G ∂ ∗ G \partial_*G ∂ ∗ G G G G H \mathcal{H} H
L \mathcal{L} L L L L H \mathcal{H} H ( ∂ ∗ L , G ) ≠ ∅ (\partial_*L,G) \ne\varnothing ( ∂ ∗ L , G ) = ∅
C \mathcal{C} C ∂ ∗ G \partial_*G ∂ ∗ G
Moreover, every infinite connected component C C C ∂ ∗ G \partial_*G ∂ ∗ G
C = ( ∂ ∗ stab ( C ) , G ) . C=(\partial_*\operatorname{stab}(C),G). C = ( ∂ ∗ stab ( C ) , G ) . Corollary A.13 (Theorem B for Cashen–Mackay Morse hierarchies). Let G G G H \mathcal{H} H G G G L L L H \mathcal{H} H ( ∂ ∗ L , G ) = ∅ (\partial_*L,G)=\varnothing ( ∂ ∗ L , G ) = ∅
Remark A.14. There are two main advantages when working with the Cashen–Mackay topology rather than the direct-limit topology. Firstly, the metric is metrizable. Secondly, each infinite connected component of the Morse boundary topologically embed into the Morse boundary of its stabilizer [16 Thm] (see Theorem A.2 ). An inconvenience as in Theorem 6.11 does not occur in the Cashen-Mackay topology. On the other hand, imposing the quasi-convexity condition makes all statements more technical, and adds the need of working with a specific model space rather than only with the group itself.
We finish this section by proving that almost all Morse hierarchies constructed in § 6 are Cashen–Mackay Morse hierarchies.
Corollary A.15. If G G G G G G
G G G
G G G
G G G
G G G
G G G
G G G
Proof. Each of the listed groups has a Morse hierarchy descried in § 6.1 . In order to show that these are also Cashen–Mackay Morse hierarchies, it suffices to show that all the listed examples satisfy the additional quasi-convexity condition 2. This is clearly true for (1) and (2). For (6), it follows from [20 Proposition 4.8.1] . For the remaining cases (3), (4), and (5), the groups that occur in the Morse hierarchies we construct in § 6.1 are fundamental groups of total spaces of graph of spaces. By Theorem A.7, it is enough to show that the edge spaces of these total spaces are quasi-convex. For (3), the edge spaces are convex by construction. The hierarchies in (6) can be view as a combination of (3) and (2), and hence satisfy the quasi-convexity condition we need. Finally, for (4), the quasi-convexity condition is given by [33 Lem 2.1] .
References
Maxwell Institute for Mathematical Sciences and Department of Mathematics, Heriot-Watt University, Edinburgh, UK Email address: m.cordes@hw.ac.uk
Institute of Mathematics, University of Wroclaw, Wroclaw, Poland Email address: annette.karrer@uwr.edu.pl
Department of Mathematics, Tufts University, Medford, MA 02155, USA Email address: kim.ruane@tufts.edu
