Every countable group embeds in a group of type $\mathrm{FP}_n$

Introduction

A group is of type $FP_n$ if the trivial module $\mathbb{Z}$ has a projective resolution over the integral group ring that is finitely generated in degrees at most $n$, and of type $F_n$ if it has an Eilenberg–Mac Lane space with finite $n$-skeleton. Higman [7] proved that a finitely generated group embeds in a finitely presented group if and only if it is recursively presented, and Leary [9 Theorem 1.1] proved that every countable group embeds in a group of type $FP_2$. We extend both embedding results to arbitrary degree $\ge2$:

Misha Gromov suggested, after a talk of the first-named author at the 2015 meeting “Growth, symbolic dynamics and combinatorics of words in groups” in Paris, that perhaps the only obstruction to embedding theorems lies in dimension 2. Corollary C gives a precise finite-degree version of this suggestion: passing from finite presentability to any fixed higher finiteness degree introduces no additional embedding obstruction.

Bestvina’s problem list [1] asks, in Question 8.7, whether the Prüfer $p$-group $\mathbb{Z}_{p^\infty}$ can be embedded in an $FP_\infty$ group, or at least an $FP_3$ group; and whether every $F_n$ group can be embedded in an $F_{n+1}$ group, if $n \ge2$. Theorem A settles the second question, while Theorem B settles the third one, both in the affirmative.

Theorem B also settles the $F_3$ part of [6 Question 1.4], and Theorem A the $FP_3$ part of [6 Question 1.4]; both hold in every degree. The target groups in both theorems can be chosen to have two generators; see Corollary 7.7.

After this manuscript was completed, OpenAI proved that every finitely presented group embeds in a group of type $F_\infty$ [11], which settles the first part of Bestvina’s question. Combining its controlled-filling argument with our construction gives the analogous statement for arbitrary countable groups and type $\mathrm{FP}_{\infty}$; see Appendix § 8:

Theorem D. Every countable group embeds in a group of type $\mathrm{FP}_{\infty}$.

The degree-raising criterion

The theorems of Higman and Leary are both proved with the Higman rope trick [10 IV.7.6], whose homological version is [9 Lemma 2.2]. Fournier-Facio and Zaremsky showed that the groups produced by the rope trick have infinite-dimensional third rational homology whenever the embedded group $F/R$ is infinite and $R \ne1$ [6 Theorem B]; in particular such groups are not of type $\mathrm{FP}_3$. We raise the finiteness degree with ascending HNN extensions instead.

Let $m \ge1$, let $B$ be a group of type $\mathrm{FP}_m$, and fix an exact sequence of $\mathbb{Z}B$-modules

in which every $F_i$ is finitely generated and free. Then $B$ is of type $\mathrm{FP}_{m+1}$ if and only if $Z$ is finitely generated [5 VIII.4.3]. Every endomorphism $\varphi$ of $B$ has chain lifts $\Phi$ to (1) (Definition 2.1), so each of them maps $Z$ into itself.

The hypothesis does not depend on the choice of the chain lift or of the finite free partial resolution; see Lemma 2.2 and Remark 2.3. It requires neither that $\Phi(Z)$ be finitely generated nor that $M$ be $\Phi$-invariant.

Outline

A fixed finite system of commuting-copy and diagonal relations produces an injective endomorphism whose action on the top cycle module has image contained in a finitely generated submodule. Theorem E then raises the finiteness degree by one. We describe the three ingredients of this construction. An oriented triple system $\mathcal{T}$ on a finite set $V$ is a set of triples $(i,j) \to k$ of distinct elements of $V$, with inputs $i,j$ and output $k$ (Definition 3.1). For a group $G$, let $G_v$ ($v \in V$) be copies of $G$, write $g_v \in G_v$ for the copy of $g \in G$, and put

with canonical homomorphisms $\iota_v \colon G \to K_{\mathcal{T}}(G)$, $g \mapsto g_v$. Thus $K_{\mathcal{T}}(G)$ is generated by the copies $G_v$ subject to the following relations, one family for each triple $(i,j) \to k$: the input copies $G_i$ and $G_j$ commute elementwise, and $g_k = g_i g_j$ for every $g \in G$, so that the output copy is identified with the diagonal in the input copies. For a single triple, $K_{\mathcal{T}}(G) = G \times G$, and $G_i$, $G_j$, $G_k$ are the two factors and the diagonal.

Suppose that $\kappa\colon K_{\mathcal{T}}(B) \to B$ is an embedding, and put $\alpha_v = \kappa\circ\iota_v$. For each triple $(i,j) \to k$, the endomorphisms $\alpha_i$ and $\alpha_j$ have commuting images and $\alpha_k(b) = \alpha_i(b)\alpha_j(b)$ for all $b \in B$. For such endomorphisms, chain lifts satisfy $\Phi_k = \Phi_i + \Phi_j$ on $\mathbb{Z}$ modulo a finitely generated submodule (Lemma 5.1). Consequently, if the standard basis vector $e_0$ of $\mathbb{Z}^V$ is an integral linear combination of the vectors $e_i + e_j - e_k$, the same combination of the maps $\Phi_v$ shows that $\Phi_0$ maps $Z$ into a finitely generated submodule (Corollary 5.3). Theorem E then applies to $\alpha_0$, provided $\alpha_0$ is injective. We show that every $\iota_v$ is injective, for every group $G$, when the incidence graph of $\mathcal{T}$ has girth at least $12$, by realising $K_{\mathcal{T}}(G)$ as the fundamental group of a non-positively curved complex of groups (Lemma 4.1). A single system $\mathcal{T}$ satisfying both conditions serves for every group and every degree; it has 15 triples and is obtained from the flags of the Fano plane (Proposition 3.2). Finally, an embedding $K_{\mathcal{T}}(B) \to B$ exists whenever $B$ is itself finitely generated and recursively presented relative to a fixed oracle, and contains every finitely generated group that is recursively presented relative to that oracle. Both properties survive the ascending HNN extension. A two-generator universal group supplies the initial case $m=1$ (Lemma 7.2). In particular, for $n=2$ the argument gives a new proof of Leary’s theorem that does not use the construction of [9]; he had asked for a simpler proof of his Theorem 1.1 in [9 Section 3].

Mitotic groups

The relations associated to a single triple in (1.2) are the factor-diagonal relations underlying a mitosis in the sense of Baumslag, Dyer and Heller [3 Section 4]. A supergroup $M$ of a group $H$ is a mitosis of $H$ if it is generated by $H$ and two elements $s,d$ such that $[h,s^{-1}hs]=1$ and $d^{-1}hd=h\cdot s^{-1}h's$ for all $h,h'\in H$. The present construction replaces the conjugations by $s$ and $d$ by a finite network of endomorphisms satisfying the girth and integrality conditions above. Additivity is used on the top cycle module of a partial resolution (Section 5), to raise a finiteness property rather than to annihilate ordinary homology.

Relation to the work of Fournier-Facio and Zaremsky

Fournier-Facio and Zaremsky proved that if every finitely generated recursively presented group embeds in a recursively presented group of type $\mathrm{FP}_n$, then every finitely presented group embeds in a group of type $\mathrm{F}_n$ [6 Theorem A]. Theorem 7.3 with the empty oracle supplies this hypothesis, so Theorem B can also be deduced from their result. We give an equally short direct proof in Subsection 7.1.

Statement on AI use

ChatGPT 6 Astra was used to assist with the proof of the main results and the preparation of an initial draft of the manuscript. Claude Opus 5.5 was used in subsequent editing. The arguments and text have been reviewed, checked, and substantially revised by the authors, who are responsible for the final manuscript.

Formalization

Theorem A has been formalized in Lean/Mathlib and is available on the Palomar repository as

https://palomar-registry.org/entry?id=PALOMAR-2026-09-30-000032.

Preliminaries

Conventions

Modules are left modules, and finite generation of a module refers to the indicated group ring. All finiteness properties are taken with integer coefficients; see [5 Chapter VIII]. A group of type $\mathrm{FP}_1$ is finitely generated. Presentations are written $\langle X:R\rangle$. For an injective endomorphism $\varphi$ of a group $B$, the ascending HNN extension $\langle B,t:t^{-1}bt=\varphi(b),\ b\in B\rangle$ contains $B$ by Britton’s lemma [10 IV.2]; its height is the homomorphism to $\mathbb{Z}$ sending $t$ to $1$ and $B$ to $0$.

Let $\theta:\Gamma\to B$ be a homomorphism of groups, extended to group rings. An additive map $f:P\to P'$ from a $\mathbb{Z}\Gamma$-module to a $\mathbb{Z}B$-module is $\theta$-semilinear if $f(\lambda x)=\theta(\lambda)f(x)$ for all $\lambda\in\mathbb{Z}\Gamma$ and $x\in P$. We shall use repeatedly the following consequence of the definition:

Chain lifts

Throughout Sections 2–6, $m\geq1$, $B$ is a group of type $\mathrm{FP}_m$, the sequence (1.1) is fixed, and $\mathbb{Z}=\ker d_m$. We write $F_*$ for the augmented complex $F_m\to\cdots\to F_0$. It satisfies

Note that $H_m(F_*)$ is the module $\mathbb{Z}$, not the group homology $H_m(B;\mathbb{Z})$.

A chain lift of $\theta$ is the same as a $\mathbb{Z}B$-linear chain map, over the identity of $\mathbb{Z}$, from $F_*$ to the complex obtained from $F_*$ by restricting scalars along $\theta$. Since the $F_i$ are free and the augmented complex is exact at $F_0,\ldots,F_{m-1}$, such a map is constructed degree by degree on bases, as in the comparison theorem [5 I.7]. In particular, chain lifts exist. They are unique up to homotopy only below the top degree; the following lemma is the substitute that we need.

Proof. Since $\varepsilon(f_0-g_0)=0$ and $\ker\varepsilon=d_1(F_1)$, choose a $\theta$-semilinear map $\sigma_0:F_0\to F_1$ with $f_0-g_0=d_1\sigma_0$. Inductively, for $1\leq i<m$, construct $\theta$-semilinear maps $\sigma_i:F_i\to F_{i+1}$ satisfying

Indeed, once the preceding maps have been constructed, the chain-map identities give

For $i=1$ the last equality follows from the choice of $\sigma_0$, and for $i>1$ from $d_{i-1}d_i=0$. Exactness at $F_i$ lets us lift the values through $d_{i+1}$ and extend $\theta$-semilinearly. Now $c=f_m-g_m-\sigma_{m-1}d_m$ satisfies $d_mc=0$, so $c(F_m)\subseteq Z$, and $c$ agrees with $f_m-g_m$ on $Z$. Since $c$ is $\theta$-semilinear and $F_m$ is finitely generated, $c(F_m)$ lies in a finitely generated submodule of $Z$ by (3). □

Remark 2.3. The hypothesis of Theorem E is also independent of the chosen finite free partial resolution. Let $F^*_*$ be another such resolution through degree $m$, put $Z'=\ker d'_m$, and choose comparison maps $u:F_*\to F'_*$ and $v:F'_*\to F_*$ lifting $\operatorname{id}_Z$. If $\Phi$ is a chain lift of $\varphi$ with $\Phi(Z)\subseteq M$, then $u\Phi v$ is a chain lift of $\varphi$ on $F'_*$ and maps $Z'$ into the finitely generated submodule $u_m(M)\subseteq Z'$. Lemma 2.2, applied to $F'_*$, gives the same conclusion for every chain lift on $F'_*$ . Interchanging the two resolutions proves independence.

Presentations relative to an oracle

Fix $O\subseteq\mathbb{N}$. All alphabets below are finite or countable, with a fixed effective enumeration, so that finite words have a computable encoding by natural numbers. A set of words is $O$-recursively enumerable ($O$-r.e.) if some Turing machine with oracle $O$ enumerates it. A presentation $\langle X:R\rangle$ is $O$-r.e. if $R$ is, and a countable group is $O$-recursively presented if it admits such a presentation. For $O=\varnothing$ these are the countable recursively presented groups.

A finitely generated $O$-recursively presented group has an $O$-r.e. presentation on any finite generating set, since the relations among finitely many words in the generators of an $O$-r.e. presentation form an $O$-r.e. set. Every countable group has an $O$-r.e. presentation for a suitable $O$: choose a countable presentation and let $O$ encode its set of relators.

Adding finitely many generators and relators to an $O$-r.e. presentation yields an $O$-r.e. presentation. In particular, if $H$ is finitely generated and $O$-recursively presented and $\varphi$ is an injective endomorphism of $H$, then $\langle H,t:t^{-1}ht=\varphi(h),\ h\in H\rangle$ is $O$-recursively presented: it suffices to impose $t^{-1}w_x t=w_x$ for $x$ in a finite generating set of $H$, where $w_x$ is any word representing $\varphi(x)$.

A finite system of triples

We shall use a system $\mathcal{T}$ with a distinguished element $0 \in V$ satisfying

Condition (3.2) means that for every abelian group $A$ and every family $(x_v)_{v\in V}$ in $A$,

Condition (3.1) is used only in Lemma 4.1, and condition (3.2) only in Corollary 5.3.

Proof. Let the points of the Fano plane be the elements of $\mathbb{Z}/7$ and its lines the sets $L_j=\{j,j+1,j+3\}$, $j\in\mathbb{Z}/7$. Its flags are the pairs $f_{j,d}=(j+d,L_j)$ with $j\in\mathbb{Z}/7$ and $d\in{0,1,3}; the flags through the point $p$ are $f_{p,0}$, $f_{p-1,1}$ and $f_{p-3,3}$. Let $V$ consist of the twenty flags other than $f_{0,3}=(3,L_0)$ and of three further indices $a$, $b$ and $0$, and let $\mathcal{T}$ consist of the fifteen triples

Thus $\pi_p$ consists of the flags through $p$ and $\lambda_j$ of the flags on $L_j$, except that the flag $(3,L_0)$ is replaced by $b$ in $\pi_3$ and by $a$ in $\lambda_0$.

Put $\nu=1$ on the triples $\pi_p$ and $\gamma$, and $\nu=-1$ on the triples $\lambda_j$. In $\nu W=\sum_p W_{\pi_p}-\sum_j W_{\lambda_j}+W_\gamma$, a flag $f_{j,0}$ or $f_{j,1}$ is an input of one triple $\pi_p$ and of one triple $\lambda_j$, and contributes $1-1=0$; a flag $f_{j,3}$ with $j\ne0$ is the output of one $\pi_p$ and one $\lambda_j$, and contributes $-1+1=0$; the index $a$ contributes $1$ from $\lambda_0$ and $-1$ from $\gamma$, the index $b$ contributes $-1$ from $\pi_3$ and $1$ from $\gamma$, and the index $0$ contributes $1$ from $\gamma$. Hence $\nu W=e_0$, which is (3.2).

Every index other than $0$ lies in exactly two triples, and $0$ lies in $\gamma$ only. Let $H$ be the graph with vertex set $\mathcal{T}$ and, for each index other than $0$, an edge joining the two triples that contain it. Then $\operatorname{girth}(\mathcal{T})$ is obtained from $H$ by subdividing every edge and attaching the pendant vertex $0$ to $\gamma$, so its girth is twice that of $H$. The graph $H$ is the incidence graph of the Fano plane, with $\pi_p$ corresponding to $p$ and $\lambda_j$ to $L_j$, in which the edge joining $3$ to $L_0$ is subdivided by the vertex $\gamma$; see Figure 1. The incidence graph of a projective plane is bipartite and contains no cycle of length 4, since two points lie on a unique line; hence it has girth at least 6, and subdividing an edge does not shorten any cycle. Therefore $\operatorname{girth}(\mathcal{T})$ has girth at least 12, which is (3.1). □

Remark 3.3. The construction is an instance of the following principle. Let $H$ be a finite graph in which every vertex has degree 3, one of the edges being a half-edge $0$ with a single endpoint, and orient $H$ so that no vertex is a source or a sink and the half-edge $0$ is directed outwards. Taking the triples to be the sets of edges at the vertices, declare the output at each vertex to be the edge whose orientation there differs from that of the other two, and put $\nu=1$ or $\nu=-1$ at vertices with one or two incoming edges respectively. Then the matrix with rows $\nu_\ell W_\ell$ is the oriented incidence matrix of $H$, with entry 1 at the tail and $-1$ at the head of each edge, so $\nu W$ vanishes on every edge with two endpoints and equals 1 on the half-edge; this is (3.2). Dually, a solution of the equations $x_k=x_i+x_j$ is a conserved flow on $H$, and conservation forces the flow $x_0$ through the half-edge to vanish. Condition (3.1) holds as soon as $H$ has girth at least 6. In the proof above, $H$ is oriented from points to lines along the flags $f_{j,0}$, $f_{j,1}$ and from lines to points along the flags $f_{j,3}$, the subdivided flag $(3,L_0)$ runs from $L_0$ through $\gamma$ to $3$, and the half-edge $0$ leaves $\gamma$.

We fix the system $\mathcal{T}$ of Proposition 3.2 for the rest of the paper.

The complex-of-groups construction

Recall the group $K_{\mathcal{T}}(G)$ from (1.2) and its canonical maps $\iota_v$. For a single triple these maps are injective because $K_{\mathcal{T}}(G)=G\times G$. For a general triple system, the following condition on the incidence graph ensures injectivity.

Proof. Let $\mathcal{P}$ be the poset with elements $o$, the elements of $V$ and the elements of $\mathcal{T}$, ordered by $\ell<v<o$ whenever $v$ is an entry of $\ell$. The monomorphisms of the complex of groups defined below go from larger to smaller elements of $\mathcal{P}$, that is, along $o\to v\to\ell$ and $o\to\ell$. Its geometric realisation $|\mathcal{P}|$ is the cone with apex $o$ on $\Gamma(\mathcal{T})$, whose 2-simplices are the triangles $\{o,v,\ell\}$ with $v$ an entry of $\ell$. In particular, $|\mathcal{P}|$ is contractible.

Let $G(\mathcal{P})$ be the simple complex of groups over $\mathcal{P}$, in the sense of [4 II.12], with local groups $G_o=1$, $G_v=G$ for $v\in V$ and $G_\ell=G\times G$ for $\ell\in\mathcal{T}$, and with the following monomorphisms: those out of $G_o$ are trivial, and for $\ell=(i,j)\to k$,

The compatibility condition on composites is vacuous, because every chain of length two in $\mathcal{P}$ begins at $o$. The direct limit of $G(\mathcal{P})$ is generated by the groups $G_v$ and $G_\ell$ subject to the identifications (4.1). Since $G\times G$ is generated by $G\times1$ and $1\times G$ subject only to the relations that these subgroups commute, eliminating the groups $G_\ell$ yields the presentation (1.2). As $|\mathcal{P}|$ is simply connected, this direct limit is the fundamental group of $G(\mathcal{P})$. It therefore suffices to show that $G(\mathcal{P})$ is developable, since the local groups of a developable complex of groups embed in its fundamental group [4 II.12].

Make every triangle $\{o,v,\ell\}$ of $|\mathcal{P}|$ a Euclidean triangle with side lengths $|ov|=\sqrt{3}$, $|v\ell|=1$ and $|o\ell|=2$. Its angles are $\pi/6$ at $o$, $\pi/2$ at $v$ and $\pi/3$ at $\ell$, and edges of the same type have the same length, so this is a piecewise Euclidean structure with a single shape. By [4 II.12.28] (see also [4 III.C.4.17]), $G(\mathcal{P})$ is developable if it is non-positively curved, that is, if its local developments are locally CAT(0). For a 2-dimensional piecewise Euclidean complex this is the link condition: every injective loop in the link of a vertex has length at least $2\pi$ [4 II.5]. The links at vertices of the local developments are as follows.

At $o$ the local group is trivial and the link is $\Gamma(\mathcal{T})$. At $v$, put $\operatorname{Inc}(v)=\{\ell\in\mathcal{T}:v\text{ is an entry of }\ell\}$. The link is the simplicial join $G * \operatorname{Inc}(v)$ of two discrete sets: the set $G_v/G_o\cong G$ of directions towards lifts of $o$, and the set of triples containing $v$. Thus it is a complete bipartite graph.

At $\ell=(i,j)\to k$, let $H_1=G\times1$, $H_2=1\times G$ and let $H_3$ be the diagonal. The link is the bipartite graph with point vertices $z\in G\times G$ and tagged coset vertices $(a,zH_a)$, $a\in\{1,2,3\}$. Each point is joined to the three tagged cosets containing it. Distinct cosets of the same subgroup are disjoint. If $a\ne b$ and two cosets of $H_a$ and $H_b$ meet at $z$, their intersection is $z(H_a\cap H_b)=\{z\}$. Consequently, two distinct tagged cosets have at most one common point. This link has no cycle of length 4, so its girth is at least 6. In each case every injective loop has length at least $2\pi$, and the link condition holds.

Thus $G(\mathcal{P})$ is non-positively curved, hence developable, and every $\iota_v$ is injective. □

Proof. Functoriality is clear from (1.2). In the group given by the displayed presentation, the relators $[x_i,y_j]$ make the images of $G_i$ and $G_j$ commute, so $g\mapsto g_i g_j$ is a homomorphism on $G$. It agrees with $g\mapsto g_k$ on $X$, hence on $G$. So all relators in (1.2) follow from the displayed ones. □

Products of endomorphisms with commuting images

For group homology, the map induced by a pointwise product of homomorphisms with commuting images differs from the sum of the induced maps by mixed terms coming from the Künneth formula; Baumslag, Dyer and Heller control these terms by an induction on the degree [3 Proposition 4.1]. At the level of the partial resolution (1.1), which has no intermediate homology, the first possible mixed tensor term occurs in degree $2m>m$.

Proof. Let $Q_* = F_* \otimes_{\mathbb{Z}} F_*$ with differential $d(x\otimes y)=dx\otimes y+(-1)^{|x|}x\otimes dy$, and with $\mathbb{Z}[B\times B]=\mathbb{Z}B\otimes\mathbb{Z}B$ acting factorwise. Each $Q_n=\bigoplus_{p+q=n}F_p\otimes F_q$ is a finitely generated free $\mathbb{Z}[B\times B]$-module. Write $\varepsilon_Q=\varepsilon\otimes\varepsilon:Q_0\to\mathbb{Z}$ for its augmentation. Let $\rho_1=\operatorname{id}\otimes\varepsilon$ and $\rho_2=\varepsilon\otimes\operatorname{id}$, where $\varepsilon$ is regarded as a chain map from $F_*$ to $\mathbb{Z}$ concentrated in degree 0. These are chain maps $Q_*\to F_*$, semilinear with respect to the two projections $B\times B\to B$, and $\rho_1(x\otimes y)=0$ whenever $y$ has positive degree (similarly for $\rho_2$). The groups $F_p$ are free abelian and $H_0(F_*)\cong\mathbb{Z}$, so the Künneth formula has no Tor terms in degrees $\le m$, and gives $H_0(Q_*) \cong\mathbb{Z}$, $H_n(Q_*) = 0$ for $0 < n < m$, and an isomorphism

In particular:

The product. Since the images commute, $\eta(b,c) = \alpha(b)\beta(c)$ is a homomorphism $B \times B \to B$. Choose $h_0$ on a basis so that $\varepsilon h_0 = \varepsilon_Q$. Since the $Q_n$ are free and the augmented complex $F_* \to\mathbb{Z}$ is exact at $F_0,\ldots,F_{m-1}$, this extends degree by degree to $\eta$-semilinear maps $h_n: Q_n \to F_n$, $0 \le n \le m$, commuting with the differentials. Put

We have $d_m h_m d = h_{m-1} d d = 0$, so $M_0 \subseteq Z$. Since $h_m d$ is $\eta$-semilinear and $Q_{m+1} = \bigoplus_{p=1}^{m} F_p \otimes_{\mathbb{Z}} F_{m+1-p}$ is finitely generated, $M_0$ is finitely generated by (2.1). The chain maps $i_1(x) = x \otimes1$ and $i_2(x) = 1 \otimes x$ from $F_*$ to $Q_*$, where $1 \in F_0 = \mathbb{Z}B$, are semilinear with respect to $b \mapsto(b,1)$ and $b \mapsto(1,b)$. Hence $f = h i_1$ and $g = h i_2$ are chain lifts of $\alpha$ and $\beta$.

The diagonal. Let $Q_{\le m}^{\mathrm{brut}}$ denote the brutal truncation of $Q_*$: its terms in degrees $0,\ldots,m$ are unchanged and its terms in higher degrees are zero. Using $\varepsilon_Q$ in degree $0$ and exactness of the augmented complex at $Q_0,\ldots,Q_{m-1}$, construct a chain map $\Delta: F_* \to Q_{\le m}^{\mathrm{brut}}$ with $\varepsilon_Q\Delta_0 = \varepsilon$, semilinear with respect to the diagonal $B \to B \times B$. We continue to use the full complex $Q_*$, including $Q_{m+1}$, when applying (5.1). Then $h\Delta$ is a chain lift of $\delta$, and $\tau_a = \rho_a\Delta$ ($a = 1,2$) is a $\mathbb{Z}B$-linear chain lift of $\operatorname{id}_B$. By Lemma 2.2 there are finitely generated submodules $Y_1,Y_2 \subseteq Z$ with $(\tau_a-\operatorname{id})(Z) \subseteq Y_a$. For $z \in Z$, the element

is a cycle, and $\rho_1(w) = \tau_1(z) - \tau_1(z) - 0 = 0$ because $\rho_1 i_1 = \operatorname{id}$ and $\rho_1 i_2$ vanishes in positive degrees; likewise $\rho_2(w) = 0$. By (5.1), $w \in d(Q_{m+1})$, so $h_m(w) \in M_0$. Since $h i_1\tau_1(z) = f(z) + f(\tau_1-\operatorname{id})(z)$ and similarly for $g$, we obtain

and the right-hand side is a finitely generated submodule of $Z$ by (2.1). Arbitrary lifts. Applying Lemma 2.2 to the pairs $(\Phi_\delta,h\Delta)$, $(\Phi_\alpha,f)$ and $(\Phi_\beta,g)$, and adding the resulting finitely generated submodules to $M_0 + \mathbb{Z}Bf(Y_1) + \mathbb{Z}Bg(Y_2)$, gives $M$.

Remark 5.2. The proof gives a bound independent of $\alpha$ and $\beta$: if $F_p$ has rank $r_p$, then $M$ can be generated by $\sum_{p=1}^{m} r_p r_{m+1-p} + 5r_m$ elements.

Proof. Choose chain lifts $\Phi_v$ of $\alpha_v$ for $v \ne0$. For each $\ell= (i,j)\to k$, Lemma 5.1 gives a finitely generated submodule $M_\ell\subseteq Z$ containing $(\Phi_i+\Phi_j-\Phi_k)(Z)$; let $M = \sum_{\ell\in\mathcal{T}} M_\ell$. By (3.2), applied to the abelian group of additive maps $Z \to Z$,

Ascending HNN extensions: proof of Theorem E

Proof of Theorem E. By Britton’s lemma, $B$ embeds in $E$. Since $\mathbb{Z}E$ is a free right $\mathbb{Z}B$-module, the functor $\mathbb{Z}E \otimes_{\mathbb{Z}B} -$ is exact. Put $P_i=\mathbb{Z}E\otimes_{\mathbb{Z}B} F_i$ and $\mathbb{Z}_E=\mathbb{Z}E\otimes_{\mathbb{Z}B}\mathbb{Z}$. Then $P_m\to\cdots\to P_0\to\mathbb{Z}[E/B]\to0$ is exact, the $P_i$ are finitely generated free $\mathbb{Z}E$-modules, and $\mathbb{Z}_E$ is the kernel of $d_m:P_m\to P_{m-1}$.

Choose a chain lift $\Phi$ of $\phi$ satisfying $\Phi(\mathbb{Z})\subseteq M$ for a finitely generated $\mathbb{Z}B$-submodule $M\subseteq\mathbb{Z}$, and define $T_i:P_i\to P_i$ by

This is well defined: for $b\in B$ we have $bt=t\varphi(b)$ in $E$, so $abt\otimes\Phi_i(x)=at\varphi(b)\otimes\Phi_i(x)=at\otimes\Phi_i(bx)$. The maps $T_i$ are $\mathbb{Z}E$-linear, commute with the differentials, and induce $T(gB)=gtB$ on $\mathbb{Z}[E/B]=\mathbb{Z}E\otimes_{\mathbb{Z}B}\mathbb{Z}$ because $\varepsilon\Phi_0=\varepsilon$. Let $\mu_1,\ldots,\mu_r$ generate $M$. If $z\in\mathbb{Z}$ and $\Phi_m(z)=\sum_p\lambda_p\mu_p$ with $\lambda_p\in\mathbb{Z}B$, then $T_m(a\otimes z)=\sum_p at\lambda_p\otimes\mu_p$. Hence

The sequence

is exact. Indeed, the height is constant on each coset $gB$ and $T$ raises it by one, so the terms of minimal height of a nonzero element of $\mathbb{Z}[E/B]$ survive in its image under $1-T$. The cokernel of $1-T$ is the quotient of $\mathbb{Z}[E/B]$ by the relations $gB=gtB$; together with $gbB=gB$ for $b\in B$, these identify $gB$ with $gxB$ for every $x\in B\cup\{t^{\pm1}\}$, hence identify all cosets, since $B$ and $t$ generate $E$.

The idea is that, modulo $(1-T_m)\mathbb{Z}_E$, every $\zeta\in\mathbb{Z}_E$ is congruent to $T_m\zeta$, which lies in the finitely generated submodule (6.1). We make this precise with a mapping cone.

Let $C_*$ be the brutal truncation in degrees $\leq m$ of the mapping cone of $1-T$: $C_0=P_0$, $C_j=P_j\oplus P_{j-1}$ for $1\leq j\leq m$, with differentials

augmented by the composite $C_0=P_0\to\mathbb{Z}[E/B]\xrightarrow{\varepsilon}\mathbb{Z}$. Since $T$ commutes with the differentials, $D_{j-1}D_j=0$. Let $\xi_1,\ldots,\xi_s$ be a basis of $F_m$. We show that $C_*$ is exact in degrees $<m$ and that $\ker D_m$ is generated by the elements

Mapping a free $\mathbb{Z}E$-module of rank $r+s$ onto $\ker D_m$ then extends $C_*$ to a partial resolution of $\mathbb{Z}$ by finitely generated free $\mathbb{Z}E$-modules in degrees $0,\ldots,m+1$, so $E$ is of type $\mathrm{FP}_{m+1}$.

Degree 0. Let $x\in P_0$ have augmentation 0. By (6.2), its image in $\mathbb{Z}[E/B]$ is $(1-T)\bar y$ for the image $\bar y$ of some $y\in P_0$. Then $x-(1-T_0)y$ lies in the kernel of $P_0\to\mathbb{Z}[E/B]$, which is $d_1(P_1)$, so $x\in D_1(C_1)$.

Degrees $1\leq j\leq m$. Let $(x,y)\in C_j$ with $D_j(x,y)=0$. We first show that $y=d_jz$ for some $z\in P_j$. If $j=1$, mapping $D_1(x,y)$ to $\mathbb{Z}[E/B]$ gives $(1-T)\bar y=0$, so $\bar y=0$ by (6.2) and $y\in d_1(P_1)$. If $j\geq2$, then $d_{j-1}y=0$ and $y$ is a boundary because $H_{j-1}(P_*)=0$. Now $d_j(x+(1-T_j)z)=0$. If $j<m$, then $x+(1-T_j)z=d_{j+1}w$ for some $w$, and $(x,y)=D_{j+1}(w,-z)$. If $j=m$, then $\zeta=x+(1-T_m)z$ lies in $\mathbb{Z}_E$ and

The elements $((1-T_m)x',-d_mx')$ with $x'\in P_m$ form the image of a $\mathbb{Z}E$-linear map $P_m\to C_m$, which is generated by the second family in (6.3). For $\zeta\in\mathbb{Z}_E$ we have $d_m\zeta=0$, hence

where the first term lies in the span of the first family in (6.3) by (6.1), and the second in the span of the second family. □

Combining Theorem E with Corollary 5.3, we obtain:

Proofs of Theorem A, Theorem B and Corollary C

We fix an oracle $\mathcal{O} \subseteq\mathbb{N}$. The first lemma is the Higman–Neumann–Neumann embedding theorem [8] (see [10 IV.3]), in a form that keeps track of presentations.

Proof. Write $G = \langle x_1,x_2,\ldots: R\rangle$, repeating generators if necessary, and let $L = \langle a,b\rangle$ be free of rank two. In $G * L$ put

The $u_i$ freely generate a free subgroup of $L$. So do the $v_i$, since the retraction $G * L \to L$ killing $G$ maps them to the free basis $\{b^{-i}ab^i:i\ge0\}$ of a subgroup of $L$. Hence $G * L$, and so $G$, embeds in

The defining relations give $a=t^{-1}bt$ and $x_i=t^{-1}a^{-i}baitb^{-i}a^i$ for $i\ge1$. Eliminating $a$ and the $x_i$ presents $G^*$ on $b,t$ with relators obtained from $R$ by substituting these words, which is an $\mathcal{O}$-r.e. set. □

Proof. For $k \ge1$ and an oracle Turing machine $\mathcal{M}$, let $G_{k,\mathcal{M}}$ be the group generated by $x_1,\ldots,x_k$ subject to those outputs of $\mathcal{M}$ with oracle $\mathcal{O}$ that are words in $x_1^{\pm1},\ldots,x_k^{\pm1}$. Every finitely generated $\mathcal{O}$-recursively presented group is isomorphic to some $G_{k,\mathcal{M}}$. Using disjoint alphabets and dovetailing the enumerations, the free product $\Omega_{\mathcal{O}}$ of all these groups has an $\mathcal{O}$-r.e. presentation and contains each of them. By Lemma 7.1, $\Omega_{\mathcal{O}}$ embeds in a two-generator $\mathcal{O}$-recursively presented group $B_{1,\mathcal{O}}$. Finite generation is equivalent to type $\mathrm{FP}_1$. □

Proof. The group $B_{1,\mathcal{O}}$ is given by Lemma 7.2. Suppose that $B=B_{m,\mathcal{O}}$ has been constructed for some $m\ge1$. By Lemma 4.2, $K_{\mathcal{T}}(B)$ is finitely generated and $\mathcal{O}$-recursively presented, so there is an embedding $\kappa:K_{\mathcal{T}}(B)\to B$. Put $\alpha_v=\kappa\circ\iota_v$ for $v\in V$. The relations (1.2) show that the family $(\alpha_v)$ satisfies the hypothesis of Corollary 5.3, and $\alpha_0$ is injective by Lemma 4.1. By Corollary 6.1,

is of type $\mathrm{FP}_{m+1}$ and contains $B$, hence every finitely generated $\mathcal{O}$-recursively presented group. It is $\mathcal{O}$-recursively presented by Subsection 2.3. □

Proof of Theorem A. Let $G$ be a countable group and choose $\mathcal{O}$ such that $G$ has an $\mathcal{O}$-r.e. presentation (see Subsection 2.3). By Lemma 7.1, $G$ embeds in a finitely generated $\mathcal{O}$-recursively presented group, hence in $B_{1,\mathcal{O}} \leq B_{n,\mathcal{O}}$, which is of type $\mathrm{FP}_{n}$ by Theorem 7.3. □

Remark 7.4. For $n=0$ every group is of type $\mathrm{FP}_{0}$, and for $n=1$ the required embedding follows from Lemma 7.1. As constructed here, the group obtained depends on $G$ and on $n$, and passing to a union does not give a group of type $\mathrm{FP}_{\infty}$: the union of the chain $B_{1,\mathcal{O}} \leq B_{2,\mathcal{O}} \leq\cdots$ is not finitely generated, since the height of $B_{m+1,\mathcal{O}}$ vanishes on $B_{m,\mathcal{O}}$ and takes the value $1$ at $t_m$, so each inclusion is proper, and a finite subset of the union lies in some $B_{m,\mathcal{O}}$. Nevertheless, $B_{2,\mathcal{O}}$ is already of type $\mathrm{FP}_{\infty}$; see Remark A.11.

Homotopical finiteness

We now turn to Theorem B and its corollary. We use the following classical result attributed to Wall [12]; see [5 VIII.7]. A short proof is given in [6 Lemma 2.1].

Proof of Theorem B. Let $\Omega_{\varnothing}$ be the recursively presented free product constructed in Lemma 7.2. By Lemma 7.1 and Higman’s embedding theorem [7], it embeds in a finitely presented group $U_{2}$. Every countable recursively presented group embeds, by Lemma 7.1, in a finitely generated recursively presented group, and the latter is isomorphic to a free factor of $\Omega_{\varnothing}$. Thus $U_{2}$ contains a copy of every countable recursively presented group.

Suppose that $U_{m}$ is finitely presented, of type $\mathrm{F}_{m}$, and contains $U_{2}$. In particular, it contains a copy of every finitely presented group. By Lemma 4.2, $K_{\mathcal{T}}(U_{m})$ is finitely presented and hence embeds in $U_{m}$. As in the proof of Theorem 7.3, composing this embedding with the maps $\iota_v$ gives endomorphisms satisfying the hypotheses of Corollary 6.1, and the resulting ascending HNN extension $U_{m+1}$ is of type $\mathrm{FP}_{m+1}$. It is finitely presented, being obtained from a finite presentation of $U_m$ by adding one generator and finitely many relators, so it is of type $\mathrm{F}_{m+1}$ by Lemma 7.5. It contains $U_m$, hence $U_2$, hence a copy of every countable recursively presented group. □

Proof of Corollary C. A group of type $\mathrm{F}_{n}$ with $n \geq2$ is finitely presented, so every subgroup of such a group embeds in a finitely presented group. Conversely, if a countable group $G$ embeds in a finitely presented group $P$, then Theorem B embeds $P$, and hence $G$, in a group of type $\mathrm{F}_{n}$. For the final assertion, a finitely generated subgroup of a finitely presented group is recursively presented: choose words for its generators and enumerate the relations among them that follow from the finite presentation. The converse is Higman’s embedding theorem [7]. □

Two-generator versions

The finite-generator version of Lemma 7.1 preserves the finiteness properties considered here.

Proof. Let $G=\langle x_{1},\ldots,x_{k}\rangle$, with $k \geq1$, and use the construction of Lemma 7.1 with only the indices $0,\ldots,k$. The subgroups generated by $u_{0},\ldots,u_{k}$ and by $v_{0},\ldots,v_{k}$ are free of rank $k+1$. Thus

contains $G$ by Britton’s lemma and is generated by $b,t$, by the same elimination formulas as before.

We recall why the finite-rank edge groups preserve type $\mathrm{FP}_{n}$. For a group $\Gamma$ that is the fundamental group of a finite graph of groups, the augmented cellular chain sequence of its Bass–Serre tree is

where one edge is chosen from each unoriented edge orbit. Induce free resolutions of $\mathbb{Z}$ over the vertex and edge groups to $\mathbb{Z}\Gamma$, lift the first map in this sequence to a chain map, and take its mapping cone. Exactness of induction and the homology sequence of the cone show that this is a free resolution of $\mathbb{Z}$. Its degree-$j$ term is the sum of the induced degree-$j$ vertex terms and degree-$(j-1)$ edge terms. Hence $\Gamma$ is of type $\mathrm{FP}_n$ if its vertex groups are of type $\mathrm{FP}_n$ and its edge groups are of type $\mathrm{FP}_{n-1}$.

Apply this first to the free product $G * \langle a,b\rangle$, whose edge group is trivial, and then to the HNN extension defining $J$, whose edge group is free of finite rank. This proves that $J$ is of type $\mathrm{FP}_n$. If $G$ is of type $\mathrm{F}_n$, it is finitely presented, and so is $J$: the construction adds finitely many generators and relators. Lemma 7.5 then shows that $J$ is of type $\mathrm{F}_n$.

Proof. Apply Lemma 7.6 to the groups supplied by Theorems A and B, which are finitely generated because $n \ge2$.

Appendix A. Embedding in a group of type $\mathrm{FP}_{\infty}$

The preprint [11], released on 6 October 2026, constructs a group of type $\mathrm{F}_{\infty}$ containing a copy of every finitely presented group. Its controlled-filling argument, combined with the triple system of Proposition 3.2 and the oracle-universal groups of Lemma 7.2, yields the following result, which contains Theorem D.

The group depends on the oracle, but not on any finiteness degree; see Remark A.11.

Relation to OAI

We adapt three ingredients of [11]: controlled additivity [11 Lemma 6.1], the inductive construction of uniform fillings [11 Proposition 6.2], and the passage to homology with coefficients in products of free modules [11 Sections 7.2–7.4]. The two idempotent diagrams of [11] are replaced by the integral identity (3.2), which annihilates the relevant classes directly; finite generation of the base group, rather than finite presentability, starts the induction; the fillings are organised as controlled partial contractions of powers of the endomorphism, which serve both in the induction and in the coefficient argument; and the geometric argument of [11 Lemma 7.1] is replaced by an algebraic one. We do not use [11 Theorem 5.1], whose hypothesis that the endomorphism factors through a finitely presented group is not available here.

The core of the argument is the following.

We fix $B$, $(\alpha_v)$, $\varphi$ and $E$ as in Proposition A.2 until its proof is complete.

Controlled chains

For a group $\Gamma$, let $C_*(\Gamma)$ be the normalized homogeneous bar complex: $C_q(\Gamma)$ is the free abelian group on the tuples $(\gamma_0,\ldots,\gamma_q)\in\Gamma^{q+1}$ with $\gamma_{i-1}\ne\gamma_i$ for all $i$, a tuple with two consecutive equal entries is read as 0, and $\partial$ is the alternating sum of the deletions. With diagonal left translation and the augmentation $\varepsilon(\gamma)=1$, it is a free $\mathbb{Z}\Gamma$-resolution of $\mathbb{Z}$ [5 I.5]. A homomorphism $\theta:\Gamma\to\Gamma'$ induces, entrywise, a $\theta$-semilinear chain map $\theta_*$. The constant map $B\to\{1\}$ likewise induces a chain map $c_*$ of $C_*(B)$; thus $c_*(b)=(1)$ in degree 0, and $c_*=0$ in positive degrees.

A bound in $\Gamma$ is a finite symmetric subset $A\subseteq\Gamma$ containing 1. Let $C_*^A(\Gamma)\subseteq C_*(\Gamma)$ be spanned by the tuples with $\gamma_i^{-1}\gamma_j\in A$ for all $i,j$. It is a subcomplex, invariant under left translation, and $C_q^A(\Gamma)$ is a free $\mathbb{Z}\Gamma$-module on the finite set $S_q(A)$ of its nondegenerate tuples with $\gamma_0=1$; this set is part of a $\mathbb{Z}\Gamma$-basis of $C_q(\Gamma)$, so $C_q^A(\Gamma)$ is a direct summand of $C_q(\Gamma)$. Every chain lies in $C_*^A(\Gamma)$ for some bound $A$, a finite union of bounds is a bound, $\theta_*(C_*^A(\Gamma))\subseteq C_*^{\theta(A)}(\Gamma')$ for every homomorphism $\theta$, and $A_1\times A_2$ is a bound in $\Gamma_1\times\Gamma_2$. An additive map defined on $C_*^A(\Gamma)$ is controlled if its image lies in $C_*^{A'}(\Gamma')$ for some bound $A'$. This restricts the shape of the simplices that occur, but neither their number, nor their coefficients, nor the position of their vertices.

Controlled additivity

Let $\operatorname{AW}:C_*(\Gamma)\to C_*(\Gamma)\otimes_{\mathbb{Z}}C_*(\Gamma)$ be the Alexander–Whitney map,

and let $\nabla:C_*(\Gamma_1)\otimes_{\mathbb{Z}}C_*(\Gamma_2)\to C_*(\Gamma_1\times\Gamma_2)$ be the shuffle map: on tuples of lengths $p+1$ and $q+1$, it is the sum, over the monotone lattice paths from $(0,0)$ to $(p,q)$, of the tuples of pairs of entries visited by the path, with sign $(-1)^N$, where $N$ is the number of pairs consisting of a vertical step and a later horizontal step. Both are natural chain maps of normalized chains, with the differential $\partial(x\otimes y)=\partial x\otimes y+(-1)^{|x|}x\otimes\partial y$ on tensor products. The map $\operatorname{AW}$ takes $C_*^A(\Gamma)$ to $C_*^A(\Gamma)\otimes C_*^A(\Gamma)$, since it only takes faces, and $\nabla$ takes $C_*^{A_1}(\Gamma_1)\otimes C_*^{A_2}(\Gamma_2)$ to $C_*^{A_1\times A_2}(\Gamma_1\times\Gamma_2).$ Write $\Delta,i_1,i_2:B\to B\times B$ for $b\mapsto(b,b)$, $(b,1)$, $(1,b)$. For every chain $w$ of $C_*(B)$,

Proof. For $x\in B\times B$ and a chain $u$, let $x*u$ be obtained by prepending $x$ to every tuple; then $\partial(x*u)=u$ whenever $u$ is a cycle of positive degree. In degree 0, both $\Delta_*$ and $\nabla\operatorname{AW}$ map $(b)$ to $((b,b))$; put $\Psi=0$ there. Let $q\ge1$, and suppose that $\Psi$ has been defined in degrees $<q$, satisfies the identity there, and maps every tuple to a chain whose entries are pairs of entries of that tuple. Let $s=(b_0,\ldots,b_q)$ be a nondegenerate tuple with set of entries $W$. The chain $u_s=\Delta_*s-\nabla\operatorname{AW}(s)-\Psi\partial s$ has entries in $W\times W$, and is a cycle by the identity in degree $q-1$. Put $\Psi(s)=(b_0,b_0)*u_s$. If $s\subset C_q^A(B)$, then $b^{-1}b'\in A$ for all $b,b'\in W$, whence the last assertion. ∎

Contractions of powers of $\varphi$

In particular, $\varphi_*^{r'+1}w_s=\partial y_s$ for some $y_s\in C_{n+1}^{A'}(B)$. Extend $s\mapsto y_s$ additively to $h_n:C_n^A(B)\to C_{n+1}^{A'}(B)$. Then $(\varphi_*^{r'+1}h_0,\ldots,\varphi_*^{r'+1}h_{n-1},h_n)$ is a controlled $n$-contraction of $\varphi^{r+r'+1}$ over $A$: in degrees $<n$ this was observed after Definition A.3, and in degree $n$

The power and the bounds depend on $n$ and $A$, but not on individual chains; this uniformity is what Lemma A.8 requires. Neither is $\varphi$ assumed to commute with the $\alpha_v$, since all relations are precomposed with the same power of $\varphi$, nor need the contractions be equivariant.

Coefficients in products of free modules

In this subsection coefficient modules are right modules. For a right $\mathbb{Z}E$-module $N$, the groups $H_*(B,N)$ are the homology of $N\otimes_{\mathbb{Z}B}C_*(B)$, on which

is a well-defined chain endomorphism, since $bt=t\varphi(b)$ in $E$.

Proof. The complex $P_*=\mathbb{Z}E\otimes_{\mathbb{Z}B}C_*(B)$ is a free $\mathbb{Z}E$-resolution of $\mathbb{Z}[E/B]$, and $\widetilde{T}(g\otimes\sigma)=gt\otimes\varphi_*\sigma$ is a $\mathbb{Z}E$-linear chain endomorphism of $P_*$ lifting $T(gB)=gtB$. By the exactness of (9), the mapping cone of $1-\widetilde{T}$ is a free $\mathbb{Z}E$-resolution of $\mathbb{Z}$. Tensoring with $N$ identifies $N\otimes_{\mathbb{Z}E}P_*$ with $N\otimes_{\mathbb{Z}B}C_*(B)$ and $1\otimes\widetilde{T}$ with $T$; the long exact sequence of the cone is the assertion.

From vanishing to finiteness

The following is a special case of the Bieri–Eckmann criterion [2 Section 1]; we include the short argument.

Proof. Write $R = \mathbb{Z}\Gamma$, and let $P_*$ be a free resolution of $\mathbb{Z}$ whose terms $P_0,\ldots,P_m$ have finite rank [5 VIII.4.3]. Let $Z_m = \ker(d_m : P_m \to P_{m-1})$, let $I$ be its underlying set, and put $N = R^I$. Since $P_m$ and $P_{m-1}$ have finite rank, $N \otimes_R P_q \cong P_q^I$ for $q = m-1,m$, compatibly with $1 \otimes d_m$. The family $\xi= (z)_{z \in Z_m}$ is therefore a cycle in $N \otimes_R P_m$, and $H_m(\Gamma,N) = 0$ gives $\xi= (1 \otimes d_{m+1})(y)$ for some $y = \sum_{a=1}^s x_a \otimes p_a$ with $x_a \in N$ and $p_a \in P_{m+1}$. In the coordinate indexed by $z$, this reads $z = \sum_a (x_a)_z d_{m+1}p_a$, so $Z_m$ is generated by $d_{m+1}p_1,\ldots,d_{m+1}p_s$. Replacing $P_{m+1}$ by a free module of rank $s$ mapping onto $Z_m$, and the higher terms by a free resolution of its kernel, yields a free resolution that agrees with $P_*$ in degrees $\le m$ and has finite rank in degree $m+1$. Iterating produces a single free resolution of finite rank in every degree. □

Proof of Proposition A.2. By Corollary 6.1 with $m = 1$, $E$ is of type $\mathrm{FP}_2$. By Lemma A.8, $H_q(E,(\mathbb{Z}E)^I) = 0$ for every set $I$ and every $q \ge2$, and Lemma A.9 with $m = 2$ concludes. □

Remark A.10. Corollary 6.1 can be avoided. Since $B$ is finitely generated, $H_0(B,(\mathbb{Z}E)^I) \cong\mathbb{Z}[E/B]^I$, with $T$ acting coordinatewise by $gB \mapsto gtB$, so $1-T$ is injective there by (6.2). With Lemmas A.7 and A.8 this gives $H_1(E,(\mathbb{Z}E)^I) = 0$, and Lemma A.9 applies with $m = 1$.

Proof of Theorem A.1

Let $B = B_{1,O}$ be the group of Lemma 7.2. By Lemma 4.2, $K_{\mathcal{T}}(B)$ is finitely generated and $\mathcal{O}$-recursively presented, so there is an embedding $\kappa: K_{\mathcal{T}}(B) \to B$. The endomorphisms $\alpha_v = \kappa\circ\iota_v$ satisfy the hypothesis of Corollary 5.3 by (1.2), and $\alpha_0$ is injective by Lemma 4.1. By Proposition A.2, the group

is of type $\mathrm{FP}_\infty$. It contains $B$, hence a copy of every countable $\mathcal{O}$-recursively presented group by Lemmas 7.1 and 7.2, and it is $\mathcal{O}$-recursively presented by Subsection 2.3.

Let $J$ be the two-generator group obtained from $E_O$ and a fixed finite generating set by the construction in the proof of Lemma 7.6. This construction does not depend on $n$, and $E_O$ is of type $\mathrm{FP}_n$ for every $n$, so $J$ is of type $\mathrm{FP}_n$ for every $n$, that is, of type $\mathrm{FP}_\infty$. It contains $E_O$, and it is $\mathcal{O}$-recursively presented, since it is obtained by adding finitely many generators and relators. This proves the first assertion. The second follows, because every countable group is $\mathcal{O}$-recursively presented for a suitable oracle (Subsection 2.3).

Remark A.11. Whatever the choice of $\kappa$, the group $E_O$ is the group $B_{2,O}$ of Theorem 7.3. Thus the first ascending HNN extension in that proof is already of type $\mathrm{FP}_\infty$: the finiteness degree enters only through the power of $\varphi$ and the bounds in Proposition A.6, not through the group. The conclusion $\mathrm{FP}_\infty$ is optimal: by Corollary C, a finitely generated group that is not recursively presented embeds in no group of type $\mathrm{F}_2$.

LB: Institut Camille Jordan, Université de Lyon 1 and Section de Mathématiques, Université de Genève

Email address: [email protected]

RM: St Petersburg State University