JSJ IV: Accessibility

June 11, 2025

JSJ IV: Accessibility

We now begin our discussion of selected results from Elia Fioravanti's groundbreaking paper. This post is based on a talk by Lawk Mineh.

Given a finitely generated group, we have been considering its actions on trees and splittings associated to those. As previously discussed, Grushko's theorem shows that there is a maximal free splitting, so one might ask more generally when there is a maximal splitting with edge groups living in some class. It is a celebrated theorem of Stallings, building on previous work of Freudental and Hopf that a group can have 0, 1,2, or infinitely many ends, that groups with infinitely many ends are precisely those that split over a finite subgroup. We can therefore ask the following:

Question: Given a finitely generated group $G$, can we fully decompose it as a graph of one-ended groups over finite subgroups?

Dunwoody gave an example to show that this isn't always possible. However, we have

Theorem (Dunwoody accessibility): Let $G$ be a group of type $\mathrm{FP}_2(\mathbb{Z}/2\mathbb{Z})$ (if this is foreign to you just think finitely presented, since Dunwoody calls this almost finitely presented). Then $G$ is accessible, i.e. there is some $N=N(G)$ such that any reduced graph of one-ended groups with finite edge groups and fundamental group (in the sense of graph of groups) has at most $N$ edges.

On the genus 2 surface there is a bound on the number of non-intersecting closed curves such that the complement has no annuli or discs. Lifting this to the universal cover gives a dual tree. For a general group, Dunwoody develops an analogue of this which he calls train tracks, and the homological finiteness condition ensures $G$ acts freely on a 2-complex with these train tracks that can then be used to build a dual tree.

For finitely presented groups, Bestvina and Feighn show that they are accessible over small subgroups (i.e. ones which don't contain a non-abelian free subgroup) by an intricate analysis of Stallings foldings.

We will be interested in developing analogous results to those discussed in previous posts so far for virtually compact special groups $G$ (i.e. those which are fundamental groups of a compact special cube complex) in order to prove a host of nice properties about $\mathrm{Out}(G)$ and the first step is to prove a suitable accessibility result. See previous JSJ posts for a discussion of how accessibility contributes to the JSJ theory. Note that in Fioravanti's paper he simply calls these 'special groups'.

Distinguished subgrousp of RAAGs

We will be interested in subgroups of right-angled Artin groups. For an introduction to RAAGs see these lecture notes.

Definition: A subgroup $H\leq G$ is a centraliser if there exists a subset $A\subseteq G$ such that $H=Z_G(A)$, where $Z_G(A):=\{g\in G\mid ga=ag,\ \forall a\in A\}$. Equivalently, we have $H=Z_G(Z_G(H))$. We do not consider $G$ itself to be a centraliser, except when it has nontrivial centre. We write
\[ \mathcal{Z}(G) := \{H\leq G\mid \text{$H$ is a centraliser in $G$}\} .\]
Note that, viewing $G$ as a subgroup of some right-angled Artin group $A_{\Gamma}$, every element of $\mathcal{Z}(G)$ is the intersection with $G$ of an element of $\mathcal{Z}(A_{\Gamma})$, but the converse does not hold: many such intersections do not lie in $\mathcal{Z}(G)$. Importantly, the family $\mathcal{Z}(G)$ is $\mathrm{Aut}(G)$--invariant and closed under taking arbitrary intersections.

Many other properties of centralisers in compact special groups $G$ follow from the fact that they are convex-cocompact and $G$--semi-parabolic, two classes of subgroups of $G$ that we discuss below.

Definition: A subgroup of a right-angled Artin group $A_{\Gamma}$ is parabolic if it is conjugate to the subgroup generated by a proper subset of the standard generators, namely $A_{\Delta}$ for some $\Delta\subsetneq\Gamma$. Consequently, if $G\leq A_{\Gamma}$ is convex-cocompact, we say that a subgroup $H\leq G$ is $G$--parabolic if $H\neq G$ and $H=G\cap P$ for a parabolic subgroup $P\leq A_{\Gamma}$.

Again, we emphasise that the data of which subgroups are $G$--parabolic depends on the chosen embedding $G\hookrightarrow A_{\Gamma}$. Since we will consider such an embedding to be fixed throughout, we will write, with an abuse of notation:
\[ \mathcal{P}(G) := \{H\leq G\mid \text{$H$ is $G$--parabolic}\} .\]
The family $\mathcal{P}(G)$ is closed under taking intersections; in particular, the trivial subgroup is $G$--parabolic. However, unlike the family of centralisers, $\mathcal{P}(G)$ does not have a purely algebraic characterisation and it fails to be $\mathrm{Aut}(G)$--invariant in general.

The main utility of $G$--parabolic subgroups comes from the following observation:

Lemma 1: There are only finitely many $G$--conjugacy classes of $G$--parabolic subgroups.

We say that a subgroup $H\leq G$ is root-closed if, whenever $g^n\in H$ for some $g\in G$ and $n\geq 2$, we actually have $g\in H$. In right-angled Artin groups, parabolic subgroups and centralisers are root-closed. As a consequence, all elements of $\mathcal{Z}(G)\cup\mathcal{P}(G)$ are root-closed in any compact special group $G$.

A subgroup $H\leq G$ is convex-cocompact, with respect to the chosen embedding $G\hookrightarrow A_{\Gamma}$, if the universal cover of the Salvetti complex $\mathcal{X}_{\Gamma}$ admits an $H$--invariant convex subcomplex on which $H$ acts cocompactly.

The simplest examples of convex-cocompact subgroups are provided by centralisers and $G$--parabolic subgroups (to be defined). The next lemma collects several preliminary properties of convex-cocompact subgroups of compact special groups.

Lemma 2: Let $G\leq A_{\Gamma}$ and $H,K\leq G$ all be convex-cocompact.

The intersection $H\cap K$ is convex-cocompact.
The normaliser $N_G(H)$ is convex-cocompact and virtually splits as $H\times P$ for some $P\in\mathcal{P}(G)$.
If $g\in G$ satisfies $gHg^{-1}\leq H$, then $gHg^{-1}=H$.
The set $\{g\in G \mid gHg^{-1}\leq K\}$ equals the product $K\cdot F\cdot N_G(H)$ for a finite subset $F\subseteq G$.
The set $\{H\cap gKg^{-1}\mid g\in G\}$ contains only finitely many $H$--conjugacy classes of subgroups.

Centralisers

Let $G\leq A_{\Gamma}$ be convex-cocompact. Centralisers are by far the most important class of subgroups of $G$ for this article: they will provide the edge groups for most of the splittings of $G$ that we will be interested in. A particularly useful property of centralisers is that they are close to being $G$--parabolic; their failure is only due to a (virtual) abelian factor. At the same time, the class $\mathcal{Z}(G)$ also has a considerable weakness: if $H\leq G$ is convex-cocompact and $Z\in\mathcal{Z}(G)$, the intersection $H\cap Z$ will not normally lie in $\mathcal{Z}(H)$.

To circumvent this issue, it is often useful to work with the following larger class of subgroups, which have a similar structure to centralisers and additionally satisfy the above stability property.

Definition: A subgroup $Q\leq G$ is $G$--semi-parabolic if it is convex-cocompact, root-closed, and there exists a $G$--parabolic subgroup $R\lhd Q$ such that $Q/R$ abelian.

For simplicity, we will say that a subgroup is co-abelian if it is normal with abelian quotient. The co-abelian $G$--parabolic subgroup $R$ is not uniquely defined but proposition 4 below shows that one can consider a more canonical parabolic part for any $G$--semi-parabolic subgroup, which will play an important role throughout the article. What is particularly useful is that parabolic parts can be defined without any explicit reference to the ambient right-angled Artin group, only requiring algebraic properties and the notion of convex-cocompactness.

Lemma 3: All elements of $\mathcal{Z}(G)$ are $G$--semi-parabolic.

Proof: Consider $Z\in\mathcal{Z}(G)$ and pick a subset $B\subseteq G$ such that $Z=Z_G(B)$. Up to conjugating $G$ by an element of $A_{\Gamma}$, the centraliser $Z_{A_{\Gamma}}(B)$ is of the form $A_{\Delta}\times A$, for a subgraph $\Delta\subseteq\Gamma$ and a convex-cocompact abelian subgroup $A\leq A_{\Delta^{\perp}}$ where $\Delta^{\perp}=\bigcap_{v\in\Delta}\mathrm{lk}(v)$. The subgroup $Z_{A_{\Gamma}}(B)$ is convex-cocompact and root-closed in $A_{\Gamma}$. Since we have $Z=G\cap Z_{A_{\Gamma}}(B)$, it follows that $Z$ is convex-cocompact and root-closed in $G$. Since $A_{\Delta}$ is co-abelian in $Z_{A_{\Gamma}}(B)$, it follows that the $G$--parabolic subgroup $G\cap A_{\Delta}$ is co-abelian in $Z$, showing that $Z$ is $G$--semi-parabolic.

Proposition 4: Let $Q\leq G$ be $G$--semi-parabolic.

There exists a unique minimal convex-cocompact, root-closed, co-abelian subgroup $P\lhd Q$.
There exists a unique minimal subgroup $P\lhd Q$ that is convex-cocompact, root-closed and co-abelian in $Q$.
The subgroup $P$ is $G$--parabolic and it has trivial centre.
If $gPg^{-1}\leq Q$ for some $g\in G$, then $gPg^{-1}=P$.
Denoting by $A$ the centre of $Q$, the subgroup $\langle P,A\rangle\cong P\times A$ has finite index in $Q$. Moreover, $A$ is root-closed, convex-cocompact and (abstractly) isomorphic to the quotient $Q/P$.

Proof: We begin with the following observation, which is the main point of the proof.
Claim: There exists a co-abelian $G$--parabolic subgroup $R_0\lhd Q$ with trivial centre.

Proof of claim. Let $R\lhd Q$ be a co-abelian $G$--parabolic subgroup. Denote by $C_R$ the centre of $R$, and note that $C_R$ is convex-cocompact as it lies in $\mathcal{Z}(R)$. Since $R$ is convex-cocompact and normal in $Q$, Lemma 2(2) shows that $Q$ virtually splits as $R\times B$ for a $Q$--parabolic group $B$, which must be abelian as $R$ is co-abelian in $Q$. Similarly, since $C_R$ is convex-cocompact and normal in $R$, we have that $R$ virtually splits as $C_R\times R_0$ for an $R$--parabolic subgroup $R_0$. Actually, since $R$ is $G$--parabolic, the fact that $R_0$ is $R$--parabolic implies that $R_0$ is $G$--parabolic as well. Summing up, we have shown that $Q$ virtually splits as $R_0\times C_R\times B$, where $R_0$ is $G$--parabolic, and $C_R$ and $B$ are abelian.

    Note that $R_0$ has trivial centre. If $g$ is an element in the centre of $R_0$, then $g$ commutes with $R_0\times C_R$. It then follows that $g$ commutes with $R$, since $Z_G(g)$ is root-closed and $R_0\times C_R$ has finite index in $R$. Thus $g$ lies the centre of $R$, and hence $g\in C_R\cap R_0=\{1\}$.

    Also note that $R_0$ is normal in $Q$. Indeed, $R_0$ is certainly normal in the product $R_0\times C_R\times B$, which has finite index in $Q$. At the same time, $R_0$ is $G$--parabolic, so $N_G(R_0)$ is $G$--parabolic by Lemma 2 (2), and in particular $N_G(R_0)$ is root-closed. Thus, it follows that $Q\leq N_G(R_0)$, as claimed.

    We are left to show that $R_0$ is co-abelian in $Q$. For this, we consider the abelian group $B':=C_R\times B$, which is central in $Q$ (again because centralisers are root-closed). The centraliser $Z_{A_{\Gamma}}(B')$ splits as $\Pi\times B''$, where $\Pi\leq A_{\Gamma}$ is parabolic and $B''\leq A_{\Gamma}$ is an abelian subgroup with $B'\leq B''$ (again by the Centraliser Theorem). Now, since $R_0$ is a convex-cocompact subgroup of the product $\Pi\times B''$, it virtually splits as the product $(R_0\cap\Pi)\times(R_0\cap B'')$, and $R_0\cap B''$ is trivial because $R_0$ has trivial centre. Thus $R_0\cap\Pi$ has finite index in $R_0$, and since $\Pi$ is root-closed, it follows that $R_0\leq\Pi$. Conversely, a similar argument shows that $\Pi\cap Q\leq R_0$, since $\Pi$ has trivial intersection with $C_R\times B$. In conclusion, $Q$ is contained in the product $\Pi\times B''$, where $B''$ is abelian and $R_0=\Pi\cap Q$. This shows that $R_0$ is co-abelian in $Q$, completing the proof of the claim.$\blacksquare$

Using the claim, we now proceed to prove the four parts of the proposition, beginning with part~(1). Let $H\lhd Q$ be convex-cocompact, root-closed and co-abelian in $Q$. Then the intersection $H\cap R_0$ is convex-cocompact and co-abelian in $R_0$. By Lemma 2(2) and the fact that $R_0$ has trivial centre, the latter implies that $H\cap R_0$ has finite index in $R_0$ and, since $H\cap R_0$ is root-closed, it follows that $H\cap R_0=R_0$. In other words, we have $R_0\leq H$, showing that $R_0$ is the unique minimal convex-cocompact, root-closed, co-abelian subgroup of $Q$.

    Setting $P:=R_0$, we have proved parts~(1) and~(2) of the proposition. Part~(4) follows from Lemma 2(2), noting that the centre of $Q$ projects injectively to a finite-index subgroup of $Q/R_0$, and that the latter is a free abelian group (for instance, because $Q/R_0$ embeds in the free abelian group $B''$ considered at the end of the proof of the claim).

    We are left to prove part~(3). Recall from the end of the proof of the claim that $Q$ is contained in a product $\Pi\times B''$, where $\Pi\leq A_{\Gamma}$ is parabolic, $B''$ is abelian, and $R_0=\Pi\cap Q$. Up to conjugating the whole $G$ by an element of $A_{\Gamma}$, we can assume that $\Pi=A_{\Delta}$ for some $\Delta\subseteq\Gamma$ and that $B''\leq A_{\Delta^{\perp}}$, where $\Delta^{\perp}=\bigcap_{v\in\Delta}\mathrm{lk}(v)$. Now, thinking of elements of $R_0$ as elements of the right-angled Artin group $A_{\Gamma}$, their cyclically reduced parts only involve standard generators in $\Delta$; the same is true of elements of any conjugate $gR_0g^{-1}$. Thus, if we have $gR_0g^{-1}\leq Q$, then $gR_0g^{-1}\leq R_0$, because $Q\leq A_{\Delta}\times A_{\Delta^{\perp}}$ and $Q\cap A_{\Delta}=R_0$. Finally, the inclusion $gR_0g^{-1}\leq R_0$ implies that $gR_0g^{-1}=R_0$ by Lemma 2(3), as required. $\blacksquare$

It is convenient to name the subgroup $P$ described in the previous proposition.

Definition: Let $Q\leq G$ be $G$--semi-parabolic. The parabolic part of $Q$ is the unique minimal subgroup $P\lhd Q$ that is convex-cocompact, root-closed and co-abelian.

The following example shows that part~(4) of Proposition 4 cannot be improved.

Example: If $Z\in\mathcal{Z}(G)$, its parabolic part $P\lhd Z$ is not a direct factor of $Z$ in general, if we do not first pass to finite index. For instance, consider $A_{\Gamma}=\langle a,b\rangle\times\langle c\rangle\cong F_2\times\mathbb{Z}$ and the index--$2$ subgroup $G=\langle a^2,b^2,ab,c^2,ac\rangle$, which has a further index--$2$ subgroup splitting as the product $\langle a^2,b^2,ab\rangle\times\langle c^2\rangle$. We have $Z_G(c^2)=G$ and its parabolic part is $P=\langle a^2,b^2,ab\rangle$. Since $G/P\cong \mathbb{Z}=\langle\overline{ac}\rangle$, we have $G=P\rtimes \mathbb{Z}$ and $P$ is not a direct factor of $G$.

Remark 5: Let $G$ be compact special and let $Q_1,Q_2\leq G$ be $G$--semi-parabolic subgroups with the same parabolic part $P$. If $Q_1$ is a proper subgroup of $Q_2$, then $Q_1$ has infinite index in $Q_2$; this is because $G$--semi-parabolic subgroups are root-closed. It follows that the free abelian group $Q_1/P$ is an infinite-index subgroup of the free abelian group $Q_2/P$, and thus it has strictly lower rank. This shows that ascending chains of $G$--semi-parabolic subgroups of $G$ with a given parabolic part contain at most $d+1$ elements, where $d$ is the largest rank of a free abelian subgroup of $G$.

    More generally, if $Q_1\leq Q_2$ are arbitrary $G$--semi-parabolic subgroups, then the parabolic part of $Q_1$ is contained in that of $Q_2$. Observing that chains of $G$--parabolic subgroups of $G$ have length at most $|\Gamma^{(0)}|$, we conclude that arbitrary chains of $G$--semi-parabolic subgroups of $G$ have length at most $|\Gamma^{(0)}|\cdot (d+1)$. Of course, this bound is far from tight.

We can now state the main results:

Theorem 6: Special groups are accessible over semi-parabolic subgroups.

The proof idea is the following:

Take $P \unlhd Q$ the parabolic part of $G$- semiparabolic $Q$
$N_G(P)/P$ is accessible over small subgroups by Bestvina-Feighn
There are finitely many $G-$conjugacy classes of $G-$parabolic subgroups, which gives a bound of the size of a fundamental domain of $G$ acting on a tree $T$ with semiparabolic edges.
Need $N_G(P)$ to act on $\mathrm{Fix}(P)$ in a way which is uniformly close to being minimal and reduced, which we do by controlling the pro-p topology of $G$.

We now elaborate on what these words mean.

Pro-$p$ topology

Definition: Let $G$ be a group and $p$ a prime number. The pro-$p$ topology on $G$ has the following as a base of open sets:
\[ \{gN \mid g\in G, \text{ } N\lhd G \text{ and $G/N$ is a finite $p$--group} \} .\]
A subgroup $H\leq G$ is $p$--separable if it is closed in the pro--$p$ topology on $G$; equivalently, for every $g\in G\setminus H$, there exists a homomorphism $f\colon G\rightarrow F$ such that $F$ is a finite $p$--group and $f(g)\not\in f(H)$. The group $G$ is residually $p$--finite if the trivial subgroup is $p$--separable; equivalently, the identity is the only element of $G$ that vanishes in all $p$--group quotients of $G$.

Remark 7 While every finite-index subgroup of $G$ contains a normal finite-index subgroup of $G$, this statement fails is we replace the two occurrences of the word ``finite'' with ``power-of--$p$''. For instance, the free group $F_2$ has non-normal subgroups of index $3$, which cannot contain any normal subgroups of index $3^n$, as maximal proper subgroups of $p$--groups are always normal. An explicit example is the subgroup $\langle x,y^3,yxy,yx^{-1}y \rangle$ within $\langle x,y\rangle\cong F_2$. We mention this to emphasise that, unlike the case of the profinite topology, it is important that the above basis for the pro--$p$ topology consists of cosets of normal subgroups, rather than cosets of general subgroups of index a power of $p$.

Our interest in pro--$p$ topologies is due to the following elementary observation.

Lemma 8: If a proper subgroup $H<G$ is $p$--separable, then $H$ is contained in a normal subgroup of $G$ of index $p$. In particular, the normal closure of $H$ is a proper subgroup of $G$.

Proof: The key point is that, if $F$ is a finite $p$--group and $F_0$ is a maximal proper subgroup of $F$, then $F_0$ is normal in $F$ (and has index $p$). See the previous post on nilpotent groups.

Now, given a proper, $p$--separable subgroup $H<G$, pick an element $g\in G\setminus H$ and a $p$--group quotient $\pi\colon G\twoheadrightarrow F$ with $\pi(g)\not\in\pi(H)$. In particular, $\pi(H)$ is a proper subgroup of $F$ and it is contained in an index--$p$ subgroup $F_0\lhd F$. The preimage $\pi^{-1}(F_0)$ is the required normal index--$p$ subgroup of $G$ containing $H$ $\blacksquare$.

Let now $G$ be a compact special group. Haglund and Wise showed that all convex-cocompact subgroups of $G$ are separable in $G$. However, many of these subgroups are not $p$--separable for any value of $p$; an example are the subgroups of $F_2$ described in Remark 7. What is important to us is that retracts, and in particular $G$--parabolic subgroups, are $p$--separable for all $p$.

Lemma 9 If $G$ is a compact special group, then each $P\in\mathcal{P}(G)$ is $p$--separable in $G$ for every prime $p$.

Proof sketch: Retracts of residually $p$--finite groups are $p$--separable. Moreover, right-angled Artin groups are residually $p$--finite for all $p$; this follows from the fact that right-angled Artin groups are residually torsion-free nilpotent.

Now, since parabolic subgroups of right-angled Artin groups are clear retracts, they are $p$--separable. If $G\leq A_{\Gamma}$ is convex-cocompact, elements of $\mathcal{P}(G)$ are by definition intersections between $G$ and parabolic subgroups of $A_{\Gamma}$, so they are $p$--separable in $G$ for all $p$. $\blacksquare$

Proposition 10: Let $G$ be a finitely generated group with a reduced splitting $G\curvearrowright T$. Suppose that there exists a subtree $U\subseteq T$, with $G$--stabiliser denoted $G_U$, such that all the following hold:

the subtrees in the family $\{gU\mid g\in G\}$ cover $T$, and distinct ones share at most one point;
$G_U$ is $p$--separable in $G$ for some prime $p$;
there exists an integer $N\geq 0$ such that either $G_U$ is elliptic in $T$ and $N=0$, or $G_U$ is $N$--accessible over the family of stabilisers of edges of $U$.

    Then the quotient $T/G$ has at most $4b_1(G;\mathbb{F}_p)+N$ edges.

The proof is a counting argument that we leave to the interested reader to derive/check.

We now deduce Theorem 6 from Proposition 10.

For this we consider a compact special group $G$, and we realise $G$ as a convex-cocompact subgroup of a right-angled Artin group $A_{\Gamma}$, in order to be able to speak of $G$--parabolic and $G$--semi-parabolic subgroups. Recall that every compact special group $G$ acts by conjugation on the set of $G$--parabolic subgroups $\mathcal{P}(G)$, and the quotient by this action $\mathcal{P}(G)/G$ is finite by Lemma 1. Also recall that, for each $P\in\mathcal{P}(G)$, the normaliser $N_G(P)$ is $G$--parabolic, and hence the quotient $N_G(P)/P$ is finitely presented (for instance, because $N_G(P)$ is finitely presented and $P$ is finitely generated).

By Bestvina and Feighn's accessibility, each finitely presented group $H$ admits a constant $\gamma(H)$ such that every reduced splitting of $H$ over abelian subgroups has at most $\gamma(H)$ edge orbits.

Corollary 11: Let $G\leq A_{\Gamma}$ be convex-cocompact. If $G\curvearrowright T$ is a reduced splitting of $G$ over $G$--semi-parabolic subgroups, then the number of edges of the quotient $T/G$ is at most:
    \[ 4b_1(G;\mathbb{Q})\cdot|\mathcal{P}(G)/G|+\sum_{[P]\in\mathcal{P}(G)/G}\gamma(N_G(P)/P) .\]

Proof: Recall that each $G$--semi-parabolic subgroup of $G$ has a parabolic part, which lies in $\mathcal{P}(G)$. Thus, for each $G$--conjugacy class of $G$--parabolic subgroups $[P]$, we can consider the edges of $T$ whose stabiliser has parabolic part in the class $[P]$, and collapse all other edges of $T$. We obtain a collapse $G\curvearrowright T_{[P]}$ that is still a reduced splitting, and has gained the property that all its edge-stabilisers are $G$--semi-parabolic with parabolic part in $[P]$. Bounding the number of edge orbits in $T_{[P]}$ for each $[P]\in\mathcal{P}(G)/G$ gives a bound on the number of edge orbits for the original tree $T$ (namely, the sum of the bounds for the $T_{[P]}$).

    Thus, we assume in the rest of the proof that there exists $P\in\mathcal{P}(G)$ such that all edge-stabilisers of $T$ have parabolic part conjugate to $P$, and our goal becomes showing that $T/G$ has at most $4b_1(G;\mathbb{Q})+\gamma(N_G(P)/P)$ edges. For this, the plan is to invoke Proposition 10 for the subtree of $P$--fixed points $\mathrm{Fix}(P)\subseteq T$, so we now proceed to check that its assumptions are satisfied.

    Every edge-stabiliser $E$ has a conjugate that has exactly $P$ as its parabolic part (rather than a conjugate of $P$). In particular, the $G$--translates of $\mathrm{Fix}(P)$ cover $P$. At the same time, each edge-stabiliser $E$ can only contain a given $G$--conjugate of $P$ if that is the parabolic part of $E$, by Proposition 4(3). This implies that no two $G$--translates of $\mathrm{Fix}(P)$ share an edge, and also that the $G$--stabiliser of $\mathrm{Fix}(P)$ equals the normaliser $N_G(P)$. The latter is $G$--parabolic in $G$ by Lemma 2 (2), and hence $p$--separable in $G$ for all primes $p$ by Lemma 9. Finally, consider the action $N_G(P)\curvearrowright\mathrm{Fix}(P)$ and note that this factors through an action $N_G(P)/P\curvearrowright\mathrm{Fix}(P)$, which has free abelian edge-stabilisers. Any reduced splitting of $N_G(P)/P$ over abelian subgroups has at most $\gamma(N_G(P)/P)$ orbits of edges by Bestvina-Feighn.

    In conclusion, Proposition 10 shows that $T/G$ has at most $4b_1(G;\mathbb{F}_p)+\gamma(N_G(P)/P)$ edges, for all primes $p$. Choosing $p$ large enough, this bound becomes $4b_1(G;\mathbb{Q})+\gamma(N_G(P)/P)$, since $G$ and its abelianisation are finitely generated. This concludes the proof of the corollary.

Posts in this series:

Search This Blog

On the shoulders of giants