## Notes of Ashot Minasyan’s talk

Residual finiteness of outer automorphism groups of relatively hyperbolic groups

Joint work with Gilbert Levitt and Denis Osin.

I like relatively hyperbolic groups, since combinatorial and geometric group theory meet there.

1. Residual finiteness

1.1. Definition

Definition 1 A group ${G}$ is residually finite if the intersection of all normal finite index subgroups of ${G}$ is trivial.

It is a useful property. Such groups have solvable word problem, they are Hopfian. Large classes of groups have it (polycyclic groups, linear groups,…). Some don’t.

Example: Baumslag-Solitar group ${BS(2,3)=\langle a,b \,|\,ba^2 b^{-1}a^{-3}\rangle}$ is not residually finite.

Question: Are all word hyperbolic groups residually finite ?

Personnally, I think the answer must be no.

Question: When is ${Out(G)}$ residually finite ?

The case of ${Aut(G)}$ is rather well understood.

Theorem 2 (Baumslag) If ${G}$ is finitely generated and residually finite, ${Aut(G)}$ is residually finite.

The situation is opposite for ${Out(G)}$.

Theorem 3 (Bumagina, Wise) Every finitely presented group is the outer automorphism group of some finitely generated and residually finite group.

So extra conditions are required.

1.2. Conjugacy separability

Definition 4 A group ${G}$ is conjugacy separable if non conjugate elements are still non conjugate in some finite quotient.

Definition 5 Say an automorphism ${\alpha}$ of ${G}$ is pointwise inner if for every ${g}$, ${\alpha(g)}$ is conjugate to ${g}$. The group of pointwise automorphisms of ${G}$ is denoted by ${Aut_{p.i}(G)}$.

Theorem 6 (Grossman) If ${G}$ is finitely generated, conjugacy separable, and ${Aut_{p.i}(G)=Inn(G)}$, then ${Out(G)}$ is residually finite.

Proof: Let ${\alpha\in Aut(G)\setminus Inn(G)}$. By assumption, there is ${g\in G}$ such that ${\alpha(g)}$ is not conjugate to ${g}$. In some finite quotient, ${\alpha(g)}$ is not conjugate to ${g}$ again. This implies that ${[\alpha]}$ is non trivial in the outer automorphism group of that finite quotient.

Example: Virtually polycyclic groups, virtually free groups, virtual surface groups (A. Martino), limit groups are conjugacy separable.

Example: ${SL(3,{\mathbb Z})}$ is not conjugacy separable.

Note that unlike residual finiteness, conjugacy separability does not pass to finite index subgroups or overgroups.

The property ${Aut_{p.i}(G)=Inn(G)}$ is easier to establish.

2. Relative hyperbolicity

2.1. Definition

Let ${G}$ be finitely generated by ${X}$, let ${H_1 ,\ldots, H_k}$ be subgrous of ${G}$. Then ${G}$ is a quotient of ${F=F_X \star H_1 \star\cdots\star H_k}$. By a finite relative presentation, I mean that the quotient is normally generated by finitely many elements.

Definition 7 Choose as an alphabet

$\displaystyle \mathcal{H}=X\cup \bigcup_{i=1}^{k}H_k .$

Assume ${G}$ has finite presentation with respect to the set of generators ${\mathcal{H}}$ with set of relators ${\mathcal{R}}$. If ${w}$ is a word in this alphabet which is trivial in ${G}$,

$\displaystyle \begin{array}{rcl} w=\prod_{i=1}^{n}f_i R_i f_i^{-1},\quad f_i \in F,\quad R_i \in \mathcal{R}, \end{array}$

define smallest possible ${n=n(w)}$ as the filling area of ${w}$.

Say that ${G}$ is hyperbolic relative to ${\{H_1 ,\ldots, H_k\}}$ if there exists a constant ${C}$ such hat for every word ${w}$ in alphabet ${\mathcal{H}}$ which is trivial in ${G}$, ${n(w)\leq C|w|}$.

Example: Limit groups are hyperbolic with respect to free abelian subgroups.

2.2. Non relatively hyperbolic groups

Definition 8 Say ${G}$ is NRH is ${G}$ is not hyperbolic relative to any family of proper subgroups.

Example: ${{\mathbb Z}^n}$, ${SL(n,{\mathbb Z})}$, ${n\geq 3}$ are NRH.

3. Results

3.1. The one ended case

Theorem 9 (Levitt, Minasyan) Suppose that ${G}$ is finitely generated, one-ended and hyperbolic relative to ${\{H_1 ,\ldots, H_k\}}$. If ${H_1 ,\ldots, H_k}$ are NRH and residually finite, then ${Out(G)}$ is residually finite.

Note that this does not assume nor imply that ${G}$ itself is residually finite.

Corollary 10 For every one-ended word hyperbolic group ${G}$, ${Out(G)}$ is residually finite.

Note that ${G}$ embeds in ${Out(G\star F_2)}$. Since ${G\star F_2}$ has infinitely many ends, one cannot conclude that ${Out(G\star F_2)}$, and thus ${G}$, is residually finite. Too bad.

3.2. The infinite ended case

Theorem 11 (Minasyan, Osin) Suppose that ${G}$ is finitely generated, residually finite group with infinitely many ends, then ${Out(G)}$ is residually finite.

The infinite ended assumption implies that ${G}$ is hyperbolic relative to a family of subgroups.

3.3. Proofs

We use graphs of groups.

Definition 12 Let ${G}$ be a graph of groups built on a finite, connected, bipartite graph ${\Gamma}$. Assume that at right vertices,

1. Vertex group ${G_v}$ is hyperbolic relative to ${\{G_{e_1},\ldots,G_{e_{k}}}$ where ${e_1 ,\ldots,e_k}$ are the edges of ${\Gamma}$ adjacent to ${v}$.
2. There is a finite subset ${S\subset G\setminus\{1\}}$ such that all finite index normal subgroups ${N_i \subset G_{e_{i}}}$ not intersecting ${S}$ satisfy ${N\cap G_{e_{i}}=N_{i}}$. Furthermore, ${G_{v}/N}$ is conjugacy separable, where ${N}$ is the subgroup normally generated by the ${N_i}$‘s.

Here is our main technical result.

Theorem 13 (Levitt, Minasyan) In addition to the assumptions above on right vertex groups, assume that ${G}$ is finitely generated, ${G}$ does not coincide with a left vertex group, ${G}$ has no nontrivial finite normal subgroups. Assume that all left vertex groups are residually finite. Then ${Out(G)}$ is residually finite.

The second theorem (infinitely many ends) follows. According to Stallings, ${G}$ splits along a finite subgroup or is an HNN extension. So ${G}$ is a graph of groups with graph with one edge. Add a vertex in the middle of the edge. The assumptions on right vertices are trivially satisfied.

In the first theorem (one end), one uses the canonical JSJ decomposition of Guirardel and Levitt. Vertices split into 3 types : elementary, surface and rigid. Put all elementary vertices on the left, surface and rigid vertices on the right. Outer automorphism groups of rigid vertex groups relative to the
incoming edge groups are finite. So up to finite index, one can
replace rigid vertices by
free products of incoming edge groups with a free group.

3.4. Corollary

The following are equivalent:

1. Every word hyperbolic group is residually finite.
2. For every word hyperbolic group, ${Out(G)}$ is residually finite.