Notes of Vincent Guirardel’s lecture

Automorphisms groups of relatively hyperbolic groups and McCool groups

0.1. Motivation

I start with an introductory problem. Given ${P, which outer automorphisms of ${P}$ do arise from outer automorphisms of ${G}$ ? We denote this subgroup by ${Out(P|G)}$.

Here is an easy way to produce such automorphisms, in a special case. Assume that ${G=A*_{E_1} P *_{E_2} B}$. Let ${a}$ be an automorphism of ${P}$ which acts trivially on ${E_1}$ and ${E_2}$ (I mean that its restrictions to ${E_i}$ coincide with inner automorphisms of ${P}$). Then ${a}$ extends to ${G}$.

Topological interpretation: ${P=}$ fundamental group of ${X}$, ${E_1}$, ${E_2}$ are fundamental groups of subsets, ${f}$ homeo of ${X}$ which is identity on these subsets. Then the outer automorphism ${a=f_\#}$ acts trivially on ${E_1}$ and ${E_2}$.

Theorem 1 Let ${G}$ be hyperbolic, ${P quasiconvex and malnormal. If ${Out(P|G)}$ is infinite, then it virtually occurs in the above manner, i.e., up to finite index,

1. ${P}$ is a vertex subgroup in a graph of groups decomposition of ${G}$.

2. Virtually all automorphisms of ${P}$ extend for that reason, i.e. ${McC(G;H)}$ has finite index in ${Out(P|G)}$, where ${H}$ is the collection of edge groups in the decomposition for edges incident to ${P}$.

Definition 2 Let ${G}$ be a group and ${H=\{H_1,...,H_n\}}$ be a collection of subgroups. ${McC(G;H)}$ is the group of outer automorphisms of ${G}$ acting trivially on each ${H_i}$.

0.2. Theme

1. McCool groups arise naturally. Many natural groups can be built from them.

2. Many results about ${Out}$ extend to ${McC}$.

3. For toral relatively hyperbolic groups, there is a descending chain condition among McCool groups.

0.3. Results

Theorem 3 (McCool) McCool groups of the free group are finitely presented.

Theorem 4 Let ${G}$ be a toral relatively hyperbolic group (${G}$ is torsion free, and parabolic groups are abelian). Then any McCool group of ${G}$ has a finite index subgroup which has a finite classifying space.

For one ended groups, results are more precise.

Theorem 5 (Sela) Let ${G}$ be hyperbolic, torsion free, one ended. Then a finite index subgroup of ${G}$ is an extension of a product of McCool groups and an abelian group.

Theorem 6 Let ${G}$ be hyperbolic. Let ${H=\{H_1,...,H_n\}}$ be a collection of subgroups. Assume ${G}$ is one ended with respect to ${H}$ (i.e. no splitting over a finite group such that ${H_i}$ becomes elliptic). Then ${McC(G;H)}$ has a finite index subgroup which is an extension of a product of McCool groups and a virtually abelian group.

Proof uses a canonical JSJ decomposition relative to ${H}$. Rigid vertex groups have finite McCool groups (Paulin-Rips).

Next we go to relatively hyperbolic group. The inspiring example is ${G}$ hyperbolic, ${P quasiconvex and malnormal. Then ${G}$ is hyperbolic relative to ${P}$.

Theorem 7 Let ${G}$ be hyperbolic relatively to ${P=\{P_1,...,P_n\}}$. Let ${H=\{H_1,...,H_n\}}$ be a collection of subgroups. Assume ${G}$ is one ended with respect to ${H}$ and ${P}$. Then ${McC(G;H)}$ has a finite index subgroup which is an extension of a product of McCool groups of ${H}$ and ${P}$ and a group of “twists” (generalizations of Dehn twists).

Here, we use a JSJ decomposition where edge groups are either virtually abelian or contained in parabolics.

Note that McCool groups appear here unexpectedly.