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<G}, 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<G} 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<G} 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.

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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