** Automorphisms groups of relatively hyperbolic groups and McCool groups **

** 0.1. Motivation **

I start with an introductory problem. Given , which outer automorphisms of do arise from outer automorphisms of ? We denote this subgroup by .

Here is an easy way to produce such automorphisms, in a special case. Assume that . Let be an automorphism of which acts trivially on and (I mean that its restrictions to coincide with inner automorphisms of ). Then extends to .

Topological interpretation: fundamental group of , , are fundamental groups of subsets, homeo of which is identity on these subsets. Then the outer automorphism acts trivially on and .

Theorem 1Let be hyperbolic, quasiconvex and malnormal. If is infinite, then it virtually occurs in the above manner, i.e., up to finite index,1. is a vertex subgroup in a graph of groups decomposition of .

2. Virtually all automorphisms of extend for that reason, i.e. has finite index in , where is the collection of edge groups in the decomposition for edges incident to .

Definition 2Let be a group and be a collection of subgroups. is the group of outer automorphisms of acting trivially on each .

** 0.2. Theme **

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

2. Many results about extend to .

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

For one ended groups, results are more precise.

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

Theorem 6Let be hyperbolic. Let be a collection of subgroups. Assume is one ended with respect to (i.e. no splitting over a finite group such that becomes elliptic). Then 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 . Rigid vertex groups have finite McCool groups (Paulin-Rips).

Next we go to relatively hyperbolic group. The inspiring example is hyperbolic, quasiconvex and malnormal. Then is hyperbolic relative to .

Theorem 7Let be hyperbolic relatively to . Let be a collection of subgroups. Assume is one ended with respect to and . Then has a finite index subgroup which is an extension of a product of McCool groups of and 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.