Automorphisms groups of relatively hyperbolic groups and McCool groups
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 1 Let 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 2 Let be a group and be a collection of subgroups. is the group of outer automorphisms of acting trivially on each .
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.
Theorem 3 (McCool) McCool groups of the free group are finitely presented.
Theorem 4 Let 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 6 Let 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 7 Let 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.