Fixed subgroups of automorphisms of relatively hyperbolic groups

TL;DR

This paper proves that fixed subgroups of automorphisms in certain relatively hyperbolic groups are relatively quasiconvex and often finitely generated or presented, especially when peripheral subgroups are slender or coherent.

Contribution

It establishes conditions under which fixed subgroups of automorphisms are relatively quasiconvex and finitely generated or presented, extending understanding of automorphism fixed points in relatively hyperbolic groups.

Findings

Fixed subgroups are relatively quasiconvex under certain conditions.

Fixed subgroups are relatively hyperbolic with respect to natural peripheral subgroups.

When peripheral subgroups are slender, fixed subgroups are finitely generated or presented.

Abstract

Let G be a finitely generated relatively hyperbolic group. We show that if no peripheral subgroup of G is hyperbolic relative to a collection of proper subgroups, then the fixed subgroup of every automorphism of G is relatively quasiconvex. It follows that the fixed subgroup is itself relatively hyperbolic with respect to a natural family of peripheral subgroups. If all peripheral subgroups of G are slender (respectively, slender and coherent), our result implies that the fixed subgroup of every automorphism of G is finitely generated (respectively, finitely presented). In particular, this happens when G is a limit group, and thus for any automorphism \phi of G, Fix(\phi) is a limit subgroup of G.