McCool groups of toral relatively hyperbolic groups

TL;DR

This paper studies McCool groups within toral relatively hyperbolic groups, proving they are of type VF and satisfy a uniform chain condition, extending known results from free groups to a broader class.

Contribution

It establishes that McCool groups in toral relatively hyperbolic groups are of type VF and satisfy a uniform chain condition, unifying two definitions of McCool groups in this context.

Findings

McCool groups are of type VF in toral relatively hyperbolic groups.

McCool groups satisfy a uniform chain condition.

The two definitions of McCool groups coincide in this setting.

Abstract

The outer automorphism group Out(G) of a group G acts on the set of conjugacy classes of elements of G. McCool proved that the stabilizer $Mc(c_1,...,c_n)$ of a finite set of conjugacy classes is finitely presented when G is free. More generally, we consider the group $Mc(H_1,...,H_n)$ of outer automorphisms $\Phi$ of G acting trivially on a family of subgroups $H_i$, in the sense that $\Phi$ has representatives $\alpha_i$ with $\alpha_i$ equal to the identity on $H_i$. When G is a toral relatively hyperbolic group, we show that these two definitions lead to the same subgroups of Out(G), which we call "McCool groups" of G. We prove that such McCool groups are of type VF (some finite index subgroup has a finite classifying space). Being of type VF also holds for the group of automorphisms of G preserving a splitting of G over abelian groups. We show that McCool groups satisfy a uniform chain condition: there is a bound, depending only on G, for the length of a strictly decreasing sequence of McCool groups of G. Similarly, fixed subgroups of automorphisms of G satisfy a uniform chain condition.