McCool groups of toral relatively hyperbolic groups

Abstract

The outer automorphism group $Out\left(G\right)$ of a group $G$ acts on the set of conjugacy classes of elements of $G$. McCool proved that the stabilizer $Mc\left(\mathcal{C}\right)$ of a finite set of conjugacy classes is finitely presented when $G$ is free. More generally, we consider the group $Mc\left(\mathcal{\mathscr{H}}\right)$ of outer automorphisms $\Phi$ of $G$ acting trivially on a family of subgroups $H_i$, in the sense that $\Phi$ has representatives ${\alpha }_i$ that are 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 $Out\left(G\right)$, 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.