#### Volume 15, issue 6 (2015)

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McCool groups of toral relatively hyperbolic groups

### Vincent Guirardel and Gilbert Levitt

Algebraic & Geometric Topology 15 (2015) 3485–3534
##### Abstract

The outer automorphism group $Out\left(G\right)$ of a group $G$ acts on the set of conjugacy classes of elements of $G\phantom{\rule{0.3em}{0ex}}$. McCool proved that the stabilizer $Mc\left(\mathsc{C}\right)$ of a finite set of conjugacy classes is finitely presented when $G$ is free. More generally, we consider the group $Mc\left(\mathsc{ℋ}\right)$ 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 $G\phantom{\rule{0.3em}{0ex}}$, for the length of a strictly decreasing sequence of McCool groups of $G\phantom{\rule{0.3em}{0ex}}$. Similarly, fixed subgroups of automorphisms of $G$ satisfy a uniform chain condition.

##### Keywords
McCool group, automorphism group, toral relatively hyperbolic group, finiteness condition, classifying space
##### Mathematical Subject Classification 2010
Primary: 20F28
Secondary: 20F65, 20F67