## Algebraic & Geometric Topology

### McCool groups of toral relatively hyperbolic groups

#### 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 $Mc(C)$ of a finite set of conjugacy classes is finitely presented when $G$ is free. More generally, we consider the group $Mc(ℋ)$ of outer automorphisms $Φ$ of $G$ acting trivially on a family of subgroups $Hi$, in the sense that $Φ$ has representatives $αi$ that are equal to the identity on $Hi$.

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.

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 6 (2015), 3485-3534.

Dates
Accepted: 13 April 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841075

Digital Object Identifier
doi:10.2140/agt.2015.15.3485

Mathematical Reviews number (MathSciNet)
MR3450769

Zentralblatt MATH identifier
1364.20020

#### Citation

Guirardel, Vincent; Levitt, Gilbert. McCool groups of toral relatively hyperbolic groups. Algebr. Geom. Topol. 15 (2015), no. 6, 3485--3534. doi:10.2140/agt.2015.15.3485. https://projecteuclid.org/euclid.agt/1510841075

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