McCool groups of toral relatively hyperbolic groups

Guirardel, Vincent; Levitt, Gilbert

doi:10.2140/agt.2015.15.3485

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McCool groups of toral relatively hyperbolic groups

Vincent Guirardel and Gilbert Levitt

Algebraic & Geometric Topology 15 (2015) 3485–3534

DOI: 10.2140/agt.2015.15.3485

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 $H_{i}$ , in the sense that $Φ$ has representatives $α_{i}$ 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 $Out (G)$ , 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.

Keywords

McCool group, automorphism group, toral relatively hyperbolic group, finiteness condition, classifying space

Mathematical Subject Classification 2010

Primary: 20F28

Secondary: 20F65, 20F67

References

Publication

Received: 15 October 2014

Accepted: 13 April 2015

Published: 12 January 2016

Authors

Vincent Guirardel
Institut de Recherche Mathématique de Rennes Université de Rennes 1 et CNRS (UMR 6625) 263 avenue du Général Leclerc CS 74205 35042 Rennes Cedex France

Gilbert Levitt
Laboratoire de Mathématiques Nicolas Oresme Université de Caen et CNRS (UMR 6139) BP 5186 14032 Caen Cedex 5 France

Volume 15, issue 6 (2015)