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McCool groups of toral relatively hyperbolic groups

Vincent Guirardel and Gilbert Levitt

Algebraic & Geometric Topology 15 (2015) 3485–3534

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() 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.

McCool group, automorphism group, toral relatively hyperbolic group, finiteness condition, classifying space
Mathematical Subject Classification 2010
Primary: 20F28
Secondary: 20F65, 20F67
Received: 15 October 2014
Accepted: 13 April 2015
Published: 12 January 2016
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
Gilbert Levitt
Laboratoire de Mathématiques Nicolas Oresme
Université de Caen et CNRS (UMR 6139)
BP 5186
14032 Caen Cedex 5