McCool groups of toral relatively hyperbolic groups

Guirardel, Vincent¹ ; Levitt, Gilbert²

¹Institut de Recherche Mathématique de Rennes, Université de Rennes 1 et CNRS (UMR 6625), 263 avenue du Général Leclerc, CS 74205, 35042 Rennes Cedex, France
²Laboratoire de Mathématiques Nicolas Oresme, Université de Caen et CNRS (UMR 6139), BP 5186, 14032 Caen Cedex 5, France

Algebraic and Geometric Topology, Tome 15 (2015) no. 6, pp. 3485-3534

Résumé

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.

Détail

DOI : 10.2140/agt.2015.15.3485

Classification : 20F28, 20F65, 20F67
Keywords: McCool group, automorphism group, toral relatively hyperbolic group, finiteness condition, classifying space

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