Algebraic & Geometric Topology

McCool groups of toral relatively hyperbolic groups

Vincent Guirardel and Gilbert Levitt

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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() 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
Received: 15 October 2014
Accepted: 13 April 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
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

Subjects
Primary: 20F28: Automorphism groups of groups [See also 20E36]
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F67: Hyperbolic groups and nonpositively curved groups

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

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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