Algebraic & Geometric Topology

McCool groups of toral relatively hyperbolic groups

Vincent Guirardel and Gilbert Levitt

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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