Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 15, Number 6 (2015), 3485-3534.
McCool groups of toral relatively hyperbolic groups
The outer automorphism group of a group acts on the set of conjugacy classes of elements of . McCool proved that the stabilizer of a finite set of conjugacy classes is finitely presented when is free. More generally, we consider the group of outer automorphisms of acting trivially on a family of subgroups , in the sense that has representatives that are equal to the identity on .
When is a toral relatively hyperbolic group, we show that these two definitions lead to the same subgroups of , which we call “McCool groups” of G. We prove that such McCool groups are of type (some finite-index subgroup has a finite classifying space). Being of type also holds for the group of automorphisms of preserving a splitting of over abelian groups.
We show that McCool groups satisfy a uniform chain condition: there is a bound, depending only on , for the length of a strictly decreasing sequence of McCool groups of . Similarly, fixed subgroups of automorphisms of satisfy a uniform chain condition.
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
Permanent link to this document
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
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