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Coarse-median preserving automorphisms

Elia Fioravanti

Geometry & Topology 28 (2024) 161–266
Abstract

This paper has three main goals.

First, we study fixed subgroups of automorphisms of right-angled Artin and Coxeter groups. If φ is an untwisted automorphism of a RAAG, or an arbitrary automorphism of a RACG, we prove that Fix φ is finitely generated and undistorted. Up to replacing φ with a power, we show that Fix φ is quasiconvex with respect to the standard word metric. This implies that Fix φ is a virtual retract and a special group in the sense of Haglund and Wise.

By contrast, there exist “twisted” automorphisms of RAAGs for which Fix φ is undistorted but not of type F (hence not special), of type F but distorted, or even infinitely generated.

Secondly, we introduce the notion of “coarse-median preserving” automorphism of a coarse median group, which plays a key role in the above results. We show that automorphisms of RAAGs are coarse-median preserving if and only if they are untwisted. On the other hand, all automorphisms of Gromov-hyperbolic groups and right-angled Coxeter groups are coarse-median preserving. These facts also yield new or more elementary proofs of Nielsen realisation for RAAGs and RACGs.

Finally, we show that, for every special group G (in the sense of Haglund and Wise), every infinite-order, coarse-median preserving outer automorphism of G can be realised as a homothety of a finite-rank median space X equipped with a “moderate” isometric G–action. This generalises the classical result, due to Paulin, that every infinite-order outer automorphism of a hyperbolic group H projectively stabilises a small H–tree.

Keywords
coarse median, median space, special group, outer automorphism, right-angled Artin group, right-angled Coxeter group, untwisted automorphism, fixed subgroup, Scott conjecture, Nielsen realisation, uniformly nonelementary, moderate action
Mathematical Subject Classification
Primary: 20F65, 20F67
Secondary: 20E36, 20F28, 20F34, 20F36, 20F55
References
Publication
Received: 25 February 2021
Revised: 10 March 2022
Accepted: 8 April 2022
Published: 27 February 2024
Proposed: Mladen Bestvina
Seconded: David Fisher, Urs Lang
Authors
Elia Fioravanti
Universität Bonn
Bonn
Germany

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