Download this article
 Download this article For screen
For printing
Recent Issues

Volume 28
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
Coarse-median preserving automorphisms

Elia Fioravanti

Geometry & Topology 28 (2024) 161–266

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.

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
Received: 25 February 2021
Revised: 10 March 2022
Accepted: 8 April 2022
Published: 27 February 2024
Proposed: Mladen Bestvina
Seconded: David Fisher, Urs Lang
Elia Fioravanti
Universität Bonn

Open Access made possible by participating institutions via Subscribe to Open.