Coarse-median preserving automorphisms

Fioravanti, Elia

doi:10.2140/gt.2024.28.161

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Coarse-median preserving automorphisms

Elia Fioravanti

Geometry & Topology 28 (2024) 161–266

DOI: 10.2140/gt.2024.28.161

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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Volume 28, issue 1 (2024)