Coarse-median preserving automorphisms

Fioravanti, Elia¹

¹ Universität Bonn, Bonn, Germany

Geometry & topology, Tome 28 (2024) no. 1, pp. 161-266.

Résumé

This paper has three main goals.

First, we study fixed subgroups of automorphisms of right-angled Artin and Coxeter groups. If $\phi$ is an untwisted automorphism of a RAAG, or an arbitrary automorphism of a RACG, we prove that $\mathrm{Fix}\phi$ is finitely generated and undistorted. Up to replacing $\phi$ with a power, we show that $\mathrm{Fix}\phi$ is quasiconvex with respect to the standard word metric. This implies that $\mathrm{Fix}\phi$ 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 $\mathrm{Fix}\phi$ 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.

Détail

DOI : 10.2140/gt.2024.28.161

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

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