Coarse-median preserving automorphisms

This paper has three main goals.First, we study fixed subgroups of automorphisms of right-angled Artin and Coxeter groups.If ' is an untwisted automorphism of a RAAG, or an arbitrary automorphism of a RACG, we prove that Fix ' is finitely generated and undistorted.Up to replacing ' with a power, we show that Fix ' is quasiconvex with respect to the standard word metric.This implies that Fix ' 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 ' 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. 20F65, 20F67; 20E36, 20F28, 20F34, 20F36, 20F55 1. Introduction 162 2. Preliminaries 174 3. Cubical convex-cocompactness 189 4. Fixed subgroups of CMP automorphisms 203 5. Invariant splittings of RAAGs 223 6. Projectively invariant metrics on finite-rank median algebras 230 7. Ultralimits and coarse-median preserving automorphisms 240 Appendix.Measurable partitions of halfspace-intervals 260