Coarse cubical rigidity

TL;DR

This paper demonstrates that many right-angled Artin and Coxeter groups have a unique coarse median structure across all cocompact cubulations, revealing a form of geometric rigidity in non-hyperbolic groups.

Contribution

It establishes the coarse cubical rigidity for broad classes of right-angled Artin and Coxeter groups, a novel property not previously known for non-hyperbolic groups.

Findings

All cocompact cubulations induce the same coarse median structure.

Automorphisms preserve the coarse median structure in these groups.

Automorphisms have fixed subgroups and satisfy Nielsen realization.

Abstract

We show that for many right-angled Artin and Coxeter groups, all cocompact cubulations coarsely look the same: they induce the same coarse median structure on the group. These are the first examples of non-hyperbolic groups with this property. For all graph products of finite groups and for Coxeter groups with no irreducible affine parabolic subgroups of rank $\geq 3$, we show that all automorphism preserve the coarse median structure induced, respectively, by the Davis complex and the Niblo-Reeves cubulation. As a consequence, automorphisms of these groups have nice fixed subgroups and satisfy Nielsen realisation.