Coarse cubical rigidity

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 $\ge 3$, 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.

42 pages, 7 figures; to appear in Journal of Topology

References in corpus (18)

• Rank rigidity for CAT(0) cube complexes
• Contracting Boundaries of CAT(0) Spaces
• Automorphisms of two-dimensional right-angled Artin groups
• Weakly modular graphs and nonpositive curvature
• The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes
• On hierarchical hyperbolicity of cubical groups
• Automorphisms of graph products of groups from a geometric perspective
• Outer space for untwisted automorphisms of right-angled Artin groups
• Finite asymptotic dimension for CAT(0) cube complexes
• Cross ratios and cubulations of hyperbolic groups
• Groups quasi-isometric to RAAG's
• Cross ratios on $\mathrm{CAT}(0)$ cube complexes and marked length-spectrum rigidity
• Superrigidity of actions on finite rank median spaces
• Coarse-median preserving automorphisms
• Cocompactly cubulated crystallographic groups
• Convex cores for actions on finite-rank median algebras
• On automorphisms and splittings of special groups
• Deforming cubulations of hyperbolic groups

Cited by in corpus (1)

• Divisible cube complexes and finite-order automorphisms of RAAGs