Cross ratios on $\mathrm{CAT}(0)$ cube complexes and marked length-spectrum rigidity

Abstract

We show that group actions on irreducible $\mathrm{CAT}(0)$ cube complexes with no free faces are uniquely determined by their ${\ell }^1$ length function. Actions are allowed to be non-proper and non-cocompact, as long as they are minimal and have no finite orbit in the visual boundary. This is, to our knowledge, the first length-spectrum rigidity result in a setting of non-positive curvature (with the exception of some particular cases in dimension 2 and symmetric spaces). As our main tool, we develop a notion of cross ratio on Roller boundaries of $\mathrm{CAT}(0)$ cube complexes. Inspired by results in negative curvature, we give a general framework reducing length-spectrum rigidity questions to the problem of extending cross-ratio preserving maps between (subsets of) Roller boundaries. The core of our work is then to show that, when there are no free faces, these cross-ratio preserving maps always extend to cubical isomorphisms. All our results equally apply to cube complexes with variable edge lengths. As a special case of our work, we construct a compactification of the Charney-Stambaugh-Vogtmann Outer Space for the group of untwisted outer automorphisms of an (irreducible) right-angled Artin group. This generalises the length function compactification of the classical Culler-Vogtmann Outer Space.

51 pages, 1 figure; minor changes to the exposition, to appear on J. Lond. Math. Soc

References in corpus (10)

• Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups
• Cross ratios and cubulations of hyperbolic groups
• Roller boundaries for median spaces and algebras
• $\mathrm{CAT}(0)$ cube complexes are determined by their boundary cross ratio
• Panel collapse and its applications
• The Tits alternative for finite rank median spaces
• Large facing tuples and a strengthened sector lemma
• Superrigidity of actions on finite rank median spaces
• Convex cores for actions on finite-rank median algebras
• Deforming cubulations of hyperbolic groups

Cited by in corpus (5)

• Cross ratios and cubulations of hyperbolic groups
• $\mathrm{CAT}(0)$ cube complexes are determined by their boundary cross ratio
• Coarse-median preserving automorphisms
• Deforming cubulations of hyperbolic groups
• Coarse cubical rigidity