The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes

Abstract

We define a bounded cohomology class, called the {\em median class}, in the second bounded cohomology -- with appropriate coefficients --of the automorphism group of a finite dimensional CAT(0) cube complex X. The median class of X behaves naturally with respect to taking products and appropriate subcomplexes and defines in turn the {\em median class of an action} by automorphisms of X. We show that the median class of a non-elementary action by automorphisms does not vanish and we show to which extent it does vanish if the action is elementary. We obtain as a corollary a superrigidity result and show for example that any irreducible lattice in the product of at least two locally compact connected groups acts on a finite dimensional CAT(0) cube complex X with a finite orbit in the Roller compactification of X. In the case of a product of Lie groups, the Appendix by Caprace allows us to deduce that the fixed point is in fact inside the complex X. In the course of the proof we construct a Γ-equivariant measurable map from a Poisson boundary of Γwith values in the non-terminating ultrafilters on the Roller boundary of X.

Minor changes that clarify some confusion have been made. Some figures have been added

References in corpus (6)

• Group actions with commensurated subsets, wallings and cubings
• Bounded cohomology with coefficients in uniformly convex Banach spaces
• Superrigidity, Weyl groups, and actions on the Circle
• Irreducible lattices, invariant means, and commensurating actions
• Coarse decompositions of boundaries for CAT(0) groups
• Isometry groups of proper CAT(0)-spaces

Cited by in corpus (26)

• On hierarchical hyperbolicity of cubical groups
• Automorphisms of graph products of groups from a geometric perspective
• Bounded cohomology with coefficients in uniformly convex Banach spaces
• Cross ratios and cubulations of hyperbolic groups
• Extending higher dimensional quasi-cocycles
• Roller boundaries for median spaces and algebras
• Cross ratios on $\mathrm{CAT}(0)$ cube complexes and marked length-spectrum rigidity
• $\mathrm{CAT}(0)$ cube complexes are determined by their boundary cross ratio
• Bounded cohomology via partial differential equations, I
• The Tits alternative for finite rank median spaces
• Measure equivalence classification of transvection-free right-angled Artin groups
• Superrigidity of actions on finite rank median spaces
• Coarse-median preserving automorphisms
• Cubulable Kähler groups
• Higher rank lattices are not coarse median
• A note on locally elliptic actions on cube complexes
• Convex cores for actions on finite-rank median algebras
• On automorphisms and splittings of special groups
• Acylindrical hyperbolicity from actions on CAT(0) cube complexes: a few criteria
• Corrigendum to "The simplicial boundary of a CAT(0) cube complex''
• Deforming cubulations of hyperbolic groups
• Homotopy equivalent boundaries of cube complexes
• Coarse cubical rigidity
• Automorphism groups of cyclic products of groups
• Cocycle superrigidity from higher rank lattices to $\mathrm{Out}(F_N)$
• Median quasimorphisms on CAT(0) cube complexes and their cup products