Superrigidity of actions on finite rank median spaces

Abstract

Finite rank median spaces are a simultaneous generalisation of finite dimensional CAT(0) cube complexes and real trees. If Γ is an irreducible lattice in a product of rank-one simple Lie groups, we show that every action of Γ on a complete, finite rank median space has a global fixed point. This is in sharp contrast with the behaviour of actions on infinite rank median spaces, where even proper cocompact actions can arise.The fixed point property is obtained as corollary to a superrigidity result; the latter holds for irreducible lattices in arbitrary products of compactly generated topological groups.We exploit Roller compactifications of median spaces; these were introduced in [48] and generalise a well-known construction in the case of cube complexes. We show that the Haagerup cocycle provides a reduced 1-cohomology class that detects group actions with a finite orbit in the Roller compactification. This is new even for CAT(0) cube complexes and has interesting consequences involving Shalom's property HFD. For instance, in Gromov's density model, random groups at low density do not have HFD.