Superrigidity of actions on finite rank median spaces

TL;DR

The paper proves a superrigidity and fixed point property for actions of certain lattices on finite rank median spaces, contrasting with infinite rank cases, and introduces new cohomological tools for analyzing group actions.

Contribution

It establishes superrigidity results for irreducible lattices acting on finite rank median spaces and introduces Roller compactifications with cohomological detection methods.

Findings

Irreducible lattices have fixed points on finite rank median spaces.

New cohomology classes detect finite orbits in Roller compactifications.

Low-density random groups lack Shalom's property H_{FD}.

Abstract

Finite rank median spaces are a simultaneous generalisation of finite dimensional ${\rm CAT}(0)$ cube complexes and real trees. If $\Gamma$ is an irreducible lattice in a product of rank one simple Lie groups, we show that every action of $\Gamma$ 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. The fixed point property is obtained as corollary to a superrigidity result; the latter holds for irreducible lattices in arbitrary products of compactly generated groups. In previous work, we introduced "Roller compactifications" of median spaces; these generalise a well-known construction in the case of cube complexes. We provide a reduced $1$-cohomology class that detects group actions with a finite orbit in the Roller compactification. Even for ${\rm CAT}(0)$ cube complexes, only second bounded cohomology classes were known with this property, due to Chatterji-Fern\'os-Iozzi. As a corollary, we observe that, in Gromov's density model, random groups at low density do not have Shalom's property $H_{FD}$.