Superrigidity of actions on finite rank median spaces

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 $H_{FD}$.