Superrigidity of actions on finite rank median spaces

Abstract

Finite rank median spaces are a simultaneous generalisation of finite dimensional $\mathrm{CAT}(0)$ 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 $\mathrm{CAT}(0)$ cube complexes, only second bounded cohomology classes were known with this property, due to Chatterji-Fernós-Iozzi. As a corollary, we observe that, in Gromov's density model, random groups at low density do not have Shalom's property $H_{FD}$.

46 pages, 3 figures; final version, to appear on Adv Math

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Cited by in corpus (7)

• Cross ratios and cubulations of hyperbolic groups
• Cross ratios on $\mathrm{CAT}(0)$ cube complexes and marked length-spectrum rigidity
• Roller boundaries for median spaces and algebras
• The Tits alternative for finite rank median spaces
• Coarse-median preserving automorphisms
• Deforming cubulations of hyperbolic groups
• Coarse cubical rigidity