Cocycle superrigidity for median spaces of finite rank

TL;DR

This paper proves cocycle superrigidity results for actions on finite rank median spaces, extending superrigidity phenomena to new geometric settings using a dynamical approach.

Contribution

It introduces a dynamical method to establish cocycle superrigidity for median spaces of finite rank, including new proofs for lattice superrigidity.

Findings

Superrigidity of Isom(X)-valued cocycles for product groups

New proof of superrigidity of homomorphisms for lattices

Extension of superrigidity to median spaces of finite rank

Abstract

We systematically investigate cocycle superrigidity in the setting of finite rank median spaces for product groups and Kazhdan groups. By employing a dynamical approach to superrigidity, we establish, for a median space X of finite rank, the superrigidity of Isom(X)-valued cocycles for a product of locally compact second countable groups. In the case of actions by irreducible lattices in such product groups, this approach yields a novel proof of the superrigidity of homomorphisms.