Superrigidity for representations of transverse measured groupoids

TL;DR

This paper establishes superrigidity results for Zariski dense representations of transverse groupoids associated with higher rank semisimple algebraic groups acting ergodically, extending rigidity phenomena to a broad class of algebraic group actions.

Contribution

It proves a superrigidity phenomenon for Zariski dense representations of transverse groupoids in higher rank semisimple algebraic groups, generalizing previous rigidity results.

Findings

Superrigidity holds for Zariski dense representations in higher rank groups.

Results apply to ergodic transverse systems with positive rank factors.

Extends rigidity phenomena to a broad class of algebraic group actions.

Abstract

For $i=1,\ldots,k$, let $\mathbf{G}_i$ be a connected, simply connected, semisimple algebraic group over some local field $\kappa_i$ of characteristic zero. Let $G_i=\mathbf{G}_i(\kappa_i)$ be the $\kappa_i$-points of $\mathbf{G}_i$ and denote by $G=\prod_{i=1}^k G_i$. If we assume that $G$ has higher rank and each factor has positive rank, given an ergodic transverse $G$-system $(X,\mu,Y)$, we prove a superrigidity phenomenon for Zariski dense representations of the transverse groupoid $(G \ltimes X)|_Y$ into either an almost simple or a reductive algebraic group.