Similarity and superrigidity for transverse measured groupoids

Abstract

Let $G$ be a locally compact, second countable, unimodular group and let $(X,\mu ,Y)$ be an ergodic integrable transverse $G$-system. We show that $G⋉X$ endowed with the product measure and the restriction $(G⋉X){|}_Y$ endowed with the transverse measure are similar ergodic groupoids. As a consequence, we prove a superrigidity phenomenon for higher rank transverse groupoids. More precisely, for $i=1,\dots ,k$, 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 integrable transverse $G$-system $(X,\mu ,Y)$, we prove that any unbounded Zariski dense morphism of the transverse groupoid $(G⋉X){|}_Y$ into either an almost simple or a reductive algebraic group is superrigid.

arXiv comment17 pages, we improved the exposition