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Superrigidity for representations of transverse measured groupoids

Filippo Sarti, Alessio Savini

Abstract

For $i=1,\ldots,k$, let $\mathbf{G}_i$ be a connected, simply connected, semisimple algebraic group over some local field $κ_i$ of characteristic zero. Let $G_i=\mathbf{G}_i(κ_i)$ be the $κ_i$-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 $(X,μ,Y)$, 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.

Superrigidity for representations of transverse measured groupoids

Abstract

For , let be a connected, simply connected, semisimple algebraic group over some local field of characteristic zero. Let be the -points of and denote by . If we assume that has higher rank and each factor has positive rank, given an ergodic transverse -system , we prove a superrigidity phenomenon for Zariski dense representations of the transverse groupoid into either an almost simple or a reductive algebraic group.
Paper Structure (8 sections, 10 theorems, 58 equations)

This paper contains 8 sections, 10 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Transverse measured groupoids
  3. Main results
  4. Measured groupoids
  5. Measured groupoids and morphisms
  6. Transverse systems
  7. Algebraic representability of ergodic groupoids
  8. Zariski dense representations

Key Result

Theorem 1

Suppose that $r_i>0$, for $i=1,\ldots, \ell$, and $r \geq 2$. Let $\kappa$ be equal either to $\mathbb{R},\mathbb{C}$ or $\mathbb{Q}_p$ and consider $H=\mathbf{H}(\kappa)$ the $\kappa$-points of an almost $\kappa$-simple $\kappa$-algebraic group. Consider a Zariski dense representation $\rho:\mathca

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2.1
  • Definition 2.2
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6: Refined Campbell theorem, ABC
  • Lemma 2.7
  • proof
  • ...and 16 more