Almost algebraic actions of algebraic groups and applications to algebraic representations

TL;DR

This paper studies the dynamics of algebraic group actions on probability measures, revealing that stabilizers are almost algebraic and applying these results to algebraic representations and super-rigidity phenomena.

Contribution

It introduces the concept of almost algebraic stabilizers in the context of algebraic group actions and applies this to advance understanding of algebraic representations and super-rigidity.

Findings

Stabilizers of measures are almost algebraic.

Orbits are separated by open invariant sets.

Applications to algebraic representations and super-rigidity.

Abstract

Let G be an algebraic group over a complete separable valued field k. We discuss the dynamics of the G-action on spaces of probability measures on algebraic G-varieties. We show that the stabilizers of measures are almost algebraic and the orbits are separated by open invariant sets. We discuss various applications, including existence results for algebraic representations of amenable ergodic actions. The latter provides an essential technical step in the recent generalization of Margulis-Zimmer super-rigidity phenomenon due to Bader and Furman.