Amenable actions of real and $p$-adic algebraic groups

TL;DR

This paper characterizes when the action of a linear algebraic group over a local field on its rational points is topologically amenable, linking it to the solvability of point stabilizers, and extends known results on group actions.

Contribution

It establishes a precise criterion for topological amenability of algebraic group actions over local fields based on stabilizer structure, combining algebraic and topological methods.

Findings

Action is topologically amenable iff all stabilizers are solvable-by-compact

Connects amenability of group actions to stabilizer properties in algebraic groups

Extends Borel-Serre results to p-adic and real algebraic groups

Abstract

Let $K$ be a locally compact field of characteristic 0. Let $G$ be a linear algebraic group defined over $K$, acting algebraically on an algebraic variety $V$. We prove that the action of $G(K)$ (the group of $K$-rational points of $G$) on $V(K)$ is topologically amenable, if and only if all points stabilizers in $G(K)$ are solvable-by-compact. This follows by combining a result by Borel-Serre \cite{BoSe} with the following fact: let $G$ be a second countable locally compact group acting continuously on a second countable locally compact space $Y$. If the action $G\curvearrowright Y$ is smooth (i.e. the Borel structure on $G\backslash Y$ is countably separated), then topological amenability of $G\curvearrowright Y$ is equivalent to amenability of all point stabilizers in $G$.