Amenable groups and Hadamard spaces with a totally disconnected isometry group

TL;DR

This paper characterizes amenable subgroups of totally disconnected groups acting on Hadamard spaces, linking their structure to topological and geometric properties, and introduces a refined boundary concept to analyze stabilizers.

Contribution

It provides a complete characterization of amenable subgroups in this setting and introduces a refined boundary to study stabilizer properties.

Findings

Amenable subgroups are (topologically locally finite)-by-(virtually abelian).

Stabilizers of points in the refined boundary are amenable.

The structure of the acting group G is closely tied to the geometric boundary.

Abstract

Let $X$ be a locally compact Hadamard space and $G$ be a totally disconnected group acting continuously, properly and cocompactly on $X$. We show that a closed subgroup of $G$ is amenable if and only if it is (topologically locally finite)-by-(virtually abelian). We are led to consider a set $\bdfine X$ which is a refinement of the visual boundary $\bd X$. For each $x \in \bdfine X$, the stabilizer $G_x$ is amenable.