The scale function for locally compact groups acting on non-positively curved spaces

TL;DR

This paper explores the scale function for totally disconnected locally compact groups acting on non-positively curved spaces, providing geometric insights into group dynamics and criteria for scale values.

Contribution

It offers geometric characterizations of the scale, parabolic and contraction groups, and tidy subgroups in the context of groups acting on CAT(0) and hyperbolic spaces, extending Willis's theory.

Findings

Geometric descriptions of parabolic and contraction groups

Criteria for elements to have scale 1

Structural insights into group actions on non-positively curved spaces

Abstract

Let $G$ be a totally disconnected, locally compact (t.d.l.c.) group. The scale $s_G(g)$ of $g \in G$ in the sense of Willis is given by the minimum value of the index $|gUg^{-1}:U \cap gUg^{-1}|$ as $U$ ranges over the compact open subgroups; the theory associated to the scale has been very successful in describing general dynamical features of automorphisms of t.d.l.c. groups. We focus on the case where $G$ acts properly and continuously by isometries on a geodesic space $X$, where $X$ is complete CAT(0) or proper and Gromov-hyperbolic, and $g \in G$ is hyperbolic. In this context, we find geometric descriptions of the parabolic and contraction groups, tidy subgroups, and structures in the $G$-action that encode the scale, including criteria for $g$ to have scale $1$.