Dynamic Asymptotic Dimension of Translation Actions on Compact Lie Groups

TL;DR

This paper introduces a method to bound the dynamic asymptotic dimension of translation actions on compact Lie groups using geometric properties and applies it to analyze the nuclear dimension of associated $C^*$-algebras.

Contribution

It develops a novel approach linking dynamic asymptotic dimension with geometric and graph-theoretic properties, specifically applied to translation actions on compact Lie groups.

Findings

Bound the dynamic asymptotic dimension using graph spaces and doubling dimension.

Characterize finite dimensionality of translation actions on compact Lie groups.

Bound the nuclear dimension of related $C^*$-algebras.

Abstract

We develop a method to bound the dynamic asymptotic dimension of isometric group actions $\Gamma\curvearrowright X$ in terms of the asymptotic dimension of a space of graphs similar to a box space of $\Gamma$ (which also determines finite-dimensionality), and a geometric property of $X$ related to the doubling dimension. We apply this method to describe the DAD of translation actions by finitely generated subgroups of compact Lie groups, characterize finite dimensionality of such actions, and consequently bound the nuclear dimension of $C^*$-algebras arising from such actions by amenable groups.