Nuclear dimension of subhomogeneous twisted groupoid C*-algebras and dynamic asymptotic dimension

TL;DR

This paper characterizes subhomogeneity in twisted étale groupoid C*-algebras, provides bounds on their nuclear dimension, and applies these results to specific group actions, including those of the infinite dihedral group.

Contribution

It removes the principality assumption in nuclear dimension bounds and relates nuclear dimension to dynamic asymptotic dimension without this assumption.

Findings

Subhomogeneity characterized for twisted étale groupoid C*-algebras.

Upper bounds on nuclear dimension established.

Infinite dihedral group actions have dynamic asymptotic dimension one.

Abstract

We characterise subhomogeneity for twisted \'etale groupoid C*-algebras and obtain an upper bound on their nuclear dimension. As an application, we remove the principality assumption in recent results on upper bounds on the nuclear dimension of a twisted \'etale groupoid C*-algebra in terms of the dynamic asymptotic dimension of the groupoid and the covering dimension of its unit space. As a non-principal example, we show that the dynamic asymptotic dimension of any minimal (not necessarily free) action of the infinite dihedral group $D_\infty$ on an infinite compact Hausdorff space $X$ is always one. So if we further assume that $X$ is second-countable and has finite covering dimension, then $C(X)\rtimes_r D_\infty$ has finite nuclear dimension and is classifiable by its Elliott invariant.