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

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.