Alexandrov groupoids and the nuclear dimension of twisted groupoid $\mathrm{C}^*$-algebras

Abstract

We consider a twist $E$ over an \'etale groupoid $G$. When $G$ is principal, we prove that the nuclear dimension of the reduced twisted groupoid $\mathrm{C}^*$-algebra is bounded by a number depending on the dynamic asymptotic dimension of $G$ and the topological covering dimension of its unit space. This generalizes an analogous theorem by Guentner, Willett, and Yu for the $\mathrm{C}^*$-algebra of $G$. Our proof uses a reduction to the unital case where $G$ has compact unit space, via a construction of ``groupoid unitizations'' $\widetilde{G}$ and $\widetilde{E}$ of $G$ and $E$ such that $\widetilde{E}$ is a twist over $\widetilde{G}$. The construction of $\widetilde G$ is for r-discrete (hence \'etale) groupoids $G$ which are not necessarily principal. When $G$ is \'etale, the dynamic asymptotic dimension of $G$ and $\widetilde{G}$ coincide. We show that the minimal unitizations of the full and reduced twisted groupoid $\mathrm{C}^*$-algebras of the twist over $G$ are isomorphic to the twisted groupoid $\mathrm{C}^*$-algebras of the twist over $\widetilde{G}$. We apply our result about the nuclear dimension of the twisted groupoid $\mathrm{C}^*$-algebra to obtain a similar bound on the nuclear dimension of the $\mathrm{C}^*$-algebra of an \'etale groupoid with closed orbits and abelian stability subgroups that vary continuously.