Type semigroups for twisted groupoids and a dichotomy for groupoid C*-algebras

TL;DR

This paper introduces a new type semigroup framework for twisted groupoids, linking it to key properties of their C*-algebras and establishing a dichotomy for their classification, especially in the context of self-similar group actions.

Contribution

It develops a general theory of type semigroups for twisted, possibly non-Hausdorff groupoids and applies it to derive a dichotomy for associated C*-algebras, including Exel-Pardo algebras.

Findings

Type semigroup relates to traces, ideals, and infiniteness properties.

C*-algebra simplicity and semigroup conditions imply a dichotomy.

Explicit calculation for groupoids from self-similar group actions.

Abstract

We develop a theory of type semigroups for arbitrary twisted, not necessarily Hausdorff \'etale groupoids. The type semigroup is a dynamical version of the Cuntz semigroup. We relate it to traces, ideals, pure infiniteness, and stable finiteness of the reduced and essential C*-algebras. If the reduced C*-algebra of a twisted groupoid is simple and the type semigroup satisfies a weak version of almost unperforation, then the C*-algebra is either stably finite or purely infinite. We apply our theory to Cartan inclusions. We calculate the type semigroup for the possibly non-Hausdorff groupoids associated to self-similar group actions on graphs and deduce a dichotomy for the resulting Exel-Pardo algebras.