On Hausdorff covers for non-Hausdorff groupoids
Kevin Aguyar Brix, Julian Gonzales, Jeremy B. Hume, Xin Li
TL;DR
The paper develops a Hausdorff-cover approach to non-Hausdorff étale groupoids, enabling a systematic reduction to the Hausdorff setting via the Hausdorff cover $ ilde{G}$. It provides complete characterisations for the vanishing of singular ideals in Steinberg algebras over arbitrary rings and conditions for the $C^*$-algebraic singular ideal to have trivial intersection with the non-Hausdorff analogue of $C_c(G)$, with finiteness hypotheses yielding further equivalences. By transferring questions to the Hausdorff cover, the work reduces complications around simplicity, the ideal intersection property, amenability and nuclearity to the well-understood Hausdorff case, and it offers decomposability criteria that describe the images of the natural embeddings. The results are illustrated through diverse examples—group bundles, Thompson groups, and self-similar groups—highlighting when singular ideals vanish and how the Hausdorff-cover perspective clarifies known phenomena and enables partial resolutions of CEPSS-type questions.
Abstract
We develop a new approach to non-Hausdorff étale groupoids and their algebras based on Timmermann's construction of Hausdorff covers. As an application, we completely characterise when singular ideals vanish in Steinberg algebras over arbitrary rings. We also completely characterise when $C^*$-algebraic singular ideals have trivial intersection with the non-Hausdorff analogue of subalgebras of continuous, compactly supported functions. This leads to a characterisation when $C^*$-algebraic singular ideals vanish for groupoids satisfying a finiteness condition. Moreover, our approach leads to further sufficient vanishing criteria for singular ideals and reduces questions about simplicity, the ideal intersection property, amenability and nuclearity for non-Hausdorff étale groupoids to the Hausdorff case.