On groupoids beyond partial actions, inner amenability, and models for Kirchberg algebras

TL;DR

This paper constructs explicit examples of non-inner amenable, locally compact Hausdorff étale groupoids that are not derived from partial group actions, addressing open questions in the field.

Contribution

It provides the first explicit examples of such groupoids, including Higson--Lafforgue--Skandalis groupoids and their variants, expanding understanding of groupoid structures beyond partial actions.

Findings

Examples include all Higson--Lafforgue--Skandalis groupoids for non-amenable residually finite groups.

Many Deaconu--Renault groupoids with connected units do not come from partial actions.

Ample transformation groupoids model all unital Kirchberg algebras in the UCT class.

Abstract

We construct the first explicit examples of locally compact Hausdorff \'etale groupoids that are not inner amenable and that do not arise as transformation groupoids associated to partial actions of discrete groups. This answers questions of Anantharaman--Delaroche and Exel. Our examples include all Higson--Lafforgue--Skandalis groupoids associated to non-amenable residually finite groups, as well as their principal variants constructed by Alekseev--Finn--Sell. These can be chosen to be second countable, ample, and in the latter case even principal. We also show that large classes of Deaconu--Renault groupoids with connected unit space do not arise from partial actions of discrete groups, including cases whose $C^*$-algebras are Kirchberg algebras in the UCT class. We contrast this with the totally disconnected case by giving ample transformation groupoid models for all unital Kirchberg algebras in the UCT class as well as many higher rank graph algebras. Finally, we characterize precisely when coarse groupoids arise from partial actions of discrete groups in terms of coarse embeddings into groups.