Ample groupoids that are neither almost finite nor purely infinite

TL;DR

This work addresses the existence of essentially principal, minimal, amenable ample groupoids that are neither almost finite nor purely infinite and do not arise from transformation groupoids. It develops a twisted topological groupoid framework, using a non-amenable group $\Gamma$ and an essentially finite groupoid $\mathcal{G}$ to form the twisted groupoid $\Gamma\mathcal{G}\rtimes\Gamma$, which is shown to be not almost finite and not purely infinite, yet can be essentially principal in the unique-invariant-measure case. The paper provides explicit non-transformation examples (e.g., with $\Gamma=\mathbb{F}_2$ and $\mathcal{G}=\mathcal{E}_n$) and thus delivers the first concrete counterexamples to Matui’s question beyond transformation groupoids. These results advance the understanding of the landscape of minimal ample groupoids and their connections to $C^*$-algebraic classification.

Abstract

We study a question of Matui and varations of it on minimal ample groupoids that are neither almost finite nor purely infinite. We first observe that there are already effective minimal ample transformation groupoids that are neither almost finite nor purely infinite. These groupoids can even be chosen to be amenable. Then we construct essentially principle ample groupoids that are neither almost finite nor purely infinite. These are based on the recent twisted topological groupoid construction of Palmer and Wu. In particular our new examples do not arise from transformation groupoids.

Theorems & Definitions (51)

• Theorem 1
• Theorem 2
• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Definition 1.6
• Proposition 1.7
• proof