Ample groupoids that are neither almost finite nor purely infinite

TL;DR

This paper constructs new examples of minimal ample groupoids that are neither almost finite nor purely infinite, expanding understanding of their properties beyond transformation groupoids.

Contribution

It introduces the first known essentially principal ample groupoids with these properties, using recent twisted topological groupoid constructions.

Findings

Existence of effective minimal ample transformation groupoids that are neither almost finite nor purely infinite.

Construction of essentially principal ample groupoids with these properties, not arising from transformation groupoids.

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.