Ample groupoids that are neither almost finite nor purely infinite

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.