The type semigroup, comparison and almost finiteness for ample groupoids

TL;DR

This paper establishes a deep connection between the algebraic structure of type semigroups and the dynamical property of comparison in ample groupoids, explores conditions for almost finiteness, and links coarse geometry with topological dynamics to generate new examples of almost finite groupoids.

Contribution

It characterizes when a minimal ample groupoid has comparison via its type semigroup and introduces a coarsely invariant property to analyze almost finiteness in non-minimal cases.

Findings

Dynamical comparison is equivalent to the type semigroup being almost unperforated in minimal ample groupoids.

Constructs new examples of almost finite principal groupoids without amenability or a-T-menability.

Links coarse geometry and topological dynamics through a new invariant, broadening understanding of groupoid properties.

Abstract

We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions.