Almost elementariness and fiberwise amenability for \'etale groupoids

TL;DR

This paper introduces almost elementariness and fiberwise amenability as new approximation properties for étale groupoids, unifying existing notions of almost finiteness and extending their applicability to purely infinite cases, with implications for C*-algebra regularity.

Contribution

It defines two new properties for étale groupoids, showing their equivalence to almost finiteness in certain contexts and linking them to C*-algebra regularity.

Findings

Almost elementary minimal groupoids yield tracially Z-stable C*-algebras.

Reduced C*-algebras of second countable amenable almost finite groupoids are Z-stable.

Fiberwise amenability relates to invariant measures and group amenability.

Abstract

In this paper, we introduce two new approximation properties for \'etale groupoids, almost elementariness and (ubiquitous) fiberwise amenability, inspired by Matui's and Kerr's notions of almost finiteness. In fact, we show that, in their respective scopes of applicability, both notions of almost finiteness are equivalent to the conjunction of our two properties. These new properties stem from viewing \'etale groupoids as coarse geometric objects in the spirit of geometric group theory. Fiberwise amenability is a coarse geometric property of \'etale groupoids that is closely related to the existence of invariant measures on unit spaces and corresponds to the amenability of the acting group in a transformation groupoid. Almost elementariness may be viewed as a better dynamical analogue of the regularity properties of C*-algebras than almost finiteness, since, unlike the latter, the former may also be applied to the purely infinite case. To support this analogy, we show almost elementary minimal groupoids give rise to tracially Z-stable reduced groupoid C*-algebras. In particular, the reduced C*-algebras of second countable amenable almost finite groupoids in Matui's sense are Z-stable.