Almost elementary étale groupoids

Abstract

Motivated by Matui and Kerr's work on almost finiteness, we introduce a new finite approximation property for (possibly non-ample) étale groupoids called {almost elementariness}, which unifies and generalizes both almost finiteness and pure infiniteness. This property serves as a dynamical analogue of regularity properties of $C^{\ast }$-algebras. In support of this view, we prove that minimal almost elementary groupoids yield tracially $\mathcal{Z}$-stable reduced groupoid $C^{\ast }$-algebras. Consequently, we obtain as a corollary that the reduced $C^{\ast }$-algebras of all minimal amenable second countable almost finite groupoids in Matui's sense are $\mathcal{Z}$-stable and thus classifiable by the Elliott invariants. Two basic ingredients underlying the definition of almost elementariness are castles in groupoids and groupoid subequivalence, both of which are developed extensively in this work. Notably, building on our flexible framework of castles, we introduce the technique of nesting of castles, which play a key role in the proof of our main theorem. Besides our main theorem on tracial $\mathcal{Z}$-stability, we also discuss in depth the relations between almost elementariness and other properties for groupoids such as effectiveness, the groupoid small boundary property, groupoid strict comparison and Matui's and Kerr's notions of almost finiteness.

87 pages. The original paper has been split into two parts, with the first part appearing separately and focusing only on fiberwise amenability. This new version contains only the second part, which focuses on almost elementariness. It has also been expanded and extensively rewritten in a new, cleaner framework of castles, though the main proof ideas stay the same

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Cited by in corpus (2)

• The type semigroup, comparison and almost finiteness for ample groupoids
• Fiberwise amenability of ample étale groupoids