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arXiv · 2610.06713

The Almost Finite--Purely Infinite Dichotomy\\ for Minimal Amenable Ample Groupoids

Gábor Elek·Ádám Timár
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Abstract

We answer, in the principal setting, a question of Matui on the almost finite--purely infinite dichotomy for minimal amenable ample groupoids. We prove that a second-countable Hausdorff minimal principal topologically amenable ample groupoid with Cantor unit space is strongly almost finite if and only if it admits an invariant probability measure; if no such measure exists, then it is purely infinite. We also prove that every bounded-degree uniformly Borel amenable Følner graph is Borel almost finite. The proofs combine the recent three-to-two comparison method of Glasner and Liu with the randomized Følner packing methods of Elek and Timár.

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Gábor Elek, Ádám Timár. 2026-10-05. The Almost Finite--Purely Infinite Dichotomy\\ for Minimal Amenable Ample Groupoids. https://arxiv.org/abs/2610.06713

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