Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Elek, Gábor, Timár, Ádám
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220719225