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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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Cite

BibTeXRIS
BibTeX
@misc{elek2026almost,
  title = {{The Almost Finite--Purely Infinite Dichotomy for Minimal Amenable Ample Groupoids}},
  author = {Gábor Elek and Ádám Timár},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06713},
  eprint = {2610.06713},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06713},
}
RIS
TY  - UNPB
TI  - The Almost Finite--Purely Infinite Dichotomy for Minimal Amenable Ample Groupoids
AU  - Gábor Elek
AU  - Ádám Timár
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.06713
UR  - https://arxiv.org/abs/2610.06713
AB  - 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.
ER  -