Matui’s almost finite–purely infinite dichotomy
For every second-countable Hausdorff minimal topologically amenable ample groupoid with Cantor unit space, is either almost finite or purely infinite?
Equivalent formulations 1Other wordings
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
Principal-case invariant-measure formulation
For every 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 it admits no invariant probability measure, then is purely infinite.
source: Elek and Timár, “The Almost Finite–Purely Infinite Dichotomy for Minimal Amenable Ample Groupoids”
References
Primary source
Additional references
- The Almost Finite--Purely Infinite Dichotomy for Minimal Amenable Ample Groupoids — arXiv — Gábor Elek, Ádám Timár
Progress summary
A new unrefereed preprint claims the classification in the principal case, but the claim has not been independently checked.
Matui’s dichotomy asks whether the relevant minimal amenable ample groupoids fall into the almost-finite or purely infinite alternatives. The reported advance addresses this question under a principalness assumption and also gives a related Borel result.
October 2026 principal-case preprint
Gábor Elek and Ádám Timár claim the dichotomy for second-countable Hausdorff minimal principal topologically amenable ample groupoids, together with a related Borel almost-finiteness theorem. The result is restricted to the principal setting and appears only as an unrefereed preprint, so it is reported here as unverified progress.
Current status (as of October 2026): The principal-case dichotomy is claimed in an unrefereed preprint, but no independently confirmed resolution was found.
Sources
Solutions 0
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