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Does every bounded-to-one action of a finitely generated commutative monoid on a Polish space generate a hyperfinite equivalence relation?

Related: Weiss hyperfiniteness theorem for amenable group actions, Slaman-Steel theorem on hyperfinite relations, Dougherty-Jackson-Kechris classification of hyperfinite Borel equivalence relations

Descriptive set theory studies the complexity of equivalence relations on Polish spaces, which are complete separable metric spaces. An equivalence relation is called hyperfinite if it can be written as an increasing union of finite Borel equivalence relations, making it the simplest non-trivial level of complexity. The paper on commutative monoids establishes hyperfiniteness for certain bounded-to-one actions, but the full picture for all finitely generated commutative monoids acting in a bounded-to-one fashion remains incomplete, particularly when the monoid has more complex algebraic structure or when the bounding condition is relaxed in natural ways.

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