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Title: The structure of hyperfinite subequivalence relations of treed equivalence relations.

Abstract: A large part of measured group theory studies structural properties of countable groups that hold "on average". This is made precise by studying the orbit equivalence relations induced by free measurable actions of these groups on a standard probability space. In this vein, the amenable groups correspond to hyperfinite equivalence relations, and the free groups to the treeable ones. In joint work with R. Tucker-Drob, we give a detailed analysis of the structure of hyperfinite subequivalence relations of a treed equivalence relation on a standard probability space, deriving the analogues of structural properties of amenable subgroups (copies of Z) of a free group. Most importantly, just like every such subgroup is contained in a unique maximal one, we show that even in the non-measure-preserving setting, every hyperfinite subequivalence relation is contained in a unique maximal one.

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