Countable Borel equivalence relations

The usual notion of cardinality entails that one set is smaller than another if there is an injection of the former into the latter. Over the last few decades, a finer notion of definable cardinality has emerged, whereby one requires that the injection is suitably definable. Broadly speaking, the goal of this project is to enhance our knowledge of the base of the definable cardinality hierarchy. In particu- lar, we focus on countable Borel equivalence relations on Polish spaces, ordered by Borel reducibility. One of our main interests concerns whether the theory of cost, as developed primarily by Gaboriau, can be extended to quasi-invariant Borel probability measures. Such an extension could well lead to a strong dynamical version of the von Neumann conjecture, as well as to a substantial connection between measure hyperfiniteness and invariant-measure hyperfiniteness. The latter would yield an optimal measure-theoretic strengthening of a recent result of Conley-Miller concerning the point in the Borel reducibility hierarchy below which Borel embeddability and Borel reducibility no longer coincide (modulo trivial counterexamples). Another focal point concerns the existence of suitably minimum and minimal non-measure- hyperfinite countable Borel equivalence relations. Recent results of Conley-Miller ensure that such questions are strongly linked to the study of Borel probability measures witnessing the failure of hyperfiniteness for orbit equivalence relations induced by natural actions of linear algebraic groups. In particular, we seek to determine whether the orbit equivalence relation induced by the usual action of the special linear group of degree two over the integers on the torus is a minimal non- measure-hyperfinite countable Borel equivalence relation (under measure reducibility). A final major interest concerns Borel combinatorics, a topic originating in the study of definable cardinality. Here we focus on a number of longstanding open questions concerning the existence of colorings and matchings of locally finite Borel graphs in the measure-theoretic context. We are also curious as to whether the underlying techniques can be employed to obtain further strengthenings of recent results of Conley-Miller concerning the base of the measure-reducibility hierarchy.

The most important results from the project include the following: (1) A characterization of the existence of quasi-invariant probability measures of a given cocycle. Such results go back to seminal work of Hopfs on non-singular transformations in the 1930s, and include generalizationsdue to Nadkarni and Becker-Kechristo Borel automorphisms and equivalence relations, roughly fifty years later. The new characterization is a substantial generalization of these results, and the underlying arguments also yield a new proof of Ditzens uniform ergodic decomposition theorem. (2) A dichotomy theorem characterizing the class of analytic graphs on Hausdorff spaces that admit a Borel two-coloring (joint with Carroy, Schrittesser, and Vidnynszky). While it is easy to see that a finite graph admits a two-coloring if and only if it does not contain an odd cycle, the existence of definable two- colorings of infinite graphs is far more subtle. We produced an acyclic two-regular Borel graph L on a Polish space with the property that an analytic graph G on a Hausdorff space admits a Borel two-coloring if and only if there is no continuous homomorphism from L to G. Under determinacy, similar arguments show that a graph G on an analytic Hausdorff space admits a two-coloring if and only if there is no homomorphism from L to G. (3) A series of theorems concerning a broad family of recurrence conditions (joint with Inselmann), highlights of which include:(a) Broad generalizations of the Glimm-Effros dichotomy, yielding countable bases for a wide variety of recurrence conditions. (b) The impossibility of using recurrence conditions to characterize the existence of invariant Borel probability measures in the descriptive set-theoretic context.(c) A new connection between the measure-theoretic and topological notions of weak mixing. (4) Applications of an infinite- dimensional analog of the open graph dichotomy to obtain a series of basis results (joint with Carroy and Soukup), including generalizations of Hurewiczs and Kechris-Louveau-Woodins characterizations of countable unions of closed sets, Lecomte-Zelenys two-dimensional analog thereof, the strengthening of the Jayne-Rogers Theorem from analytic to separable metric spaces under the axiom of determinacy, and basis theorems for Borel functions and sets at the second level of the Borel hierarchy. (5) The existence of countably-infinite bases of minimal counterexamples to the generalizations of the Feldman-Moore theorem, Glimm- Effros dichotomy, and Lusin-Novikov uniformization theorem from Polish spaces to their quotients by Borel equivalence relations, answering questions arising from recent results due to Kechris.