Analytic structures

Dichotomy theorems have played a pivotal role in descriptive set theory since the inception of the subject. Recent work has revealed a graph-theoretic approach to establishing such theorems. The goal of this project is to further examine the extent of the applicability of this approach, in the process establishing theorems both new and old. One of our main interests concerns Borel reducibility of Borel equivalence relations. In particular, we would like to establish a dichotomy theorem characterizing essential countability of such relations, both in general and in well-studied special cases. Another focal point concerns the extent to which the graph-theoretic approach can be used to establish well-known dichotomy theorems. A few examples of results we aim to prove include the Harrington-Marker-Shelah Dilworth-style theorem for Borel quasi-orders, dichotomies of Hjorth-Kechris, results of Lecomte-Zeleny on countable colorings of bounded complexity, results of Louveau-Saint Raymond and Lecomte on (potential) complexity of Borel sets, and dichotomy theorems of Pawlikowski-Sabok and Solecki characterizing sigma continuous Borel functions. A final interest concerns descriptive Kakutani equivalence of Borel functions, a topic originating in ergodic theory. Here we focus on the primary question left open by Miller Rosendal in the countable-to-one case, concerning whether there is a dichotomy theorem characterizing essentially finite-to-one Borel functions. We are also interested in the nature of descriptive Kakutani equivalence outside of the essentially countable-to-one case, as well as generalizations of known results beyond Borel functions.

The most important results from the project include the following: (1) In their seminal paper "The classification of hypersmooth Borel equivalence relations," Kechris-Louveau established a dichotomy characterizing the circumstances under which a hypersmooth Borel equivalence relation is Borel reducible to a hyperfinite Borel equivalence relation. De Rancourt and I established a family of new dichotomy theorems of this form. (2) In their seminal papers "Counting the number of equivalence classes of Borel and coanalytic equivalence relations" and "Borel orderings," Silver and Harrington-Marker-Shelah established dichotomy theorems characterizing the circumstances under which a co-analytic equivalence relation has countably-many classes and the space of a Borel quasi-order is a countable union of Borel chains. Vidnynszky and I established a common generalization of these results and obtained the first classical proof of the latter. (3) In his dissertation "Une version borélienne du théorème de Dilworth," Kada established a strengthening of Dilworth's theorem for Borel quasi-orders. Carroy, Vidnynszky, and I established several generalizations of this result and obtained the first classical proof of the original theorem. (4) In their paper "Recent developments in the theory of Borel reducibility," Hjorth-Kechris established a dichotomy characterizing the circumstances under which a Borel orbit equivalence relation induced by a Borel action of a tsi Polish group is Borel reducible to a countable Borel equivalence relation. I produced the first classical proof of this result, in the process isolating a fundamental property that yields the desired reduction. (5) Carroy, Schrittesser, Vidnynszky, and I discovered a natural class of Borel graphs L with the property that any given analytic graph admits a Borel two-coloring if and only if it does not contain a Borel homomorphic image of L. (6) De Rancourt and I discovered countably infinite bases of minimal counterexamples to generalizations of results of Feldman-Moore, Glimm-Effros, and Lusin-Novikov to quotients of Polish spaces by Borel orbit equivalence relations, in the process generalizing earlier work of Conley-Miller and Marks-Miller and answering a pair of open questions due to Kechris. (7) Assuming that every (Z * Z)-orderable Borel equivalence relation is hyperfinite, I established several algebraic results concerning the group G of permutations of R / Q whose graphs are Borel when viewed as subsets of R x R, including the fact that every element of G is the composition of three involutions and a commutator, as well as the fact that G has the Bergman property and exactly four proper normal subgroups. The first result answers a question posed by Kechris. (8) Geschke, Grebk, and I showed that Li-Yorke chaos of a Polish dynamical system implies the existence of a scrambled Cantor set, answering a question posed by Blanchard-Huang-Snoha.