Uniformity, Universality, and Recursion Theory

We prove a number of results motivated by global questions of uniformity in recursion theory, and some longstanding open questions about universality of countable Borel equivalence relations. Our main technical tool is a class of games for constructing functions on free products of countable groups. These games show the existence of refinements of Martin’s ultrafilter on Turing invariant sets to the quotient space of equivalence relations that are much finer than Turing equivalence. This in turn implies a number of structural properties for these equivalence relations and for Martin’s ultrafilter. We also investigate uniform universality: a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. Here we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with uniform universality, and for many natural classes of countable Borel equivalence we can also classify exactly which are uniformly universal. We end with some connections between uniformity problems and Borel combinatorics. This is a prelimary draft. Corrections and comments are greatly appreciated, and may be sent to [email protected]. ∗The author is partially supported by the National Science Foundation under grant DMS-1204907, and the Turing Centenary research project ”Mind, Mechanism and Mathematics”, funded by the John Templeton Foundation under Award No. 15619.