Uniformity, Universality, and Computability Theory

TL;DR

This paper explores uniformity and universality in computability theory and Borel equivalence relations, introducing new tools and classifications to understand their structure and interactions.

Contribution

It introduces a game-based method for constructing functions on free products of groups and investigates uniform universality, providing classifications and new ultrafilters for Borel equivalence relations.

Findings

Uniform universality may be equivalent to standard universality.

Classified which countable Borel equivalence relations are uniformly universal.

Constructed ultrafilters finer than Turing equivalence for certain orbit relations.

Abstract

We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalb\'an, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this additional uniformity hypothesis, we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with universality. For many natural classes of countable Borel equivalence relations, we can also classify exactly which are uniformly universal. We also show the existence of refinements of Martin's ultrafilter on Turing invariant Borel sets to the invariant Borel sets of equivalence relations that are much finer than Turing equivalence. For example, we construct such an ultrafilter for the orbit equivalence relation of the shift action of the free group on countably many generators. These ultrafilters imply a number of structural properties for these equivalence relations.