Universal equivalence relations on X N generated by permutation actions of countable subgroups of S ∞ Andrew Marks and

Let S∞ be the group of all permutations of N and X be a standard Borel space. Then the space XN of functions from N to X is a standard Borel space, and S∞ acts on this space by permutation where given x ∈ XN and g ∈ S∞, g · x(n) = x(g−1(n)). Given any countable subgroup G of S∞, we can likewise restrict this action to G and consider the induced orbit equivalence relation on XN, which we will note EX N G . Hjorth has asked [1] [2] whether given countable Borel equivalence relations E ⊆ F , if E is universal must F be universal? We are interested in this question in the setting of equivalence relations of the form EX N G above. However, we will modify the question and instead ask for a stronger form of universality known as uniform universality for which we currently have more traction to prove theorems. Suppose G and H are countable groups equipped with Borel actions on the standard Borel spaces X and Y with associated orbit equivalence relations relations EX G and E Y H . Then say a homomorphism f from E X G to EY H is uniform if there exists a function u : G → H such that for all g ∈ G and x ∈ X, we have f(g · x) = u(g) · f(x). This uniformity function u need not be a homomorphism from G to H in general, but in the case where G is a free group, then u may be assumed to be a homomorphism. We say that a Borel action of a countable group H on a standard Borel space Y generates a uniformly universal equivalence relation EY H if given any countable Borel equivalence relation EX G there exists a uniform Borel reduction from EX G to E Y H . We say that E Y H is uniformly weakly universal if there exists an injective uniform Borel homomorphism from every EX G to E Y H . For example, the usual argument that E(Fω, ωω) is universal also shows that it is uniformly universal. Of course, it is important to specify the action as well as the equivalence relation when we are discussing uniform universality. Note for example, that any universal countable Borel equivalence relation is uniformly universal for some Borel action of a countable group that generates it. This follows from the fact that if E is universal, then F vB E for every countable Borel equivalence relation F [4]. Thus, if E is universal and generated by a Borel action of G, we can embed E(Fω, ωω) into E, and and use this embedding to obtain a Borel action of Fω ∗G that generates E for