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Essential countability of treeable equivalence relations
We establish a dichotomy theorem characterizing the circumstances under which
a treeable Borel equivalence relation E is essentially countable. Under
additional topological assumptions on the treeing, we in fact show that E is
essentially countable if and only if there is no continuous embedding of E1
into E. Our techniques also yield the first classical proof of the analogous
result for hypersmooth equivalence relations, and allow us to show that up to
continuous Kakutani embeddability, there is a minimum Borel function which is
not essentially countable-to-one
Locally Equivalent Correspondences
Given a pair of number fields with isomorphic rings of adeles, we construct
bijections between objects associated to the pair. For instance we construct an
isomorphism of Brauer groups that commutes with restriction. We additionally
construct bijections between central simple algebras, maximal orders, various
Galois cohomology sets, and commensurability classes of arithmetic lattices in
simple, inner algebraic groups. We show that under certain conditions, lattices
corresponding to one another under our bijections have the same covolume and
pro-congruence completion. We also make effective a finiteness result of Prasad
and Rapinchuk.Comment: Final Version. To appear in Ann. Inst. Fourie
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