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Logic Seminar

Wednesday, July 1, 2020
12:00pm to 1:00pm
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New jump operators on Borel equivalence relations
John Clemens, Department of Mathematics, Boise State University,

We introduce a new class of jump operators on Borel equivalence relations, associated to countable groups. For each countable group Gamma, we define the Gamma-jump of an equivalence relation E and produce an analysis of these jumps analogous to the situation of the Friedman--Stanley jump with respect to actions of S_infty. In particular, we show that for many (but not all) groups the Gamma-jump of E is strictly above E and iterates of the Gamma-jump produce a hierarchy of equivalence relations cofinal in terms of potential Borel complexity. We also produce new examples of equivalence relations strictly between E_0^\omega and F_2, and give an application to the complexity of the isomorphism problem for countable scattered linear orders. This is joint work with Sam Coskey.

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