| Time: | 3:30pm - 4:30 pm |
| Room: |
Wean Hall 8220
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| Speaker: |
Assaf Shani CMU |
| Title: |
Above countable products of countable equivalence relations
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| Abstract: |
We study some Borel equivalence relations very low in the hierarchy, above the countable products of countable Borel equivalence relations.
Clemens and Coskey have recently defined new jump operations as follows. Given an equivalence relation E on X and a countable
group G, the G-jump of E is defined on the product space XG by declaring x and y equivalent if and only if there is some g in G such that after shifting x by g,
it is pointwise E-equivalent to y. Given a countable Borel equivalence relation E they showed that the
G-jumps of E are above the infinite power Eω in the Borel reducibility hierarchy, and asked whether the F2 jump is above the Z-jump.
We show in fact that the Z2-jump is strictly above the Z-jump. This should be viewed in contrast to the fact that a countable
Borel equivalence relation induced by an action of Z2 is Borel reducible to a Z-action. We establish a characterization of strong ergodicity between Borel
equivalence relations in terms of symmetric models. The proof then relies on analyzing symmetric models in which very weak forms of choice fail.
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