Strong ergodicity around countable products of countable equivalence relations [article]

Assaf Shani
2019 arXiv   pre-print
This paper deals with countable products of countable Borel equivalence relations and equivalence relations "just above" those in the Borel reducibility hierarchy. We show that if E is strongly ergodic with respect to μ then E^N is strongly ergodic with respect to μ^N. We answer questions of Clemens and Coskey regarding their recently defined Γ-jump operations, in particular showing that the Z^2-jump of E_∞ is strictly above the Z-jump of E_∞. We study a notion of equivalence relations which
more » ... be classified by infinite sequences of "definably countable sets". In particular, we define an interesting example of such equivalence relation which is strictly above E_∞^N, strictly below =^+, and is incomparable with the Γ-jumps of countable equivalence relations. We establish a characterization of strong ergodicity between Borel equivalence relations in terms of symmetric models. The proofs then rely on a fine analysis of the very weak choice principles "every sequence of E-classes admits a choice sequence", for various countable Borel equivalence relations E.
arXiv:1910.08188v1 fatcat:o7w5ygorjnagrcuwvct32i24iy