Strong ergodicity for the Gamma-jump operator and for actions of Polish wreath products

TL;DR

This paper demonstrates strong ergodic properties of orbit equivalence relations induced by Polish wreath product group actions, establishing generic ergodicity between different jump operators and addressing a question in the field.

Contribution

It introduces new results on generic ergodicity for orbit equivalence relations generated by Polish wreath product groups, extending previous work on jump operators.

Findings

Existence of an orbit equivalence relation with generic ergodicity properties

Establishment of generic ergodicity between Gamma-jumps and iterated Delta-jumps

Answering a question of Clemens and Coskey on ergodic properties

Abstract

Let $\Gamma$ and $\Delta$ be sufficiently distinct countable groups. We show that there is an orbit equivalence relation $E$, induced by an action of the Polish wreath product group $\Gamma\wr\Gamma$, so that $E$ is generically $F$-ergodic for any orbit equivalence relation $F$ induced by an action of $\Delta\wr\Delta$. More generally, we establish generic ergodicity between $\Gamma$-jumps and the iterated $\Delta$-jumps, answering a question of Clemens and Coskey. The proofs follow a translation between Borel homomorphisms and definable pins.