Strong ergodicity for the $\varGamma$-jump operator and for actions of Polish wreath products
Volume 274 / 2026
Fundamenta Mathematicae 274 (2026), 99-118
MSC: Primary 03E15; Secondary 03E75, 37A20, 03E40
DOI: 10.4064/fm220404-18-5
Published online: 3 September 2026
Abstract
Let $\varGamma $ and $\varDelta $ 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 $\varGamma \wr \varGamma $, so that $E$ is generically $F$-ergodic for any orbit equivalence relation $F$ induced by an action of $\varDelta \wr \varDelta $. More generally, we establish generic ergodicity between $\varGamma $-jumps and the iterated $\varDelta $-jumps, answering a question of Clemens and Coskey (2022). The proofs follow a translation between Borel homomorphisms and definable pins, extending ideas of Shani (2024) and Larson and Zapletal (2020).
