Abelian group actions and hypersmooth equivalence relations

TL;DR

This paper demonstrates that Borel actions of certain abelian groups induce hypersmooth or essentially hyperfinite orbit equivalence relations, extending previous results and answering open questions in descriptive set theory.

Contribution

It establishes that Borel actions of locally compact abelian groups produce orbit equivalence relations that are hypersmooth or hyperfinite, broadening understanding of group actions in descriptive set theory.

Findings

Borel actions of a sum of abelian groups induce hypersmooth relations.

Borel actions of Polish LCA groups induce essentially hyperfinite relations.

Extends previous results by Gao, Jackson, Ding, and Gao.

Abstract

We show that a Borel action of a standard Borel group which is isomorphic to a sum of a countable abelian group with a countable sum of real lines and circles induces an orbit equivalence relation which is hypersmooth, i.e., Borel reducible to eventual agreement on sequences of reals, and it follows from this result along with the structure theory for locally compact abelian groups that Borel actions of Polish LCA groups induce orbit equivalence relations which are essentially hyperfinite, extending a result of Gao and Jackson and answering a question of Ding and Gao.