Non-Archimedean Abelian Polish Groups and Their Actions

TL;DR

This paper characterizes tame non-archimedean abelian Polish groups by their structure and explores the complexity of their orbit equivalence relations, showing they are essentially hyperfinite if countable and providing an upper bound in the Borel hierarchy.

Contribution

It provides a structural characterization of tame non-archimedean abelian Polish groups and analyzes the Borel complexity of their orbit equivalence relations.

Findings

Tame groups do not involve $ ext{Z}^ ext{omega}$ or $( ext{Z}(p)^{< ext{omega}})^ ext{omega}$.

Countable orbit equivalence relations are essentially hyperfinite.

An upper bound in the Borel hierarchy for these groups' orbit relations.

Abstract

In this paper we consider non-archimedean abelian Polish groups whose orbit equivalence relations are all Borel. Such groups are called tame. We show that a non-archimedean abelian Polish group is tame if and only if it does not involve $\Z^\omega$ or $(\Z(p)^{<\omega})^\omega$ for any prime $p$. In addition to determining the structure of tame groups, we also consider the actions of such groups and study the complexity of their orbit equivalence relations in the Borel reducibility hierarchy. It is shown that if such an orbit equivalence relation is essentially countable, then it must be essentially hyperfinite. We also find an upper bound in the Borel reducibility hierarchy for the orbit equivalence relations of all tame non-archimedean abelian Polish groups.