Lifts of Borel actions on quotient spaces

TL;DR

This paper investigates conditions under which Borel group actions on quotient spaces can be lifted to actions on the original space, addressing a fundamental problem in descriptive set theory.

Contribution

It provides new criteria and methods for lifting Borel actions from quotient spaces to original spaces in the context of countable Borel equivalence relations.

Findings

Established necessary and sufficient conditions for liftability.

Developed new techniques for constructing Borel actions.

Connected lifting problems to properties of countable groups and equivalence relations.

Abstract

Given a countable Borel equivalence relation E and a countable group G, we study the problem of when a Borel action of G on X/E can be lifted to a Borel action of G on X.