The isomorphism problem for group actions

TL;DR

This paper investigates the complexity of the isomorphism problem for ergodic group actions, demonstrating that the conjugacy relation is not Borel for certain ergodic measure-preserving actions of indicable groups.

Contribution

It establishes that the conjugacy relation for ergodic measure-preserving actions of indicable groups is not Borel, highlighting a complexity barrier in classifying such actions.

Findings

Conjugacy relation is not Borel for ergodic measure-preserving actions of indicable groups.

Shows limitations in classifying ergodic actions using Borel complexity.

Provides insights into the descriptive set-theoretic complexity of the isomorphism problem.

Abstract

We discuss the isomorphism problem for ergodic actions of locally compact groups. In particular we show that the conjugacy relation is not Borel for ergodic measure preserving actions of indicable groups.