Strong Orbit Equivalence in Cantor dynamics and simple locally finite groups

TL;DR

This paper explores the relationship between minimal homeomorphisms and locally finite groups, establishing a universal isomorphism relation and providing a dynamical perspective on strong orbit equivalence.

Contribution

It introduces a new dynamical approach to characterizing strong orbit equivalence and demonstrates the universality of the isomorphism relation for certain locally finite groups.

Findings

The isomorphism relation on simple, countable, locally finite groups is universal.

A dynamical approach to strong orbit equivalence is developed.

The work connects group theory with dynamical systems in a novel way.

Abstract

We study certain countable locally finite groups attached to minimal homeomorphisms, and prove that the isomorphism relation on simple, countable, locally finite groups is a universal relation arising from a Borel $S_\infty$-action. This work also provides a dynamical approach to a result of Giordano, Putnam and Skau characterizing strong orbit equivalence.