On Algebraic Properties of Topological Full Groups

TL;DR

This paper explores the algebraic structure of topological full groups associated with Cantor minimal systems, revealing their local embeddability, subgroup decompositions, and invariant random subgroups, thus advancing understanding of their algebraic and dynamical properties.

Contribution

It introduces a structural description of topological full groups as unions of permutational wreath products, proves their local embeddability into finite groups, and analyzes their subgroup and invariant properties.

Findings

Topological full groups are unions of permutational wreath products of Z.

They are locally embeddable into finite groups.

The commutator subgroup is decomposable into a product of two locally finite groups.

Abstract

In the paper we discuss the algebraic structure of topological full group $[[T]]$ of a Cantor minimal system $(X,T)$. We show that the topological full group $[[T]]$ has the structure similar to a union of permutational wreath products of group $\mathbb Z$. This allows us to prove that the topological full groups are locally embeddable into finite groups; give an elmentary proof of the fact that group $[[T]]'$ is infinitely presented; and provide explicit examples of maximal locally finite subgroups of $[[T]]$. We also show that the commutator subgroup $[[T]]'$, which is simple and finitely-generated for minimal subshifts, is decomposable into a product of two locally finite groups and that the groups $[[T]]$ and $[[T]]'$ possess continuous ergodic invariant random subgroups.