On Algebraic Properties of Topological Full 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 $[[T]{]}^{\prime }$ is infinitely presented; and provide explicit examples of maximal locally finite subgroups of $[[T]]$. We also show that the commutator subgroup $[[T]{]}^{\prime }$, 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 $[[T]{]}^{\prime }$ possess continuous ergodic invariant random subgroups.

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