A Tits alternative for topological full groups

TL;DR

This paper establishes a Tits alternative for topological full groups of minimal actions, showing they are either amenable or contain free groups, depending on the nature of the acting group.

Contribution

It generalizes the Tits alternative to a broad class of topological full groups, extending previous results from $ extbf{Z}$-actions to more general finitely generated groups.

Findings

Topological full groups of virtually cyclic groups are amenable.

Non-virtually cyclic groups can produce free subgroups in their topological full groups.

The paper extends the Tits alternative to topological full groups of minimal actions.

Abstract

We prove a Tits alternative for topological full groups of minimal actions of finitely generated groups. On the one hand, we show that topological full groups of minimal actions of virtually cyclic groups are amenable. By doing so, we generalize the result of Juschenko and Monod for $\mathbf{Z}$-actions. On the other hand, when a finitely generated group $G$ is not virtually cyclic, then we construct a minimal free action of $G$ on a Cantor space such that the topological full group contains a non-abelian free group.