On the topological full group of a minimal Cantor Z^2-system

TL;DR

This paper demonstrates that the topological full group of a minimal Cantor Z^2-action can contain a non-abelian free group, providing a counterexample to a previously posed question about amenability.

Contribution

It constructs a specific minimal Cantor Z^2-action whose topological full group is non-amenable, contrasting prior results for Z-actions.

Findings

Topological full group of certain Z^2-actions contains free groups

Counterexample to amenability for general amenable groups

Answers a question posed by Grigorchuk and Medynets

Abstract

Grigorchuk and Medynets recently announced that the topological full group of a minimal Cantor Z-action is amenable. They asked whether the statement holds for all minimal Cantor actions of general amenable groups as well. We answer in the negative by producing a minimal Cantor Z^2-action for which the topological full group contains a non-abelian free group.