On the topological full group of a minimal Cantor 𝐙²-system

Abstract

Grigorchuk and Medynets recently conjectured that the topological full group of a minimal Cantor Z \mathbf {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 \mathbf {Z}^2 -action for which the topological full group contains a non-abelian free group.