On uniformly recurrent subgroups of finitely generated groups

Abstract: We prove that if $G$ is a finitely generated group and $Z$ is a uniformly recurrent subgroup of $G$ then there exists a minimal system $(X,G)$ with $Z$ as its stability system. This answers a query of Glasner and Weiss \cite{GW} in the case of finitely generated groups. Using the same method (introduced by Alon, Grytczuk, Haluszczak and Riordan \cite{AGHR}) we will prove that finitely generated sofic groups have free Bernoulli-subshifts admitting an invariant probability measure.