Realization of aperiodic subshifts and uniform densities in groups

Abstract

A theorem of Gao, Jackson and Seward, originally conjectured to be false by Glasner and Uspenskij, asserts that every countable group admits a $2$-coloring. A direct consequence of this result is that every countable group has a strongly aperiodic subshift on the alphabet $\{0,1\}$. In this article, we use Lov\'asz local lemma to first give a new simple proof of said theorem, and second to prove the existence of a $G$-effectively closed strongly aperiodic subshift for any finitely generated group $G$. We also study the problem of constructing subshifts which generalize a property of Sturmian sequences to finitely generated groups. More precisely, a subshift over the alphabet $\{0,1\}$ has uniform density $\alpha \in [0,1]$ if for every configuration the density of $1$'s in any increasing sequence of balls converges to $\alpha$. We show a slightly more general result which implies that these subshifts always exist in the case of groups of subexponential growth.