Realization of aperiodic subshifts and uniform densities in groups

TL;DR

This paper proves the existence of strongly aperiodic subshifts and uniform density subshifts in groups, using Lovász local lemma, extending results to finitely generated groups and groups of subexponential growth.

Contribution

It provides a new simple proof of a theorem on colorings in groups and constructs subshifts with uniform densities in finitely generated and subexponential groups.

Findings

Every countable group admits a 2-coloring.

Existence of G-effectively closed strongly aperiodic subshifts.

Construction of subshifts with uniform density in groups of subexponential growth.

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.