Realization of aperiodic subshifts and uniform densities in groups

  • Nathalie Aubrun

    École Normale Supérieure de Lyon, France
  • Sebastián Barbieri

    École Normale Supérieure de Lyon, France
  • Stéphan Thomassé

    École Normale Supérieure de Lyon, France
Realization of aperiodic subshifts and uniform densities in groups cover
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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 . In this article, we use Lovász local lemma to first give a new simple proof of said theorem, and second to prove the existence of a -effectively closed strongly aperiodic subshift for any finitely generated group . 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 has uniform density if for every configuration the density of 1's in any increasing sequence of balls converges to . We show a slightly more general result which implies that these subshifts always exist in the case of groups of subexponential growth.

Cite this article

Nathalie Aubrun, Sebastián Barbieri, Stéphan Thomassé, Realization of aperiodic subshifts and uniform densities in groups. Groups Geom. Dyn. 13 (2019), no. 1, pp. 107–129

DOI 10.4171/GGD/487