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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 Lovász 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 α ∈ [0, 1] if for every configuration the density of 1's in any increasing sequence of balls converges to α. We show a slightly more general result which implies that these subshifts always exist in the case of groups of subexponential growth.
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Submitted on : Wednesday, January 23, 2019 - 3:53:18 PM
Last modification on : Saturday, September 11, 2021 - 3:19:08 AM

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Nathalie Aubrun, Sebastián Barbieri, Stéphan Thomassé. Realization of aperiodic subshifts and uniform densities in groups. Groups, Geometry, and Dynamics, European Mathematical Society, 2019, 13 (1), pp.107-129. ⟨hal-01989763⟩

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