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Author Aubrun, Nathalie ♦ Barbieri, Sebastián ♦ Thomassé, Stéphan
Source Hyper Articles en Ligne (HAL)
Content type Text
Publisher European Mathematical Society
File Format PDF
Language English
Subject Keyword Countable groups ♦ Stur-mian sequences ♦ Amenable groups ♦ Aperiodic subshift ♦ Symbolic dynamics ♦ math ♦ Mathematics [math]/Dynamical Systems [math.DS] ♦ Mathematics [math]/Group Theory [math.GR]
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.
ISSN 16617207
Educational Use Research
Learning Resource Type Article
Publisher Date 2019-01-01
e-ISSN 16617215
Journal Groups Geometry and Dynamics
Volume Number 13
Issue Number 1
Page Count 23
Starting Page 107
Ending Page 129