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.

Uniformly recurrent subgroups and simple C⁎-algebras

“…Indeed, if a labeled automorphism of a Schreier graph fixes one vertex, it must fix all the other vertices as well. Now we proceed similarly as in the proof of Lemma 2 [3] or in the proof of Theorem 2.6 [2].…”

“…Proof. Since the proof in [3] is about edge-colorings and the proof in [2] is in slightly different setting, for completeness we give a proof using Lovász's Local Lemma, that closely follows the proof in [3]. Now, let us state the Local Lemma.…”

“…Let H ∈ Z and consider the Schreier graph S = S Q Γ (H). Following [2] and [3] we call a coloring c : Γ/H → K nonrepetitive if for any path (x 1 , x 2 , . .…”

“…In the proof we will use the Lovász Local Lemma technique of Alon, Grytczuk, Haluszczak and Riordan [3] to construct a minimal action on the space of rooted colored Γ-Schreier graphs. This approach has already been used to construct free Γ-Bernoulli subshifts by Aubrun, Barbieri and Thomassé [2] . Very recently, Matte Bon and Tsankov [24] completely answered the query of Glasner and Weiss for uniformly recurrent subgroups of discrete and locally compact groups.…”

A generalization of the simulation theorem for semidirect products

“…Proof. In [4] an effectively closed H-subshift is constructed for all finitely generated groups H with decidable word problem. One way to construct this object is as follows: a Theorem [1] of Alon, Grytczuk, Haluszczak and Riordan uses Lovász local lemma to show that every finite regular graph of degree ∆ can be vertex-colored with at most (2e 16 + 1)∆ 2 colors such that the sequence of colors in any non-intersecting path does not contain a square.…”