Free Subshifts with Invariant Measures from the Lovász Local Lemma

Abstract

Gao, Jackson, and Seward (see arXiv:1201.0513) proved that every countably infinite group $\Gamma$ admits a nonempty free subshift $X\subseteq \{0,1{\}}^{\Gamma }$. Furthermore, a theorem of Seward and Tucker-Drob (see arXiv:1402.4184) implies that every countably infinite group $\Gamma$ admits a free subshift $X\subseteq \{0,1{\}}^{\Gamma }$ that supports an invariant probability measure. Aubrun, Barbieri, and Thomassé (see arXiv:1507.03369) used the Lovász Local Lemma to give a short alternative proof of the Gao--Jackson--Seward theorem. Recently, Elek (see arXiv:1702.01631) followed another approach involving the Lovász Local Lemma to obtain a different proof of the existence of free subshifts with invariant probability measures for finitely generated sofic groups. Using the measurable version of the Lovász Local Lemma for shift actions established by the author (see arXiv:1604.07349), we give a short alternative proof of the existence of such subshifts for arbitrary groups. Moreover, we can find such subshifts in any nonempty invariant open set.

5 pages; v5: a minor typo fixed; v4: a minor typo fixed; v3: a minor typo fixed; v2: several new references added

References in corpus (1)

• On uniformly recurrent subgroups of finitely generated groups

Cited by in corpus (1)

• On minimal actions of countable groups