Free Subshifts with Invariant Measures from the Lov\'asz 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 Thomass\'{e} (see arXiv:1507.03369) used the Lov\'{a}sz 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 Lov\'{a}sz 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 Lov\'{a}sz 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.