Free Subshifts with Invariant Measures from the Lov\'asz Local Lemma

TL;DR

This paper provides a new, concise proof for the existence of free subshifts with invariant measures in arbitrary groups using the measurable Lovász Local Lemma, extending previous results and methods.

Contribution

It introduces a short alternative proof leveraging the measurable Lovász Local Lemma for arbitrary groups, and shows such subshifts exist in any nonempty invariant open set.

Findings

Existence of free subshifts with invariant measures for all groups.

Subshifts can be found in any nonempty invariant open set.

The proof simplifies previous approaches using 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 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.