Building Large Free Subshifts Using the Local Lemma

Abstract

Gao, Jackson, and Seward proved that every countably infinite group $\Gamma$ admits a nonempty free subshift $X \subseteq 2^\Gamma$. Here we strengthen this result by showing that free subshifts can be "large" in various senses. Specifically, we prove that for any $k \geqslant 2$ and $h < \log_2 k$, there exists a free subshift $X \subseteq k^\Gamma$ of Hausdorff dimension and, if $\Gamma$ is sofic, entropy at least $h$, answering two questions attributed by Gao, Jackson, and Seward to Juan Souto. Furthermore, we establish a general lower bound on the largest "size" of a free subshift $X'$ contained inside a given subshift $X$. A central role in our arguments is played by the Lov\'{a}sz Local Lemma, an important tool in probabilistic combinatorics, whose relevance to the problem of finding free subshifts was first recognized by Aubrun, Barbieri, and Thomass\'{e}.