Building large free subshifts using the Local Lemma

Abstract

Gao, Jackson, and Seward [12] 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 $k\ge 2$ 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 [13]. Furthermore, we establish a general lower bound on the largest "size" of a free subshift $X^{\prime }$ contained inside a given subshift $X$. A central role in our arguments is played the Lovász Local Lemma, an important tool in probabilistic combinatorics, whose relevance to the problem of finding free subshifts was first recognized by Aubrun, Barbieri, and Thomassé [3].