Soficity of free extensions of effective subshifts

TL;DR

This paper investigates when free extensions of effective subshifts over certain groups are sofic, showing conditions under which they are or are not sofic, with implications for group actions and symbolic dynamics.

Contribution

It characterizes when free extensions of effective subshifts are sofic for groups G=H×K, depending on properties like amenability and decidability, and introduces new applications in symbolic dynamics.

Findings

If K is nonamenable and H has decidable word problem, free extensions of effectively closed H-subshifts are sofic.

If both H and K are amenable, some effectively closed H-subshifts have non-sofic free extensions.

Introduces a new simulation theorem and identifies groups with strongly aperiodic SFTs.

Abstract

Let $G$ be a group and $H\leqslant G$ a subgroup. The free extension of an $H$-subshift $X$ to $G$ is the $G$-subshift $\widetilde{X}$ whose configurations are those for which the restriction to every coset of $H$ is a configuration from $X$. We study the case of $G = H \times K$ for infinite and finitely generated groups $H$ and $K$: on the one hand we show that if $K$ is nonamenable and $H$ has decidable word problem, then the free extension to $G$ of any $H$-subshift which is effectively closed is a sofic $G$-subshift. On the other hand we prove that if both $H$ and $K$ are amenable, there are always $H$-subshifts which are effectively closed by patterns whose free extension to $G$ is non-sofic. We also present a few applications in the form of a new simulation theorem and a new class of groups which admit strongly aperiodic SFTs.