Soficity of free extensions of effective subshifts
Sebastián Barbieri, Mathieu Sablik, Ville Salo · Discrete and Continuous Dynamical Systems · 2024
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.