Soficity of Free Extensions of Effective Subshifts

Let G be a group and H ≤ G a subgroup. The free extension of an H-subshift X to G is the G-subshift X͂ 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 × 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. © 2025 American Institute of Mathematical Sciences. All rights reserved.