Soficity of free extensions of effective subshifts

: Barbieri, Sebastian; Sablik, Mathieu; Salo, Ville

Publisher: AMER INST MATHEMATICAL SCIENCES-AIMS

: SPRINGFIELD

: 2025

: Discrete and continuous dynamical systems: series a

: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS

: DISCRETE CONT DYN-A

: 45

: 4

: 1117

: 1149

: 33

: 1078-0947

: 1553-5231

DOI: https://doi.org/10.3934/dcds.2024125

: https://www.aimsciences.org//article/doi/10.3934/dcds.2024125

: https://arxiv.org/pdf/2309.02620

: https://arxiv.org/abs/2309.02620

Let G be a group and H <= G a subgroup. The free extension of an H-subshift X to G is the G-subshift Xe 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 x 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.

:
S. Barbieri was supported by the FONDECYT grants 11200037 and 1240085. M. Sablik was supported by ANR project Difference (ANR- 20-CE48-0002) and the project Computability of asymptotic properties of dynamical systems from CIMI Labex (ANR-11-LABX-0040). V. Salo was supported by the Academy of Finland project 2608073211.