Gate lattices

Ville Salo

Published 2022-04-01Version 1

A reversible gate on a subshift on a residually finite group $G$ can be applied on any sparse enough finite-index subgroup $H$, to obtain what we call a gate lattice. Gate lattices are automorphisms of the shift action of $H$, thus generate a subgroup of the Hartman-Kra-Schmieding stabilized automorphism group. We show that for subshifts of finite type with a gluing property we call the eventual filling property, the subgroup generated by even gate lattices is simple. Under some conditions, even gate lattices generate all gate lattices, and in the case of a one-dimensional mixing SFT, they generate the inert part of the stabilized automorphism group, thus we obtain that this group is simple. In the case of a full shift this has been previously shown by Hartman, Kra and Schmieding.