Gate lattices and the stabilized automorphism group

TL;DR

This paper investigates the structure of the stabilized automorphism group of certain subshifts of finite type, introducing gate lattices to describe its monolith and proving simplicity in specific cases like the integer group.

Contribution

It introduces gate lattices to characterize the monolith of the stabilized automorphism group and proves its simplicity for one-dimensional cases.

Findings

The stabilized automorphism group is simply monolithic with a unique simple monolith.

Gate lattices generate the monolith and, under certain conditions, generate a perfect group.

The stabilized inert automorphism group of a one-dimensional mixing subshift of finite type is simple.

Abstract

We study the stabilized automorphism group of a subshift of finite type with a certain gluing property called the eventual filling property, on a residually finite group $G$. We show that the stabilized automorphism group is simply monolithic, i.e. it has a unique minimal non-trivial normal subgroup -- the monolith -- which is additionally simple. To describe the monolith, we introduce gate lattices, which apply (reversible logical) gates on finite-index subgroups of $G$. The monolith is then precisely the commutator subgroup of the group generated by gate lattices. If the subshift and the group $G$ have some additional properties, then the gate lattices generate a perfect group, thus they generate the monolith. In particular, this is always the case when the acting group is the integers. In this case we can also show that gate lattices generate the inert part of the stabilized automorphism group. Thus we obtain that the stabilized inert automorphism group of a one-dimensional mixing subshift of finite type is simple.