If G is a residually finite countably infinite group, and X is a subshift over G, then we associate to X a large countably infinite discrete group, whose generators are homeomorphisms induced by "reversible logical gates" applied at every element of a finite-index subgroup of G. We call the resulting group L. We show that under two dynamical assumptions on X (namely that X is of finite type, and has a gluing property we call EFP), the commutator [L, L] is simple, and forms the monolith (unique minimal nontrivial normal subgroup) of L. We also outline the dynamical context for this result: Hartman, Kra and Schmieding define the stabilized automorphism group of a subshift as the union of automorphism groups of finite-index subactions. They have shown that for full shifts on integers, the "inert part" of this group is simple. It turns out that this inert part coincides with [L, L], so our result generalizes theirs in several directions.
