Stable finiteness of twisted group rings and noisy linear cellular automata

Abstract

AbstractFor linear nonuniform cellular automata (NUCA) which are local perturbations of linear CA over a group universeGand a finite-dimensional vector space alphabetVover an arbitrary fieldk, we investigate their Dedekind finiteness property, also known as the direct finiteness property, i.e., left or right invertibility implies invertibility. We say that the groupGis$L^1$-surjunctive, resp. finitely$L^1$-surjunctive, if all such linear NUCA are automatically surjective whenever they are stably injective, resp. when in additionkis finite. In parallel, we introduce the ring$D^1(k[G])$which is the Cartesian product$k[G] \times (k[G])[G]$as an additive group but the multiplication is twisted in the second component. The ring$D^1(k[G])$contains naturally the group ring$k[G]$and we obtain a dynamical characterization of its stable finiteness for every fieldkin terms of the finite$L^1$-surjunctivity of the groupG, which holds, for example, whenGis residually finite or initially subamenable. Our results extend known results in the case of CA.