Stable finiteness of twisted group rings and noisy linear cellular automata

TL;DR

This paper explores the algebraic properties of twisted group rings and their connection to the surjunctivity of linear cellular automata over groups, extending classical results to more general settings.

Contribution

It introduces the ring $D^1(k[G])$ with a twisted multiplication and links its stable finiteness to the $L^1$-surjunctivity of groups, broadening the understanding of cellular automata algebraic structures.

Findings

Stable finiteness of $D^1(k[G])$ characterizes $L^1$-surjunctivity of $G$.

Residually finite and initially subamenable groups are $L^1$-surjunctive.

Results extend classical CA properties to twisted group rings and NUCA.

Abstract

For linear non-uniform cellular automata (NUCA) which are local perturbations of linear CA over a group universe $G$ and a finite-dimensional vector space alphabet $V$ over an arbitrary field $k$, 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 group $G$ is $L^1$-surjunctive, resp. finitely $L^1$-surjunctive, if all such linear NUCA are automatically surjective whenever they are stably injective, resp. when in addition $k$ is 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 field $k$ in terms of the finite $L^1$-surjunctivity of the group $G$, which holds for example when $G$ is residually finite or initially subamenable. Our results extend known results in the case of CA.