Stable finiteness of twisted group rings and noisy linear cellular automata

Abstract

For linear nonuniform 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.

On the Garden of Eden theorem for non-uniform cellular automata

“…also obtained for endomorphisms of symbolic varieties to deal with more general alphabet structures [8,29]. Other recent developments of the Garden of Eden theorems are related to the surjunctivity conjecture of Gottschalk [17] for CA and the Kaplansky conjecture [19] for group rings (see [26,[31][32][33][34]). When the uniformity in the definition of CA breaks down, the cells in the Universe of a CA can follow different transition rules.…”