Stable finiteness of twisted group rings and noisy linear cellular automata

TL;DR

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.

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.

Theorems & Definitions (35)

• Definition 1.1
• Definition 1.2
• Theorem A
• Definition 1.3
• Theorem B
• Theorem C
• Corollary 1.4
• Lemma 4.1
• proof
• Theorem 4.2