# Stable finiteness of twisted group rings and noisy linear cellular automata

arXiv: https://arxiv.org/abs/2209.06002 · HTML: https://mlsanity.com/paper/2209.06002/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Sep 2022.

## Built on by (most cited first, 1 of 1)

- [Some pointwise and decidable properties of non-uniform cellular automata](https://mlsanity.com/paper/2210.00676/) (2022)

## Abstract (arXiv)

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 L1-surjunctive, resp. finitely L1-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 D1(k[G]) which is the Cartesian product k[G] × (k[G])[G] as an additive group but the multiplication is twisted in the second component. The ring D1(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 L1-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.