Stable finiteness of twisted group rings and noisy linear cellular automata
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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] × (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.
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