Abstract:We prove that a group $G$ is locally finite if and only if every linear cellular automaton $V^G\to V^G$, over every vector-space alphabet $V$, has closed image in the prodiscrete topology. This gives an affirmative answer to Open Problem 7 of Ceccherini-Silberstein and Coornaert. More precisely, if $G$ is not locally finite, then for every field $K$ and every infinite-dimensional $K$-vector space $V$ there is a finite-memory linear cellular automaton $\tau\colon V^G\to V^G$ whose image is not closed. The construction uses an infinite ray in a locally finite Cayley graph of a finitely generated infinite subgroup of $G$; the labels of the ray are encoded into the internal coordinates of a countable direct-sum subspace of $V$. In particular, the first Grigorchuk group admits such an infinite-dimensional linear cellular automaton, giving a concrete example in the periodic non-locally finite case.
From: Jiang Yang [view email]
[v1]
Mon, 22 Jun 2026 00:59:20 UTC (9 KB)
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