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arXiv · 1709.00858

No Tits alternative for cellular automata

Ville Salo
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Abstract

We show that the automorphism group of a one-dimensional full shift (the group of reversible cellular automata) does not satisfy the Tits alternative. That is, we construct a finitely-generated subgroup which is not virtually solvable yet does not contain a free group on two generators. We give constructions both in the two-sided case (spatially acting group $\Z$) and the one-sided case (spatially acting monoid $\N$, alphabet size at least eight). Lack of Tits alternative follows for several groups of symbolic (dynamical) origin: automorphism groups of two-sided one-dimensional uncountable sofic shifts, automorphism groups of multidimensional subshifts of finite type with positive entropy and dense minimal points, automorphism groups of full shifts over non-periodic groups, and the mapping class groups of two-sided one-dimensional transitive SFTs. We also show that the classical Tits alternative applies to one-dimensional (multi-track) reversible linear cellular automata over a finite field.

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Ville Salo. 2019-01-29. No Tits alternative for cellular automata. https://arxiv.org/abs/1709.00858

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