No Tits alternative for cellular automata

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. * email: vosalo@utu.fi 1 Mentioned open by Day in [24]. 2 A group G is periodic, or torsion, if ∀g ∈ G : ∃n ≥ 1 : g n = 1.

Universal groups of cellular automata

“…This provides a new proof of the two-sided case of [46]. This motivates the study of the groups P (B Z , H), especially when H is a symmetric group.…”

“…For |A| ≥ 8, Aut(A N ) is non-linear, as it does not even satisfy the Tits alternative [46]. By the previous proposition, Aut(A N ) is also non-linear for |A| = 6.…”

“…Introduction. Automorphism groups of subshifts have been a topic of much interest in recent years [38,49,43,17,13,16,19,20,15,22,44,46,3], with most results dealing with either the case of highly constrained subshifts such as minimal and low-complexity subshifts, or the case of weakly constrained subshifts such as SFT's. This paper is about the second case.…”

“…The set of finitely generated subgroups of Aut(A Z ) is relatively big: It is closed under direct and free products and finite extensions [44], contains the graph groups (a.k.a. right-angled Artin groups) [31], and contains a group not satisfying the Tits alternative [46] (we give another proof in Proposition 4.21). In the extended version of [3] we prove that there is an f.g. subgroup with undecidable torsion problem.…”

“…We state one corollary obtained in the symbolic dynamics setting (other embedding theorems are surveyed in [46]). A sofic shift is a subshift defined by a regular language of forbidden patterns; in particular all full shifts A Z are trivially sofic.…”