No Tits alternative for cellular automata

TL;DR

This paper demonstrates that the automorphism groups of certain one-dimensional cellular automata do not satisfy the Tits alternative, providing explicit constructions and extending the result to various symbolic dynamical groups.

Contribution

It constructs finitely-generated subgroups of automorphism groups that are not virtually solvable yet contain no free subgroup, showing the Tits alternative fails in this context.

Findings

Automorphism groups of one-dimensional full shifts lack the Tits alternative.

Explicit constructions of non-virtually solvable subgroups without free groups.

The classical Tits alternative holds for linear cellular automata over finite fields.

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.