Universal CA groups with few generators

TL;DR

This paper constructs universal cellular automata groups with a minimal number of generators, confirming conjectures about their quotient structures from free groups and finite cyclic groups.

Contribution

It demonstrates the existence of f.g.-universal cellular automata groups as quotients of specific small free products, advancing understanding of their algebraic structure.

Findings

Existence of f.g.-universal cellular automata groups as quotients of $ ext{Z} * ext{Z}_2$

Existence of such groups as quotients of $ ext{Z}_2 * ext{Z}_2 * ext{Z}_2$

Confirmation of previous conjectures about their algebraic structure

Abstract

There exist f.g.-universal cellular automata groups which are quotients of $\mathbb{Z} * \mathbb{Z}_2$ or $\mathbb{Z}_2 * \mathbb{Z}_2 * \mathbb{Z}_2$, as previously conjectured by the author.