Minimal sofic shift on a group that is not finitely-generated

TL;DR

This paper constructs a specific non-finitely generated group that admits a minimal sofic shift, answering an open question and expanding understanding of symbolic dynamics on complex groups.

Contribution

It demonstrates the existence of a non-finitely generated group with a minimal sofic shift, using novel techniques involving Thompson's V and simulation theory.

Findings

Existence of a non-finitely generated group with a minimal sofic shift

Construction based on Thompson's V and simulation theory

Answers an open question in symbolic dynamics on groups

Abstract

We prove that there exists a group which is not finitely generated, but admits a minimal sofic shift. This answers a question of Doucha, Melleray and Tsankov. The group is of the form $(F_4 \times F_2) \rtimes F_{\infty}$. The construction itself is based on simulation theory and properties of Thompson's~$V$.