Freeness of Sch\"utzenberger groups of primitive substitutions

TL;DR

This paper investigates the conditions under which Sch"utzenberger groups of primitive substitutions are free, introduces a simple test for freeness, and provides a counterexample to a previous conjecture, also exploring relative freeness.

Contribution

It presents a new simple freeness test for Sch"utzenberger groups and provides the first known counterexample to a longstanding conjecture, advancing understanding of their algebraic properties.

Findings

Developed a simple freeness test for Sch"utzenberger groups

Constructed a primitive invertible substitution with a non-free Sch"utzenberger group

Provided initial results on the relative freeness of these groups

Abstract

Our main goal is to study the freeness of Sch\"utzenberger groups defined by primitive substitutions. Our findings include a simple freeness test for these groups, which is applied to exhibit a primitive invertible substitution with corresponding non-free Sch\"utzenberger group. This constitutes a counterexample to a result of Almeida dating back to 2005. We also give some early results concerning relative freeness of Sch\"utzenberger groups, a question which remains largely unexplored.