Conjugate subgroups and overgroups of $V_n$

TL;DR

This paper investigates the subgroups and overgroups of the generalized Thompson groups $V_n$ formed through conjugation by rational homeomorphisms, especially those induced by synchronizing transducers, revealing how these transformations affect the group's structure.

Contribution

It provides a detailed description of subgroups and overgroups of $V_n$ arising from conjugation by rational homeomorphisms, particularly focusing on synchronizing transducers and their inverses.

Findings

Characterization of subgroups and overgroups via properties of conjugating transducers

Identification of how conjugation restricts or extends the action of $V_n$

Insights into the structure of $V_n$ under conjugation by rational homeomorphisms

Abstract

We describe subgroups and overgroups of the generalised Thompson groups $V_n$ which arise via conjugation by rational homeomorphisms of Cantor space. We specifically consider conjugating $V_n$ by homeomorphisms induced by synchronizing transducers and their inverses. Our descriptions of the subgroups and overgroups use properties of the conjugating transducer to either restrict or augment the action of $V_n$ on Cantor space.