Conjugate subgroups and overgroups of Vn

Abstract

We describe subgroups and overgroups of the generalized Thompson groups [Formula: see text] which arise via conjugation by rational homeomorphisms of Cantor space. We specifically consider conjugating [Formula: see text] 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 [Formula: see text] on Cantor space. We also consider a class [Formula: see text] of transducers containing all invertible, initial transducers. We associate to every transducer [Formula: see text] in this class, a group [Formula: see text]. We show that for a certain subclass of [Formula: see text], the groups [Formula: see text] are simple.