The further chameleon groups of Richard Thompson and Graham Higman: Automorphisms via dynamics for the Higman groups $G_{n,r}$

Abstract

We describe, through the use of Rubin's theorem, the automorphism groups of the Higman-Thompson groups $G_{n,r}$ as groups of specific homeomorphisms of Cantor spaces $\mathfrak{C}_{n,r}$. This continues a thread of research begun by Brin, and extended later by Brin and Guzm\'an: to characterise the automorphism groups of the `Chameleon groups of Richard Thompson,' as Brin referred to them in 1996. The work here completes the first stage of that twenty-year-old program, containing (amongst other things) a characterisation of the automorphism group of $V$, which was the `last chameleon.' The homeomorphisms which arise fit naturally into the framework of Grigorchuk, Nekrashevich, and Suschanskii's rational group $\mathscr{R}$: they are exactly those homeomorphisms which are induced by bi-sychronizing transducers, which we define in the paper. This result appears to offer insight into the nature of Brin and Guzman's exotic automorphisms, while also uncovering connections with the theory of reset words for automata (arising in the Road Colouring Problem) and with the theory of automorphism groups of the full shift.