Automorphism towers of groups of homeomorphisms of Cantor space

Abstract

We show that for any full and sufficiently transitive (i.e. \textit{flexible}) group $G$ of homeomorphisms of Cantor space, $\mathrm{Aut}(\mathrm{Aut}(G))=\mathrm{Aut}(G)$. This class contains many generalisations of the Higman-Thompson groups $G_{n,r}$, and the Rational group ${\mathcal{R}}_2$ of Grigorchuk, Nekrashevych, and Suchanski{\u ı}. We also demonstrate that for generalisations $T_{n,r}$ of R. Thompson's group $T$, $\mathrm{Aut}(\mathrm{Aut}(T_{n,r}))=\mathrm{Aut}(T_{n,r})$. In the case of the groups $G_{n,r}$ and $T_{n,r}$ our results extend results of Brin and Guzm{\' a}n for Thompson's group $T$, and generalisations of Thompson's group $F$.

11 pages; significant changes to improve readability and correct typos; main result proved in a more general context. The author is very grateful to anonymous referees for their comments on early drafts