# Automorphism towers of groups of homeomorphisms of Cantor space

## TL;DR

This paper proves that for a broad class of highly transitive groups of homeomorphisms of Cantor space, the automorphism group of their automorphism group coincides with the automorphism group itself, extending known results for Thompson-like groups.

## Contribution

It establishes the automorphism tower property for many groups of homeomorphisms of Cantor space, including generalizations of Higman-Thompson and R. Thompson groups.

## Key findings

- Aut(Aut(G)) = Aut(G) for flexible groups G of homeomorphisms of Cantor space
- Extends automorphism results to groups T_{n,r} and G_{n,r}
- Generalizes previous results for Thompson's groups T and F

## 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 \i}. 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 $F$.