Automorphism towers of groups of homeomorphisms of Cantor space

Abstract: We show that for any full and sufficiently transitive (i.e. extit{flexible}) group

G

of homeomorphisms of Cantor space,

m a t h r m A u t (m a t h r m A u t (G)) = m a t h r m A u t (G)

. This class contains many generalisations of the Higman-Thompson groups

G_{n, r}

, and the Rational group

m a t h c a l R_{2}

of Grigorchuk, Nekrashevych, and Suchanski{u i}. We also demonstrate that for generalisations

T_{n, r}

of R. Thompson's group

T

,

m a t h r m A u t (m a t h r m A u t (T_{n, r})) = m a t h r m A u t (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

.

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