The further chameleon groups of Richard Thompson and Graham Higman : automorphisms via dynamics for the Higman groups Gn,r

Abstract

We characterise the automorphism groups of the Higman groups Gn,r as groups of specific homeomorphisms of Cantor spaces Cn,r, through the use of Rubin's theorem. This continues a thread of research begun by Brin, and extended later by Brin and Guzmán: 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.' As it happens, the homeomorphisms which arise naturally fit into the framework of Grigorchuk, Nekrashevich, and Suschanskiī's rational group of transducers, and exhibit fascinating connections with the theory of reset words for automata (arising in the Road Colouring Problem), while also appearing to offer insight into the nature of Brin and Guzmán's exotic automorphisms.