Automorphisms of the generalised Thompson's group Tn,r

Abstract

The recent paper The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups Gn,r of Bleak, Cameron, Maissel, Navas and Olukoya (BCMNO) characterizes the automorphisms of the Higman-Thompson groups Gn,r. his characterization is as the specific subgroup of the rational group Rn,r of Grigorchuk, Nekrashevych and Suchanski{\u i}'s consisting of those elements which have the additional property of being bi-synchronizing. This article extends the arguments of BCMNO to characterize the automorphism group of Tn,r as a subgroup of Aut(Gn,r). We naturally also study the outer automorphism groups Out(Tn,r) . We show that each group Out(Tn,r) can be realized a subgroup of Out(Tn,n−1). Extending results of Brin and Guzman, we also show that the groups Out(Tn,r), for n>2, are all infinite and contain an isomorphic copy of Thompson's group F. Our techniques for studying the groups Out(Tn,r) work equally well for Out(Gn,r) and we are able to prove some results for both families of groups. In particular, for X ∈ {T,G}, we show that the groups Out(Xn,r) fit in a lattice where Out(Xn,1) ⊴ Out(Xn,r) for all 1 ≤ r ≤n−1 and Out(Xn,r) ⊴ Out(Xn,n−1). This gives a partial answer to a question in BCMNO concerning the normal subgroup structureof Out(Gn,n−1). Furthermore, we deduce that for 1 ≤ j,d ≤ n−1 such that d = gcd (j,n−1), Out(Xn,j) = Out(Xn,d) extending a result of BCMNO for the groups Gn,r to the groups Tn,r. We give a negative answer to the question in BCMNO which asks whether or not Out(Gn,r) ≅ Out(Gn,s) if and only if gcd (n−1,r) = gcd (n−1,s). Lastly, we show that the groups Tn,r have the R∞ property. This extends a result of Burillo, Matucci and Ventura and, independently, Gonçalves and Sankaran, for Thompson's group T.