Automorphisms of the generalised Thompson's group T_n,r

The recent paper The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups G_n,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 R_n,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 T_n,r as a subgroup of Aut(G_n,r). We naturally also study the outer automorphism groups Out(T_n,r) . We show that each group Out(T_n,r) can be realized a subgroup of Out(T_n,n−1). Extending results of Brin and Guzman, we also show that the groups Out(T_n,r), for n>2, are all infinite and contain an isomorphic copy of Thompson's group F.  Our techniques for studying the groups Out(T_n,r) work equally well for Out(G_n,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(X_n,r) fit in a lattice where Out(X_n,1) ⊴ Out(X_n,r) for all 1 ≤ r ≤n−1 and Out(X_n,r) ⊴ Out(X_n,n−1). This gives a partial answer to a question in BCMNO concerning the normal subgroup structureof Out(G_n,n−1). Furthermore, we deduce that for 1 ≤ j,d ≤ n−1 such that d = gcd (j,n−1), Out(X_n,j) = Out(X_n,d) extending a result of BCMNO for the groups G_n,r to the groups T_n,r. We give a negative answer to the question in BCMNO which asks whether or not Out(G_n,r) ≅ Out(G_n,s) if and only if gcd (n−1,r) = gcd (n−1,s). Lastly, we show that the groups T_n,r have the R∞ property. This extends a result of Burillo, Matucci and Ventura and, independently, Gonçalves and Sankaran, for Thompson's group T.