Automorphisms of the generalized Thompson's group Tn,r$T_{n,r}$

The recent paper The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups Gn,r$G_{n,r}$ of Bleak, Cameron, Maissel, Navas and Olukoya (BCMNO) characterizes the automorphisms of the Higman–Thompson groups Gn,r$G_{n,r}$ . This characterization is as the specific subgroup of the rational group Rn,r$\mathcal {R}_{n,r}$ of Grigorchuk, Nekrashevych and Suchanskiĭ consisting of 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$T_{n,r}$ as a subgroup of Aut(Gn,r)$\mathop {\mathrm{Aut}}({G_{n,r}})$ . We naturally also study the outer automorphism groups Out(Tn,r)$\mathop {\mathrm{Out}}({T_{n,r}})$ . We show that each group Out(Tn,r)$\mathop {\mathrm{Out}}({T_{n,r}})$ can be realized a subgroup of the group Out(Tn,n−1)$\mathop {\mathrm{Out}}({T_{n,n-1}})$ . Extending results of Brin and Guzman, we also show that the groups Out(Tn,r)$\mathop {\mathrm{Out}}({T_{n,r}})$ , for n>2$n\,{>}\,2$ , are all infinite and contain an isomorphic copy of Thompson's group F$F$ . Our techniques for studying the groups Out(Tn,r)$\mathop {\mathrm{Out}}({T_{n,r}})$ work equally well for Out(Gn,r)$\mathop {\mathrm{Out}}({G_{n,r}})$ and we are able to prove some results for both families of groups. In particular, for X∈{T,G}$X \in \lbrace T,G\rbrace$ , we show that the groups Out(Xn,r)$\mathop {\mathrm{Out}}({X_{n,r}})$ fit in a lattice structure where Out(Xn,1)⊴Out(Xn,r)$\mathop {\mathrm{Out}}({X_{n,1}}) \unlhd \mathop {\mathrm{Out}}({X_{n,r}})$ for all 1⩽r⩽n−1$1 \leqslant r \leqslant n-1$ and Out(Xn,r)⊴Out(Xn,n−1)$\mathop {\mathrm{Out}}({X_{n,r}}) \unlhd \mathop {\mathrm{Out}}({X_{n,n-1}})$ . This gives a partial answer to a question in BCMNO concerning the normal subgroup structure of Out(Gn,n−1)$\mathop {\mathrm{Out}}({G_{n,n-1}})$ . Furthermore, we deduce that for 1⩽j,d⩽n−1$1\leqslant j,d \leqslant n-1$ such that d=gcd(j,n−1)$d = \gcd (j, n-1)$ , Out(Xn,j)=Out(Xn,d)$\mathop {\mathrm{Out}}({X_{n,j}}) = \mathop {\mathrm{Out}}({X_{n,d}})$ extending a result of BCMNO for the groups Gn,r$G_{n,r}$ to the groups Tn,r$T_{n,r}$ . We give a negative answer to the question in BCMNO which asks whether Out(Gn,r)≅Out(Gn,s)$\mathop {\mathrm{Out}}({G_{n,r}}) \cong \mathop {\mathrm{Out}}({G_{n,s}})$ if and only if gcd(n−1,r)=gcd(n−1,s)$\gcd (n-1,r) = \gcd (n-1,s)$ . Lastly, we show that the groups Tn,r$T_{n,r}$ have the R∞$R_{\infty }$ property. This extends a result of Burillo, Matucci and Ventura and, independently, Gonçalves and Sankaran, for Thompson's group T$T$ .