Automorphisms of the generalized Thompson's group T n , r $T_{n,r}$

The recent paper The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groupsG n , r $G_{n,r}$ of Bleak, Cameron, Maissel, Navas and Olukoya (BCMNO) characterizes the automorphisms of the Higman–Thompson groupsG n , r $G_{n,r}$ . This characterization is as the specific subgroup of the rational groupR n , 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 ofT n , r $T_{n,r}$ as a subgroup ofAut ( G n , r ) $\mathop {\mathrm{Aut}}({G_{n,r}})$ . We naturally also study the outer automorphism groupsOut ( T n , r ) $\mathop {\mathrm{Out}}({T_{n,r}})$ . We show that each groupOut ( T n , r ) $\mathop {\mathrm{Out}}({T_{n,r}})$ can be realized a subgroup of the groupOut ( T n , n − 1 ) $\mathop {\mathrm{Out}}({T_{n,n-1}})$ . Extending results of Brin and Guzman, we also show that the groupsOut ( T n , r ) $\mathop {\mathrm{Out}}({T_{n,r}})$ , forn > 2 $n\,{>}\,2$ , are all infinite and contain an isomorphic copy of Thompson's group F $F$ . Our techniques for studying the groupsOut ( T n , r ) $\mathop {\mathrm{Out}}({T_{n,r}})$ work equally well forOut ( G n , r ) $\mathop {\mathrm{Out}}({G_{n,r}})$ and we are able to prove some results for both families of groups. In particular, forX ∈ { T , G } $X \in \lbrace T,G\rbrace$ , we show that the groupsOut ( X n , r ) $\mathop {\mathrm{Out}}({X_{n,r}})$ fit in a lattice structure whereOut ( X n , 1 ) ⊴ Out ( X n , r ) $\mathop {\mathrm{Out}}({X_{n,1}}) \unlhd \mathop {\mathrm{Out}}({X_{n,r}})$ for all1 ⩽ r ⩽ n − 1 $1 \leqslant r \leqslant n-1$ andOut ( X n , r ) ⊴ Out ( X n , 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 ofOut ( G n , n − 1 ) $\mathop {\mathrm{Out}}({G_{n,n-1}})$ . Furthermore, we deduce that for1 ⩽ j , d ⩽ n − 1 $1\leqslant j,d \leqslant n-1$ such thatd = gcd ( j , n − 1 ) $d = \gcd (j, n-1)$ ,Out ( X n , j ) = Out ( X n , d ) $\mathop {\mathrm{Out}}({X_{n,j}}) = \mathop {\mathrm{Out}}({X_{n,d}})$ extending a result of BCMNO for the groupsG n , r $G_{n,r}$ to the groupsT n , r $T_{n,r}$ . We give a negative answer to the question in BCMNO which asks whetherOut ( G n , r ) ≅ Out ( G n , s ) $\mathop {\mathrm{Out}}({G_{n,r}}) \cong \mathop {\mathrm{Out}}({G_{n,s}})$ if and only ifgcd ( n − 1 , r ) = gcd ( n − 1 , s ) $\gcd (n-1,r) = \gcd (n-1,s)$ . Lastly, we show that the groupsT n , r $T_{n,r}$ have theR ∞ $R_{\infty }$ property. This extends a result of Burillo, Matucci and Ventura and, independently, Gonçalves and Sankaran, for Thompson's group T $T$ .