Automorphisms of the generalized Thompson's group Tn, r$T_{n, r}$. (15th August 2022)

• Record Type: Journal Article
• Authors: Olukoya, Feyishayo
• Is Part Of: Transactions of the London Mathematical Society. Volume 9:Number 1(2022)
• Page Start: 86
• Page End: 135
• Publication Date: 2022-08-15
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1112/tlm3.12044
• Languages: English
• ISSNs: 2052-4986

Abstract

Abstract: The recent paper The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups G n, r $G_{n, r}$ of Bleak, Cameron, Maissel, Navas and Olukoya (BCMNO) characterizes the automorphisms of the Higman–Thompson groups G n, r $G_{n, r}$ . This characterization is as the specific subgroup of the rational group R 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 of T n, r $T_{n, r}$ as a subgroup of Aut ( G n, r ) $\mathop {\mathrm{Aut}}({G_{n, r}})$ . We naturally also study the outer automorphism groups Out ( T n, r ) $\mathop {\mathrm{Out}}({T_{n, r}})$ . We show that each group Out ( T n, r ) $\mathop {\mathrm{Out}}({T_{n, r}})$ can be realized a subgroup of the group Out ( T n, n − 1 ) $\mathop {\mathrm{Out}}({T_{n, n-1}})$ . Extending results of Brin and Guzman, we also show that the groups Out ( T n, 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 ( T n, r ) $\mathop {\mathrm{Out}}({T_{n, r}})$ work equally well for Out ( G n, 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 ( X n, r ) $\mathop {\mathrm{Out}}({X_{n, r}})$ fit in a lattice structure where Out ( X n, 1 ) ⊴ Out ( X n, 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 ( 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 of Out ( G n, 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 ( 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 groups G n, r $G_{n, r}$ to the groups T n, r $T_{n, r}$ . We give a negative answer to the question in BCMNO which asks whether Out ( G n, r ) ≅ Out ( G n, 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 T n, 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 $T$ .