Automorphisms of Generalized Thompson Groups

0.1. Results. We study the automorphisms of some generalizations of Thompson’s groups and their underlying structures. The automorphism groups of two of Thompson’s original groups were analyzed in [2] and were shown to be “small” and “unexotic.” Our results differ sharply from [2] in that we show that the automorphism groups of the generalizations are “large” and have “exotic” elements. The term exotic is from [1] and is explained later in this introduction. Richard J. Thompson introduced the triple of infinite groups F ⊆ T ⊆ G in the 1960s and showed that they have several interesting properties. These groups were later generalized to infinite groups Fn,∞ ⊆ Fn ⊆ Tn,r ⊆ Gn,r for n ≥ 2 and r ≥ 1 for which F = F2,∞ = F2, T = T2,1 and G = G2,1. Further background is given in the second part of this introduction. The automorphism groups Aut(F ) and Aut(T ) were analyzed in [2]. In this paper, we study Aut(Fn,∞), Aut(Fn) and Aut(Tn,n−1) for n > 2. We find that all three contain exotic elements, and that the complexity increases as n increases. We also find that the structures of Aut(Fn) and Aut(Tn,n−1) are closely related, while differing significantly from the structure of Aut(Fn,∞). This difference increases as n increases. Our analysis is only partial in that we discover large subgroups of these groups containing exotic elements, but we do not know whether the subgroups are proper. Automorphism groups of a class of groups that include the Thompson groups were studied in the unpublished notes [1] where the structures were analyzed modulo the unanswered question of whether exotic automorphisms could exist. We now give more details. Throughout the paper, functions will be written to the right of their arguments and composition will proceed from left to right.