Structure of a Conjugating Automorphism Group

Abstract

We examine the automorphism group Aut(F n ) of a free group F n of rank n⩾ 2 on free generators x1, x2,...,x n . It is known that Aut(F2) can be built from cyclic subgroups using a free and semidirect product. A question remains open as to whether this result can be extended to the case n > 2. Every automorphism of Aut(F n ) sending a generator x i to an element f i -1 xπ(i)f i , where f i ∈ F n and π is some permutation on a symmetric group S n , is called a conjugating automorphism. The conjugating automorphism group is denoted C n . A set of automorphisms for which π is the identity permutation form a basis-conjugating automorphism group, denoted Cb n . It is proved that Cb n can be factored into a semidirect product of some groups. As a consequence we obtain a normal form for words in C n . For n ⩾ 4, C n and Cb n have an undecidable occurrence problem in finitely generated subgroups. It is also shown that C n , n ⩾ 2, is generated by at most four elements, and we find its respective genetic code, and that Cb n , n⩾ 2, has no proper verbal subgroups of finite width.