Generic properties of Whitehead's Algorithm, stabilizers in Aut(Fk) and one-relator groups

We show that the " hard " part of Whitehead's algorithm for solving the automorphism problem in a fixed free group F k terminates in linear time (in terms of the length of an input) on an exponentially generic set of input pairs and thus the algorithm has strongly linear-time generic-case complexity. We also prove that the stabilizers of generic elements of F k in Aut(F k) are cyclic groups generated by inner auto-morphisms. We apply these results to one-relator groups and show that one-relator groups are generically complete groups, that is, they have trivial center and trivial outer automorphism group. We prove that the number In of isomorphism types of k-generator one-relator groups with defining relators of length n satisfies const1 n (2k − 1) n ≤ In ≤ const2 n (2k − 1) n. Thus In grows in essentially the same manner as the number of cyclic words of length n.