Generic properties of Whitehead's Algorithm and isomorphism rigidity of random one-relator groups

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We prove that Whitehead's algorithm for solving the automorphism problem in a fixed free group Fₖ has strongly linear time generic-case complexity. This is done by showing that the ``hard'' part of the algorithm terminates in linear time on an exponentially generic set of input pairs. We then apply these results to one-relator groups. We obtain a Mostow-type isomorphism rigidity result for random one-relator groups: If two such groups are isomorphic then their Cayley graphs on the given generating sets are isometric. Although no nontrivial examples were previously known, we prove that one-relator groups are generically complete groups, that is, they have trivial center and trivial outer automorphism group. We also prove that the stabilizers of generic elements of Fₖ in Aut(Fₖ) are cyclic groups generated by inner automorphisms and that Aut(Fₖ)-orbits are uniformly small in the sense of their growth entropy. We further prove that the number Iₖ(n) of isomorphism types of k-generator one-relator groups with defining relators of length n satisfies c₁/n (2k-1)ⁿ ≤ Iₖ(n)≤ c₂/n (2k-1)ⁿ, where c₁=c₁(k)>0, c₂=c₂(k)>0 are some constants independent of n. Thus Iₖ(n) grows in essentially the same manner as the number of cyclic words of length n.