Generic properties of Whitehead’s algorithm and isomorphism rigidity of random one-relator groups

Ilya Kapovich, Paul Schupp and Vladimir Shpilrain

Vol. 223 (2006), No. 1, 113–140

DOI: 10.2140/pjm.2006.223.113

Abstract

We prove that Whitehead’s algorithm for solving the automorphism problem in a fixed free group F_k 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 F_k in Aut(F_k) are cyclic groups generated by inner automorphisms and that Aut(F_k)-orbits are uniformly small in the sense of their growth entropy. We further prove that the number I_k(n) of isomorphism types of k-generator one-relator groups with defining relators of length n satisfies

where c₁,c₂ are positive constants depending on k but not on n. Thus I_k(n) grows in essentially the same manner as the number of cyclic words of length n.

Keywords

one-relator groups, Whitehead algorithm, generic-case complexity

Mathematical Subject Classification 2000

Primary: 20P05

Secondary: 03D15, 20F36, 57M05, 68W40

Milestones

Received: 24 April 2004

Revised: 11 August 2004

Accepted: 24 August 2004

Published: 1 January 2006