2003/03/31 by Ilya Kapovich, Paul Schupp, Kapovich, Ilya +4 · 2 citations
Computer Science · Mathematics · #20F36 #Computational Complexity (cs.CC) #FOS: Computer and information sciences #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Topological and Geometric Data Analysis #cs.CC #math.GR #math.GT #msc:20F36
paper · pdf · doi:10.48550/arxiv.math/0303386
final revised version, to appear in Pacific J. Math
openalex publication_date 2003/03/31 · arxiv created 2004/08/27 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.