Generic properties of whitehead's algorithm and isomorphism rigidity of random one-relator groups

Ilya Kapovich, Paul Schupp, Vladimir Shpilrain

Research output: Contribution to journalArticle

Abstract

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(F k)-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) ≤c 2(2k-1)n, where c1, c2 are positive constants depending on k but not on n. Thus Ik(n) grows in essentially the same manner as the number of cyclic words of length n.

Original languageEnglish (US)
Pages (from-to)113-140
Number of pages28
JournalPacific Journal of Mathematics
Volume223
Issue number1
DOIs
StatePublished - Jan 2006

Keywords

  • Generic-case complexity
  • One-relator groups
  • Whitehead algorithm

ASJC Scopus subject areas

  • Mathematics(all)

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