Mathematische Annalen

, Volume 331, Issue 1, pp 1-19

First online:

Genericity, the Arzhantseva-Ol’shanskii method and the isomorphism problem for one-relator groups

  • Ilya KapovichAffiliated withDepartment of Mathematics, University of Illinois at Urbana-Champaign Email author 
  • , Paul SchuppAffiliated withDepartment of Mathematics, University of Illinois at Urbana-Champaign

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We show that the isomorphism problem is solvable in at most exponential time for a class of one-relator groups which is exponentially generic in the sense of Ol’shanskii. This is obtained by applying the Arzhantseva-Ol’shanskii graph minimization method to prove the general result that for fixed integers m≥2 and n≥1 there is an exponentially generic class of non-free m-generator n-relator groups with the property that there is only one Nielsen equivalence class of m-tuples which generate a non-free subgroup. In particular, every m-generated subgroup in such a generic group G is either free or is equal to G itself and such groups are thus co-Hopfian. These results are obtained by elementary methods without using the deep results of Sela about co-Hopficity and the isomorphism problem for torsion-free hyperbolic groups.