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Mathematische Annalen

, Volume 331, Issue 1, pp 1–19 | Cite as

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

  • Ilya KapovichEmail author
  • Paul Schupp
Article

Abstract.

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.

Keywords

Equivalence Class General Result Generic Class Generic Group Minimization Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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