## 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.

Original language | English (US) |
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Pages (from-to) | 1-19 |

Number of pages | 19 |

Journal | Mathematische Annalen |

Volume | 331 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2005 |

## ASJC Scopus subject areas

- Mathematics(all)