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

Ilya Kapovich, Paul Schupp

Research output: Contribution to journalArticlepeer-review

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 languageEnglish (US)
Pages (from-to)1-19
Number of pages19
JournalMathematische Annalen
Volume331
Issue number1
DOIs
StatePublished - Jan 2005

ASJC Scopus subject areas

  • Mathematics(all)

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