### Abstract

The Hanna Neumann conjecture states that the intersection of two nontrivial subgroups of rank k + 1 and l + 1 of a free group has rank at most kl + 1. In a recent paper [3] W. Dicks proved that a strengthened form of this conjecture is equivalent to his amalgamated graph conjecture. He used this equivalence to reprove all known upper bounds on the rank of the intersection. We use his method to improve these bounds. In particular we prove an upper bound of 2kl - k - l + 1 for the rank of the intersection above (k, l ≧ 2) improving the earlier 2kl - min(k, l) bound of [1]. We prove a special case of the amalgamated graph conjecture in the hope that it may lead to a proof of the general case and thus of the strengthened Hanna Neumann conjecture.

Original language | English |
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Pages (from-to) | 95-104 |

Number of pages | 10 |

Journal | Inventiones Mathematicae |

Volume | 123 |

Issue number | 1 |

Publication status | Published - Jan 1996 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Inventiones Mathematicae*,

*123*(1), 95-104.

**Towards the Hanna Neumann conjecture using Dicks' method.** / Tardos, G.

Research output: Contribution to journal › Article

*Inventiones Mathematicae*, vol. 123, no. 1, pp. 95-104.

}

TY - JOUR

T1 - Towards the Hanna Neumann conjecture using Dicks' method

AU - Tardos, G.

PY - 1996/1

Y1 - 1996/1

N2 - The Hanna Neumann conjecture states that the intersection of two nontrivial subgroups of rank k + 1 and l + 1 of a free group has rank at most kl + 1. In a recent paper [3] W. Dicks proved that a strengthened form of this conjecture is equivalent to his amalgamated graph conjecture. He used this equivalence to reprove all known upper bounds on the rank of the intersection. We use his method to improve these bounds. In particular we prove an upper bound of 2kl - k - l + 1 for the rank of the intersection above (k, l ≧ 2) improving the earlier 2kl - min(k, l) bound of [1]. We prove a special case of the amalgamated graph conjecture in the hope that it may lead to a proof of the general case and thus of the strengthened Hanna Neumann conjecture.

AB - The Hanna Neumann conjecture states that the intersection of two nontrivial subgroups of rank k + 1 and l + 1 of a free group has rank at most kl + 1. In a recent paper [3] W. Dicks proved that a strengthened form of this conjecture is equivalent to his amalgamated graph conjecture. He used this equivalence to reprove all known upper bounds on the rank of the intersection. We use his method to improve these bounds. In particular we prove an upper bound of 2kl - k - l + 1 for the rank of the intersection above (k, l ≧ 2) improving the earlier 2kl - min(k, l) bound of [1]. We prove a special case of the amalgamated graph conjecture in the hope that it may lead to a proof of the general case and thus of the strengthened Hanna Neumann conjecture.

UR - http://www.scopus.com/inward/record.url?scp=0030526910&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030526910&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0030526910

VL - 123

SP - 95

EP - 104

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 1

ER -