Towards the Hanna Neumann conjecture using Dicks' method

Research output: Contribution to journalArticle

9 Citations (Scopus)

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 languageEnglish
Pages (from-to)95-104
Number of pages10
JournalInventiones Mathematicae
Volume123
Issue number1
Publication statusPublished - Jan 1996

Fingerprint

Intersection
Upper bound
Graph in graph theory
Free Group
Equivalence
Subgroup
Form

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Inventiones Mathematicae, Vol. 123, No. 1, 01.1996, p. 95-104.

Research output: Contribution to journalArticle

@article{9edf813528554f49b346ad69ab6eda75,
title = "Towards the Hanna Neumann conjecture using Dicks' method",
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.",
author = "G. Tardos",
year = "1996",
month = "1",
language = "English",
volume = "123",
pages = "95--104",
journal = "Inventiones Mathematicae",
issn = "0020-9910",
publisher = "Springer New York",
number = "1",

}

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 -