Algebraic & Geometric Topology

Intersections and joins of free groups

Richard Peabody Kent

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Abstract

Let H and K be subgroups of a free group of ranks h and kh, respectively. We prove the following strong form of Burns’ inequality:

rank ( H K ) 1 2 ( h 1 ) ( k 1 ) ( h 1 ) rank ( H K ) 1 .

A corollary of this, also obtained by L Louder and D B McReynolds, has been used by M Culler and P Shalen to obtain information regarding the volumes of hyperbolic 3–manifolds.

We also prove the following particular case of the Hanna Neumann Conjecture, which has also been obtained by Louder. If HK has rank at least (h+k+1)2, then HK has rank no more than (h1)(k1)+1.

Article information

Source
Algebr. Geom. Topol., Volume 9, Number 1 (2009), 305-325.

Dates
Received: 31 January 2008
Revised: 18 August 2008
Accepted: 28 January 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796967

Digital Object Identifier
doi:10.2140/agt.2009.9.305

Mathematical Reviews number (MathSciNet)
MR2482079

Zentralblatt MATH identifier
1170.20017

Subjects
Primary: 20E05: Free nonabelian groups
Secondary: 57M50: Geometric structures on low-dimensional manifolds

Keywords
free group rank intersection join Hanna Neumann Conjecture

Citation

Kent, Richard Peabody. Intersections and joins of free groups. Algebr. Geom. Topol. 9 (2009), no. 1, 305--325. doi:10.2140/agt.2009.9.305. https://projecteuclid.org/euclid.agt/1513796967


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References

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