Heegaard splittings and 1-relator groups

2006/08/25 by Joseph D. Masters, Masters, Joseph D.
Mathematics · #57M05 #57M07 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #math.GR #math.GT #msc:57M05 #msc:57M07

paper · pdf · doi:10.48550/arxiv.math/0608635

Paper has been withdrawn by the author due to a gap in the proof of main theorem

arxiv created 2012/04/21 · arxiv updated 2012/04/24

Abstract

We show that if M is a fibered, orientable 3-manifold, and if π₁ M has 1-relator presentation, then the presentation is induced by a Heegaard splitting of M. A corollary is that, for these manifolds, the rank of π₁ M is equal to the "restricted" Heegaard genus of M. We also explore the analogy between 1-relator groups and Haken 3-manifolds, showing that every 1-relator group possesses a "1-relator hierarchy".

Related

• One-sided Heegaard splittings of non-Haken 3-manifolds
• Heegaard splittings of compact 3-manifolds
• Heegaard splittings of sufficiently complicated 3-manifolds II: Amalgamation
• One-sided and two-sided Heegaard splittings
• Heegaard splittings, the virtually Haken conjecture and Property tau