Heegaard splittings and 1-relator groups

Joseph D. Masters

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arXiv · math/0608635

Heegaard splittings and 1-relator groups

Joseph D. Masters

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Abstract

We show that if $M$ is a fibered, orientable 3-manifold, and if $\pi_1 M$ 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 $\pi_1 M$ 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".

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BibTeX RIS

Joseph D. Masters. 2006-08-25. Heegaard splittings and 1-relator groups. https://arxiv.org/abs/math/0608635

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