Heegaard splittings and 1-relator groups

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".

Paper has been withdrawn by the author due to a gap in the proof of main theorem

References in corpus (1)

• Rank gradient, cost of groups and the rank versus Heegaard genus problem