Heegaard splittings and 1-relator groups
  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". 
Joseph D. Masters