The structure of 3-manifolds with 2-generated fundamental group

The main purpose of this article is to describe compact orientable irreducible 3-manifolds that have a non-trivial JSJ-decomposition and 2-generated fundamental group. The rank of a group is the minimal number of elements needed to generate it. A natural question is whether the Heegaard genus of such a manifold is equal to 2. When the JSJ-decomposition is empty, there are examples of closed 3manifolds of Heegaard genus 3 that have 2-generated fundamental group [BZg]. These are Seifert fibered manifolds. Furthermore the second named author [W2] has recently found graph manifolds with the same property. At this point these are the only known examples of 3-manifolds that have 2-generated fundamental group but are not of Heegaard genus 2. In particular, it is still an open problem to find such examples that admit a complete hyperbolic structure. There are also examples of Seifert manifolds with Heegaard genus g + 1 ≥ 3 and fundamental group of rank g [MSc]. When the JSJ-decomposition is non-trivial, our main result with respect to this question is the following; we will denote the base spaces by their topological type, followed by a list with the orders of their cone points. We denote the Möbius band by Mö, the disk by D, the annulus by A and the 2-punctured disk by Σ. We furthermore denote by Q the orientable circle bundle over the Möbius band.