The structure of 3-manifolds with two-generated fundamental group (Q1767608)

The JSJ-decomposision of a compact orientable irreducible 3-manifold \(M\) is the canonical splitting of \(M\) along a finite (maybe empty) collection of disjoint and non-parallel nor boundary-parallel incompressible embedded tori into Seifert fibred or atoroidal compact submanifolds. The authors prove a structure theorem for compact orientable irreducible 3-manifolds \(M\) with rank \(\pi _1(M)=2\) which explicitly describes all the possible pieces of such a decomposition. Combining this theorem with other work they obtain results on the classification and possible geometries of rank-two \(3\)-manifolds.