(3,1)-Surfaces via Branched Surfaces

Loosely speaking, a (n,1)-surface is a very nicely immersed π₁-injective surface in a 3-manifold. Its concept was born around 1981 by Peter Scott in his work on Seifet fibered spaces. It has been shown that if a 3-manifold M contains a (4,1)-surface, then its universal cover is R³ and π₁(M) determines M up to homeomorphism. Homotopic homeomorphisms are isotopic on a 3-manifold containing a (3,1)-surface. On the other hand, some class of 3-manifolds, such as manifolds with nonpositive cubing, by Aitchison and Rubinstein, are known to contain (4,1)-surfaces. One natural question, then, is how 'big' is the set of 3-manifolds with (4,1)-surfaces in the set of all 3-manifolds. Similar question for embedded π₁-injective surfaces, called incompressible surfaces, has been answered in a work of Floyd and Oertel around 1980. They showed th a t the set of incompressible surfaces in a 3-manifold is carried by a finite number of branched surfaces. Combining this with a theorem of Hatcher, one can reasonably argue that 3-manifolds containing incompressible surfaces, called Haken manifolds, are scarce. In this paper we prove a similar result in the context of (3,1)-surfaces and non Haken 3-manifolds. Theorem 1 If M is a non Haken 3-manifold, then the set o f (3,1)-surfaces in M are embeddedly carried by a finite number of branched surfaces. 'Embeddedly carried' is a precise generalization of 'carried' in the context of immersed surfaces. Careful examination of when the theorem is not true will lead one to obtain a sequence of least area embedded disks in M that limits to an essential measured lamination of M. Such lamination always approximates an incompressible surface in M . In some cases euler characteristic of the lamination is zero, hence M has an essential torus. We strongly suspect this is actually true in all cases. We hope that this method generalizes to the context of (4,1)-surfaces in any 3-manifold. This would establish some kind of finiteness property for (4,1)-surfaces in a 3-manifold, as in the case of incompressible surfaces.