Subgroup separability of 3-manifold groups.

Abstract

This dissertation examines the subgroup separability (LERF-ness) of 3-manifold groups and its relationship to incompressible immersed surfaces in those 3-manifolds. Burns, Karrass, and Solitar, in 1987, gave the first example of a 3-manifold M whose fundamental group is not LERF. This manifold is carefully examined, and it is shown that the non-separable subgroup in the example is not geometric. It is also proved that the closed surfaces in M must lift to an embedding in a finite cover of M while some non-closed surfaces cannot. The recently proved criterion on whether some surfaces can lift to embeddings (by Rubinstein and Wang) is presented and used to construct such "non-virtually-embedded" surfaces. Finally, it is shown that there exist cubed 3-manifolds with non-LERF fundamental groups. Several squarings of surfaces, along with some results on squared surfaces of non-positive curvature, are given in the last chapter.