Immersions and Embeddings of Totally Geodesic Surfaces

The work of Waldhausen, Thurston and others has shown that the existence of an embedding of a closed, orientable, incompressible surface in a 3-manifold is a great help in the understanding of that manifold. Unfortunately many examples exist of manifolds which contain no such embedding. However, it does seem at least conjecturally possible that any irreducible manifold with infinite fundamental group could contain an immersion of such a surface, and this has motivated the study of the question of whether such a surface can always be lifted to an embedding in some finite covering of the 3-manifold. The general question seems to be some way from resolution; the purpose of this note is to give an affirmative answer in a very special case. THEOREM 1. Let M be a closed, hyperbolic 3-manifold which contains a totally geodesic immersion of a closed surface. Then: (a) there is a finite covering of M which contains an embedded closed, orientable, totally geodesic surface; (b) there is a finite covering ofM which contains an embedded non-separating, closed, orientable, totally geodesic surface. The restriction to the closed case is unnecessary, and is made only to save some words. Standard results and terminology of hyperbolic geometry will be used without proof; good references can be found in [3] or [5]. By a slight adaptation of the technique used to prove Theorem 1, we are also able to show the following. THEOREM 2. Let G be the fundamental group of a closed, hyperbolic 3-manifold. Then the cyclic subgroups of G are virtually separable. The traditional group-theoretic condition used to deal with immersion to embedding problems is subgroup separability or LERF. We now define this despite the fact we shall use a somewhat weaker property. If G is a group and H < G, let us define H * = f] {K \ H ^ K ^ G and K has finite index in G). Then G is said to be LERF if for each finitely generated / f ^ G w e have H = H*. The reason for introducing the notation H * is that such a strong condition as LERF is often not necessary; for many purposes the knowledge that the index [H *: H] is finite suffices. In [2], this condition was called virtual separability