Essential Surfaces and Tameness of Covers

摘要

Suppose that M is a closed orientable 3-manifold. An essential surface S in M is a π1-inJective map of a closed surface S to M. Throughout this paper, we will testrict ourselves to the case of S being orientable. The surface cover Ms of S is said to be topologically tame if it is homeomorphic to S x R . Equivalently, there is a compactification of the surface cover homeomorphic to S x [-l, 1]. In this paper we establish tameness of surface covers for two natural classes of essential surfaces. The first class we call topologicallyfinite. The defining properties of such surfaces are conditions that are easily seen to be satisfied by quasi-Fuchsian surfaces in hyperbolic 3-manifolds (cf. [RSI] ). Using geometric techniques, it is straightforward to check that quasi-Fuchsian surfaces in a hyperbolic 3-manifold have topologically tame surface covers. In fact, Ms can be compactified by adding the quotient of the domain of discontinuity by the action of f* (π1(S)) at infinity (see e.g. [Th] ). We give an altemate topological proof of tameness by using three important properties of geometrically finite surfaces. The proof is reminiscent of an argument in [HRS] that establishes this tameness for the case when S is a torus.