Hyperbolic 3-Manifolds. First, we give an informal review of covering spaces and their correspondence with subgroups of the fundamental group of the space. A covering space is a topological space along with a continuous map that surjects onto the “covered” space. Every point in the covered space has a neighborhood whose inverse image under the cover- ing map consists of disjoint open sets that map homeomorphically onto the specified neighborhood. An elementary result is that the fundamental group of the covering space injects into the fundamental group of the covered space by the group homomor- phism induced by the covering map. Thus, the fundamental group of the covering space is isomorphic to a subgroup of the covered space. Furthermore, the index of the injected subgroup is the cardinality of the inverse image of a point in the cov- ered space. If the cardinality is finite, then we call the covering space finite-sheeted. The classification of covering spaces theorem states that after fixing a base-point in the covered space, conjugacy classes of subgroups of the fundamental group are in direct correspondence with covering spaces up to isometry. Simply put, we may use conjugacy classes of subgroups of the fundamental group and covering spaces interchangeably. Specifically, we will be concerned about covering spaces that are finite-sheeted. Shortly, we will use this terminology to develop an equivalence rela- tion between two covering spaces (or subgroups). The following material comes from Chapter 1 of [16]. A hyperbolic 3-manifold is a manifold which locally looks like hyperbolic 3-space, i.e., the manifold is equipped with a Riemannian metric such that every point has a neighborhood that is isometric to a ball in H3. If Γ is a torsion-free Kleinian group, then Γ acts freely and properly discontinuously on H3. So, the quotient H3/Γ is an orientable hyperbolic 3-manifold. For the converse, we have a theorem below, but, first we require a few definitions.
