Coarse hyperbolic models for 3-manifolds

The tameness and ending lamination conjectures together tell us that a hyperbolic 3-manifold with finitely generated fundamental group is determined by its topology and a finite number of “end invariants”. In this paper we describe how some of this theory generalises to a much broader class of metrics. To simplify the discussion, we focus on the particular case where the 3-manifold is homotopy equivalent to a compact surface. We will state the main result (Theorem 0.7) in the “doubly degenerate” case. This case illustrates the main features of the argument, though further generalisations are possible, as we will briefly discuss. The ending lamination conjecture is closely related to the large scale geometry of Teichmüller space, and one of the main motivations for this study is its potential applications in that direction, for example to the Weil-Petersson metric. To be more precise, let Σ be a compact orientable surface. Let M be an orientable complete riemannian 3-manifold, with a preferred homotopy equivalence M −→ Σ. We give some hypotheses under which such a manifold will serve as a “model” of a (constant curvature) hyperbolic 3-manifold. The main requirements can be paraphrased by saying that M has locally bounded geometry and a thick-thin decomposition with “standard” thin part, and that the universal cover is Gromov hyperbolic. The last assumption turns out to be equivalent to asserting that the thick part is hyperbolic relative to the thin part. There are many variations on the hypotheses that would work as well, as we will elaborate in Section 2. We begin with a more precise formulation. Given x ∈ M , we define the essential systole of M at x, denoted sys(M,x), to be the length of the shortest homotopically non-trivial loop in M passing through x. A free homotopy class of closed curves in M is parabolic if it has arbitrarily short representatives in M . It peripheral if its image in Σ can be homotoped into ∂Σ. We denote by D the unit euclidean n-ball. We make the following assumptions on M :