The classification of finitely generated Kleinian groups and their associated quotient hyperbolic 3-manifolds has generated significant new tools for studying many problems at the interfaces of Teichmüller theory, Kleinian groups and low-dimensional topology. In particular, the existence of a model manifold for a hyperbolic 3-manifold M that is uniformly bi-Lipschitz to M has foregrounded the extent to which many problems in the study of closed and finite volume hyperbolic 3- manifolds can be understood in terms of combinatorial structures associated to surfaces. Likewise, large-scale questions in the geometry of Teichmüller space have come into relief in terms of a new understanding of these combinatorics: the asymptotic geometry of geodesics in various metrics has been reconstituted and understood in a new language, yet the structure of the classical Weil-Petersson metric from this point of view remains largely unclear and tantalizingly open. Our proposed research will demonstrate how model manifolds serve as building blocks for hyperbolic structures on closed manifolds via Heegaard splittings, to develop control on the synthetic geometry and dynamics of the Weil-Petersson metric on Teichmüller space via the complex of curves, and to continue to reveal applications of the model manifolds to the topology of deformation spaces of hyperbolic 3-manifolds.
The idea of a "coarse model" in geometry proposes that one might sacrifice a certain degree of precision in the interest of capturing more large-scale structure. Frequently a coarse model plays a similar role to DNA in biology: it can determine fine features of a space despite its apparently coarse nature. In a recent result of the P.I. with R. Canary and Y. Minsky, such models were used to classify all `constantly negatively curved,' or `hyperbolic' three-dimensional spaces of infinite volume that are `tame' in a certain sense. The classification result solved a long-standing conjecture of William Thurston, and opened the door to developing a more detailed and complete picture of geometries on manifolds previously considered understood. After Perelman's solution to Thurston's geometrization conjecture and the famous Poincaré conjecture, the groundwork is in place for a fundamental investigation of algebraic, geometric and topological properties of all spaces of 3-dimensions and how these properties interrelate.
