Geometry and Combinatorics of Hyperbolic 3-manifolds

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DMS-0203976Yair MinskyIn this project Minsky will study connections between the geometricstructure of hyperbolic 3-manifolds, and the combinatorial structureof surfaces. The main focus of the project is a program to solveThurston's Ending Lamination Conjecture, which states that ahyperbolic 3-manifold is uniquely determined by its topological typeand a list of invariants describing the asymptotic geometry of itsends. This conjecture is central in the field and would settle anumber of outstanding questions about the structure of the deformationspace of all hyperbolic structures on a given manifold. The principaltool in this approach is the Complex of Curves, a combinatorial objectwhich describes the set of all essential homotopy classes of simpleloops on a given surface. Such a complex, viewed as a metric space,is hyperbolic in the sense of Cannon and Gromov, and the invariantswhich describe the geometry of a 3-manifold turn out to be "points atinfinity" for the complex. A study of geodesics within the complex ofcurves then allows us to construct a model for the geometry of the3-manifold which depends only on its end invariants. Part of theprogram is joint work with Brock and Canary. The complex of curves isalso instrumental in studying the Teichmuller space of hyperbolicstructures on a surface, and its symmetry group, known as the MappingClass Group. Hyperbolicity of the complex implies a type of relativehyperbolicity property for the Teichmuller space and the Mapping ClassGroup, and in collaboration with Brock and Masur, Minsky hopes toapply this to study the geometric properties of Teichmuller space, andto answer a number of group-theoretic and algorithmic questions aboutthe mapping class group.Topological spaces appear in mathematics at varying levels ofabstraction. The earth, for example, can be thought of as a sphere,spinning around its axis. The set of all positions of this sphere,perhaps as parameterized by the time of day in New York, can itself bevisualized as a circle. The positions of the earth in its orbitaround the sun also make up a circle, and all this informationtogether can be described by a 2-dimensional surface, called a torus,so that each point in the torus corresponds to exactly one position ofthe earth-sun system, and our motion in space gives rise to atrajectory winding around the torus. Once a surface is being studied,we often give it various structures: it can have a metric, which is away of measuring distance in it; it can also be filled up in differentways by loops. The set of all possible metrics can itself be studiedas a "space", where motion in this space corresponds to changingshapes of the surface. The set of all loops in the surface givesrise to another space, of a more combinatorial character. Afascinating phenomenon, not uncommon in mathematics, is the appearanceof analogies between these different levels of abstraction. Forexample, a property of a single metric on a surface called "hyperbolicity"appears again, in another guise, as a property of the space of allmetrics on the surface and the space of all loops. The interactionbetween these different levels yields interesting phenomena and toolswhich can be applied to solve a number of problems in