Hyperbolic Geometry and Combinatorial Surface Topology

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PI: Yair MinskyTitle: Hyperbolic Geometry and Combinatorial Surface Proposal: 9971596TopologyAbstract: Minsky will investigate phenomena of rigidity, deformation and classification in the related theories of hyperbolic 3-manifolds (Kleinian groups) and mapping class groups of surfaces. In Kleinian groups Minsky plans to continue work on Thurston's Ending LaminationConjecture, which states that a hyperbolic 3-manifold is uniquely determined by its topological type and a list of invariants that describe the asymptotic geometry of its ends. Consequences of this conjecture are a rigidity theorem for Kleinian group actions on the sphere, and a description of parameter spaces of isomorphic Kleinian groups. In collaboration with H. Masur and B. Farb, Minsky plans to consider the question of quasi-isometric rigidity of the mapping class group of a surface, which says that all quasi-isometries of this group must in fact respect its group structure up to a bounded error. A common tool to both sets of questions is the "complex of curves" on a surface, which is a simplicial complex encoding the combinatorial structure of the set of isotopy classes of simple curves on the surface. Masur and Minsky have previously established a hyperbolicity property for this complex, with applications to the conjugacy problem for mapping class groups. Minsky has found explicit connections between geometric properties of this complex and those of Kleinian representations of surface groups, and this work should have applications to the ending lamination problem. Jointly with R. Canary and J. Brock, he will also investigate the structure of geometric limits of Kleinian groups, which can be quite intricate. A good description of the set of all geometric limits is an important tool for obtaining uniform estimates on individual groups.The interactions between geometry, topology and dynamics have been a beautiful and powerful feature of mathematics and physics for more than a hundred years. Dynamics is the study of time-evolution of mathematical or physical systems, whereas geometry and topology involve "static" objects such as surfaces or higher-dimensional analogues, often the background for a dynamical process. Henri Poincare already knew that the standard round sphere, the setting of classical analysis and geometry, functioned also as a "horizon at infinity" for an exotic non-Euclidean geometry that we now call Hyperbolic space. Dynamical properties of transformations of the sphere translate to geometric properties of rigid motions of this space, and give rise to families of symmetric tilings whose structure we can study by geometric and topological methods. The complexity of these systems can constrain them so much that a combinatorial (or topological) description suffices to determine them uniquely, and this is what we call rigidity. This phenomenon occurs in many guises throughout geometry and dynamics, and is relevant to issues such as classification of systems, mapping out regions of stability and instability, deformation and bifurcation of families of systems, and probabilistic properties such as ergodicity, all of which have significance in both pure and applied mathematics. The particular aspects studied in this project are typical in some ways and special in others. A better understanding of them promises to shed light on other parts of the field, both by way of examples and by the formulation of organizing principles.