3 Dimensional Geometry, Heegaard Splittings and Rank of the Fundamental Group

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After the work of Perelman, most 3-manifolds are known to admit a hyperbolic metric, i.e. a metric of constant negative curvature. Unfortunately, with few exceptions, no further properties of this metric are known. In this project the P.I. will study how the topology of the 3-manifold and the properties of the hyperbolic geometry are related to each other. A first goal is to obtain explicite estimates of geometric data in terms of topological and combinatorial information such as the rank of the fundamental group of the Heegaard genus of the manifold. The second goal is to use these geometric information to reconstruct, under suitable assumptions, the hyperbolic metric itself.A 3-manifold is a mathematical object of fundamental interest. For example, the space we live in is a 3-manifold. A recent trend in the study of 3-manifolds is to encode as much of their fine, infinitely complicated, geometric structure in finite combinatorial models. A certain amount of information is lost in the process. The goal of the project is to quantify how much information actually gets lost. Obtaining concrete a priori estimates is then crucial. For example, they open the door to predictions in terms of finite models of phenomena occurig in 3-manifolds. Surprisingly, it seems very likely that in many situations sufficiently precise a priori estimates will allow to recover all the geometric information. Concrete estimates will make possible to have accurate computer simulations.