Three-Manifolds and Geometry

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Three-manifolds are spaces closely related to the universe we live in. To develop a better understanding of such spaces one calls upon methods from several research areas, such as geometric topology, algebra, number theory, and theoretical physics. An invariant of a manifold is an associated object that depends only on the manifold type, and carries useful information. This NSF funded project addresses several important, unsolved problems related to classification of three-manifolds and realization of number-theoretic manifold invariants. Bi-Lipschitz geometry is a natural framework for the geometric study of universal covers of compact manifolds, where it connects geometric topology with geometric group theory. It is also the natural framework for the study of the local geometry of algebraic sets. In both situations, three-dimensional manifolds play an important role. One aim of the project is to complete the quasi-isometric classification of 3-manifold groups, a project on which a large number of researchers have been working for two decades, and on which the PI and Behrstock made very significant progress over the last several years. Another is to apply Lipschitz geometry of complex singularities to significant open questions, such as the geometric meaning of Zariski equisingularity in higher dimensions (the PI and Anne Pichon recently resolved the 2-dimensional case), the Zariski multiplicity conjecture, classification of complex map germs and more. At the other end of the spectrum, for hyperbolic manifolds there are also postulated connections between geometric, arithmetic, representation-theoretic and quantum based invariants of manifolds, on which the PI and his students have made progress, and which is continuing.