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Geometric structures on prime 3-manifolds

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Alice Kwon (CUNY)

Abstract: Geometrization of 3-manifolds is stated in terms of Thurston's eight locally homogeneous Riemannian geometries for certain prime 3-manifolds which are topological building blocks for all 3-manifolds. These homogeneous Riemannian geometries, as Klein envisioned, are included in more general homogeneous geometries like projective geometry or any locally homogeneous geometry defined by a Lie group. These Lie geometries have pictorial dynamical aspects related to developing maps and holonomy representations. This picture was elegantly formulated by Charles Ehresmann and Elie Cartan around 1950s using (G,X) structures. In joint work with Dennis Sullivan, we extend the Cartan-Ehresmann concept to the notion of a "finite Lie diagram" geometry and use it to prove that all prime 3-manifolds have a geometric structure in this sense. The motivation was to relate the topological gluing data for 3-manifolds with subsets of Thurston's Lie groups in order to define a geometric structure on prime 3-manifolds not carrying one pure Thurston geometry.