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Explore the fundamental concepts of 3-manifold topology and delve into Thurston's geometric perspective on 3-manifolds in this comprehensive lecture. Begin with a basic overview of 3-manifold topology before examining Thurston's geometric viewpoint, including an in-depth discussion of Perelman's Geometrization Theorem. Discover the pervasive role of hyperbolic geometry in addressing purely topological questions in three dimensions. Investigate the algorithmic and group-theoretic implications of Geometrization, focusing on the solvability of the homeomorphism problem and residual finiteness of fundamental groups in compact 3-manifolds. Gain practical insights into identifying hyperbolic structures on 3-manifolds and learn how to apply these concepts in large-scale computational experiments. This 58-minute talk, presented by Nathan Dunfield at the Centre de recherches mathématiques (CRM) Workshop on Groups around 3-Manifolds, offers a comprehensive exploration of geometrization and its far-reaching consequences in the field of 3-manifold topology.
