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Hyperbolic 3-manifolds
Article By:
Meyerhoff, Robert Department of Mathematics, Boston College, Chestnut Hill, Massachusetts.
Storm, Peter A. Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania.
Last reviewed:2010
DOI:https://doi.org/10.1036/1097-8542.YB100181
Hyperbolic geometry is the non-Euclidean geometry discovered by János Bolyai, N. I. Lobachevsky, and K. F. Gauss about 200 years ago. Work of George Daniel Mostow, William Thurston, and Gregory Perelman shows that hyperbolic 3-manifolds are the most important class of three-dimensional (3-D) manifolds. Therefore, it is of great importance to come to a deep understanding of hyperbolic 3-manifolds. The first part of this article begins with basic ideas, but ultimately focuses on the most natural tool for analyzing a hyperbolic 3-manifold: its volume. The second part of the article investigates the geometric rigidity of hyperbolic 3-manifolds.
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