| Hyperbolic 3-manifold |
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Information AboutHyperbolic 3-manifold |
| CATEGORIES ABOUT HYPERBOLIC 3-MANIFOLD | |
| 3-manifolds | |
| hyperbolic geometry | |
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Its Thick-thin Decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and/or ends which are the product of a Euclidean surface and the closed half-ray. The manifold is of finite volume if and only if its thick part is compact. In this case, the ends are of the form torus cross the closed half-ray and are called cusps. A '''cusped hyperbolic 3-manifold''' is a hyperbolic ''3''-manifold with at least one cusp. One way to generate many cusped hyperbolic 3-manifolds is to take the complement of hyperbolic knots and links, e.g. the Figure-eight Knot , Borromean Rings , and many 2-bridge Knot s. Thurston 's theorem on Hyperbolic Dehn Surgery states that most Dehn Filling s on hyperbolic links and all but finitely many Dehn fillings on hyperbolic knots result in Closed hyperbolic 3-manifolds. One can sometimes manually construct a hyperbolic 3-manifold, such as with of the bundle should be Pseudo-Anosov . This is part of his celebrated Geometrization Theorem for Haken Manifold s. According to Thurston's Geometrization Conjecture , any closed, Irreducible , Atoroidal 3-manifold with infinite Fundamental Group is hyperbolic. There is an analogous statement for 3-manifolds with boundary. (Notice that hyperbolic 3-manifolds satisfy these properties.) Heuristically, this means that "many" 3-manifolds are in fact hyperbolic. SEE ALSO |
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