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Hyperbolic 3-ManifoldA hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannnian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously. Its thick-thin decomposition has a thin part consisting of tubular neighborhoods of short 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. 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 knots. Thurston's theorem on hyperbolic Dehn surgery states that most Dehn fillings 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 Seifert-Weber space, but more often, they result from constructions such as the above-mentioned Dehn filling method. Also, Thurston gave a necessary and sufficient criterion for a surface bundle over the circle to be hyperbolic: the monodromy of the bundle should be pseudo-Anosov. This is part of his celebrated geometrization theorem for Haken manifolds. 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.
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