Hyperbolic 3-manifold

Hyperbolic 3-manifold

A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian 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. See also Kleinian model.

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 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.

ee also

*hyperbolic manifold
*Mostow rigidity theorem

References

* W. Thurston, "The geometry and topology of 3-manifolds," Princeton lecture notes (1980). Available via MSRI: http://www.msri.org/publications/books/gt3m/

* W. Thurston, "3-dimensional geometry and topology," Princeton University Press. 1997.

* W. Thurston, "Three-dimensional manifolds, Kleinian group and hyperbolic geometry," Bull. AMS 6 (1982) no 3. 357--381.


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