Hyperbolic 3-manifold
From Wikipedia, the free encyclopedia
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.
[edit] Constructions
The first cusped hyperbolic 3-manifold to be discovered was the Gieseking manifold, in 1912. It is constructed by gluing faces of an ideal hyperbolic tetrahedron together.
The complements of knots and links in the 3-sphere are frequently cusped hyperbolic manifolds. Examples include the figure-eight knot and the Borromean rings. More generally, geometrization implies that a knot which not a satellite knot not a torus knot is a hyperbolic knot.
Thurston's theorem on hyperbolic Dehn surgery states that, provided a finite collection of filling slopes are avoided, the remaining Dehn fillings on hyperbolic links are hyperbolic 3-manifolds.
The Seifert-Weber space is a compact hyperbolic 3-manifold, obtained by gluing opposite faces of a dodecahedron together.
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.
[edit] See also
[edit] 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.
