# 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.

## 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 complements of the figure-eight knot and the Borromean rings and the Whitehead link. More generally, geometrization implies that a knot which is neither a satellite knot nor 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.

The Weeks manifold has the smallest volume of any closed orientable hyperbolic 3-manifold.

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 hyperbolization theorem for Haken manifolds.

According to Thurston's geometrization conjecture, proved by Perelman, any closed, irreducible, atoroidal 3-manifold with infinite fundamental group is hyperbolic. There is an analogous statement for 3-manifolds with boundary.