Hyperbolic 3-manifold

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Hyperbolic 3-manifold

A hyperbolic 3-manifold is a 3-manifold

3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds....
equipped with a complete

Complete space

In mathematical analysis, a metric space M is said to be complete if every Cauchy sequence of points in M has a limit that is also in M or alternatively if every Cauchy sequence in M converges in M....
Riemannian metric of constant sectional curvature

Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature depends on a two-dimensional plane in the tangent space at p....
-1. In other words, it is the quotient of three-dimensional hyperbolic space

Hyperbolic space

In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1....
by a subgroup of hyperbolic isometries acting freely and properly discontinuously. See also Kleinian model

Kleinian model

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Encyclopedia

A hyperbolic 3-manifold is a 3-manifold

3-manifold

Complete space

Sectional curvature

Hyperbolic space

Kleinian model

In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space where Γ is a discrete group of PSL....
.

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

Figure-eight knot (mathematics)

In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot and trefoil knot....
, Borromean rings

Borromean rings

In mathematics, the Borromean rings consist of three topological circles which are link ed and form a Brunnian link, i.e., removing any ring results in two unlinked rings....
, and many 2-bridge knot

2-bridge knot

In the mathematical field of knot theory, a 2-bridge knot is a knot which can be isotopy so that the natural height function given by the z-coordinate has only two maxima and two minima as critical points....
s. Thurston

William Thurston

William Paul Thurston is an United States mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields medal for the depth and originality of his contributions to mathematics....
's theorem on hyperbolic Dehn surgery

Hyperbolic Dehn surgery

In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold....
states that most Dehn fillings on hyperbolic links and all but finitely many Dehn fillings on hyperbolic knots result in closed

Closed

Closed may refer to:Math* Closure * Closed manifold* Orbit #Closed Orbits* Closed set* One-formOther* Cloister, a closed walkway...
hyperbolic 3-manifolds.

One can sometimes manually construct a hyperbolic 3-manifold, such as with Seifert-Weber space

Seifert-Weber space

In mathematics, Herbert_Seifert-Weber space is a closed hyperbolic 3-manifold. It is also known as Seifert-Weber dodecahedral space and hyperbolic dodecahedral 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

Surface bundle over the circle

In mathematics, a surface bundle over the circle is a fiber bundle with base space a circle, and with fiber space a surface. Therefore the total space has dimension 2 + 1 = 3....
to be hyperbolic: the monodromy

Monodromy

In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic geometry and differential geometry behave as they 'run round' a Mathematical singularity....
of the bundle should be pseudo-Anosov. This is part of his celebrated geometrization theorem for Haken manifold

Haken manifold

In mathematics, a Haken manifold is a compact space, P?-irreducible 3-manifold that contains a 2-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface....
s.

According to Thurston's geometrization conjecture, any closed, irreducible, atoroidal

Atoroidal

In mathematics, there are three definitions for atoroidal as applied to 3-manifolds:*A 3-manifold is atoroidal if it does not contain an embedding, non-boundary parallel, Incompressible surface torus....
3-manifold with infinite fundamental group

Fundamental group

In mathematics, more specifically algebraic topology, the fundamental group or Poincar? group is a group associated to any given pointed space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other....
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.