Hyperbolic 3-manifold

Hyperbolic 3-manifold

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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 made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...

 equipped with a complete
Complete space
In mathematical analysis, a metric space M is called 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 K depends on a two-dimensional plane σp in the tangent space at p...

 -1. In other words, it is the quotient of three-dimensional hyperbolic space
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...

 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
Gieseking manifold
In mathematics, the Gieseking manifold, is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately 1.01494161...

, in 1912. It is constructed by gluing faces of an ideal hyperbolic tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

 together.

The complements of knots
Knot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...

 and links
Knot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...

 in the 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...

 are frequently cusped hyperbolic manifolds. Examples include the complements of 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...

 and the Borromean rings
Borromean rings
In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link, i.e., removing any ring results in two unlinked rings.- Mathematical properties :...

 and the Whitehead link
Whitehead link
In knot theory, the Whitehead link, discovered by J.H.C. Whitehead, is one of the most basic links.J.H.C. Whitehead spent much of the 1930s looking for a proof of the Poincaré conjecture...

. More generally, geometrization implies that a knot which is neither a satellite knot
Satellite knot
In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non-boundary parallel torus in its complement. The class of satellite knots include composite knots, cable knots and Whitehead doubles. A satellite knot K can be picturesquely described as follows:...

 nor a torus knot
Torus knot
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q...

 is a hyperbolic knot.

Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...

'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, 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
Weeks manifold
In mathematics, the Weeks manifold, sometimes called the Fomenko–Matveev–Weeks manifold, is a closed hyperbolic 3-manifold obtained by and Dehn surgeries on the Whitehead link. It has volume approximately equal to 0.9427... and showed that it has the smallest volume of any closed orientable...

 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
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. In general, fiber bundles over the circle are a special case of mapping tori....

 to be hyperbolic: the monodromy
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...

 of the bundle should be pseudo-Anosov. This is part of his celebrated hyperbolization theorem
Hyperbolization theorem
In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture.-Statement:...

 for Haken manifold
Haken manifold
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface...

s.

According to Thurston's geometrization conjecture, proved by Perelman, 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 embedded, non-boundary parallel, incompressible torus....

 3-manifold with infinite fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological 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.

See also

  • Hyperbolic manifold
    Hyperbolic manifold
    In mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold...

  • Kleinian group
    Kleinian group
    In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...

  • Mostow rigidity theorem
    Mostow rigidity theorem
    In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique...

  • Arithmetic hyperbolic 3-manifold
    Arithmetic hyperbolic 3-manifold
    In mathematics, an arithmetic hyperbolic 3-manifold is a hyperbolic 3-manifold whose fundamental group is an arithmetic group as a subgroup of PGL...