In mathematics, more precisely in topology and differential geometry, a **hyperbolic 3–manifold** is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries (a Kleinian group).

- Importance in topology
- Structure
- Manifolds of finite volume
- Geometrically finite manifolds
- Manifolds with finitely generated fundamental group
- Construction of hyperbolic 3–manifolds of finite volume
- Hyperbolic polyhedra, reflection groups
- Gluing ideal tetrahedra and hyperbolic Dehn surgery
- Arithmetic constructions
- The hyperbolisation theorem
- Virtual properties
- The space of all hyperbolic 3–manifolds
- Geometric convergence
- Jørgensen–Thurston theory
- Quasi-Fuchsian groups
- Notes
- References

Hyperbolic 3–manifolds of finite volume have a particular importance in 3–dimensional topology as follows from Thurston's geometrisation conjecture proved by Perelman. The study of Kleinian groups is also an important topic in geometric group theory.

Hyperbolic geometry is the most rich and least understood of the eight geometries in dimension 3 (for example, for all other geometries it is not hard to give an explicit enumeration of the finite-volume manifolds with this geometry, while this is far from being the case for hyperbolic manifolds). After the proof of the Geometrisation conjecture, understanding the topological properties of hyperbolic 3–manifolds is thus a major goal of 3-dimensional topology. Recent breakthroughs of Kahn–Markovic, Wise, Agol and others have answered most long-standing open questions on the topic but there are still many less prominent ones which have not been solved.^{ [1] }

In dimension 2 almost all closed surfaces are hyperbolic (all but the sphere, projective plane, torus and Klein bottle). In dimension 3 this is far from true: there are many ways to construct infinitely many non-hyperbolic closed manifolds. On the other hand the heuristic statement that "a generic 3–manifold tends to be hyperbolic" is verified in many contexts. For example, any knot which is not either a satellite knot or a torus knot is hyperbolic.^{ [2] } Moreover almost all Dehn surgeries on a hyperbolic knot yield a hyperbolic manifold. A similar result is true of links (Thurston's hyperbolic Dehn surgery theorem), and since all 3–manifolds are obtained as surgeries on a link in the 3–sphere this gives a more precise sense to the informal statement. Another sense in which "almost all" manifolds are hyperbolic in dimension 3 is that of random models. For example random Heegaard splittings of genus at least 2 are almost surely hyperbolic (when the complexity of the gluing map goes to infinity).^{ [3] }

The relevance of the hyperbolic geometry of a 3–manifold to its topology also comes from the Mostow rigidity theorem, which states that the hyperbolic structure of a hyperbolic 3–manifold of finite volume is uniquely determined by its homotopy type. In particular geometric invariant such as the volume can be used to define new topological invariants.

In this case one important tool to understand the geometry of a manifold is the thick-thin decomposition. It states that a hyperbolic 3–manifold of finite volume has a decomposition into two parts:

- the
*thick*part, where the injectivity radius is larger than an absolute constant; - and its complement, the
*thin*part, which is a disjoint union of solid tori and cusps.

The thick-thin decomposition is valid for all hyperbolic 3–manifolds, though in general the thin part is not as described above. A hyperbolic 3–manifold is said to be geometrically finite if it contains a convex submanifold (its *convex core*) onto which it retracts, and whose thick part is compact (note that all manifolds have a convex core, but in general it is not compact).^{ [4] } The simplest case is when the manifold does not have "cusps" (i.e. the fundamental group does not contain parabolic elements), in which case the manifold is geometrically finite if and only if it is the quotient of a closed, convex subset of hyperbolic space by a group acting cocompactly on this subset.

This is the larger class of hyperbolic 3–manifolds for which there is a satisfying structure theory. It rests on two theorems:

- The tameness theorem which states that such a manifold is homeomorphic to the interior of a compact manifold with boundary;
- The ending lamination theorem which provides a classification of hyperbolic structure on the interior of a compact manifold by its "end invariants".

The oldest construction of hyperbolic manifolds, which dates back at least to Poincaré, goes as follows: start with a finite collection of 3-dimensional hyperbolic finite polytopes. Suppose that there is a side-paring between the 2-dimensional faces of these polyhedra (i.e. each such face is paired with another, distinct, one so that they are isometric to each other as 2-dimensional hyperbolic polygons), and consider the space obtained by gluing the paired faces together (formally this is obtained as a quotient space). It carries a hyperbolic metric which is well-defined outside of the image of the 1-skeletons of the polyhedra. This metric extends to a hyperbolic metric on the whole space if the two following conditions are satisfied:^{ [5] }

- for each (non-ideal) vertex in the gluing the sum of the solid angles of the polyhedra to which it belongs is equal to ;
- for each edge in the gluing the sum of the dihedral angles of the polyhedra to which it belongs is equal to .

A notable example of this construction is the Seifert–Weber space which is obtained by gluing opposite faces of a regular dodecahedron.

A variation on this construction is by using hyperbolic Coxeter polytopes (polytopes whose dihedral angles are of the form ). Such a polytope gives rise to a Kleinian reflection group, which is a discrete subgroup of isometries of hyperbolic space. Taking a torsion-free finite-index subgroup one obtains a hyperbolic manifold (which can be recovered by the previous construction, gluing copies of the original Coxeter polytope in a manner prescribed by an appropriate Schreier coset graph).

In the previous construction the manifolds obtained are always compact. To obtain manifolds with cusps one has to use polytopes which have ideal vertices (i.e. vertices which lie on the sphere at infinity). In this setting the gluing construction does not always yield a complete manifold. Completeness is detected by a system of equations involving the dihedral angles around the edges adjacent to an ideal vertex, which are commonly called Thurston's gluing equations. In case the gluing is complete the ideal vertices become cusps in the manifold. An example of a noncompact, finite volume hyperbolic manifold obtained in this way is the Gieseking manifold which is constructed by glueing faces of a regular ideal hyperbolic tetrahedron together.

It is also possible to construct a finite-volume, complete hyperbolic manifold when the gluing is not complete. In this case the completion of the metric space obtained is a manifold with a torus boundary and under some (not generic) conditions it is possible to glue a hyperbolic solid torus on each boundary component so that the resulting space has a complete hyperbolic metric. Topologically, the manifold is obtained by hyperbolic Dehn surgery on the complete hyperbolic manifold which would result from a complete gluing.

It is not known whether all hyperbolic 3–manifolds of finite volume can be constructed in this way.^{ [6] } In practice however this is how computational software (such as SnapPea or Regina) stores hyperbolic manifolds.^{ [7] }

The construction of arithmetic Kleinian groups from quaternion algebras gives rise to particularly interesting hyperbolic manifolds. On the other hand they are in some sense "rare" among hyperbolic 3-manifolds (for example hyperbolic Dehn surgery on a fixed manifold results in a non-arithmetic manifold for almost all parameters).

In contrast to the explicit constructions above it is possible to deduce the existence of a complete hyperbolic structure on a 3–manifold purely from topological information. This is a consequence of the Geometrisation conjecture and can be stated as follows (a statement sometimes referred to as the "hyperbolisation theorem", which was proven by Thurston in the special case of Haken manifolds):

If a compact 3-manifold with toric boundary is irreducible and algebraically atoroidal (meaning that every -injectively immersed torus is homotopic to a boundary component) then its interior carries a complete hyperbolic metric of finite volume.

A particular case is that of a surface bundle over the circle: such manifolds are always irreducible, and they carry a complete hyperbolic metric if and only if the monodromy is a pseudo-Anosov map.

Another consequence of the Geometrisation conjecture is that any closed 3-manifold which admits a Riemannian metric with negative sectional curvatures admits in fact a Riemannian metric with constant sectional curvature -1. This is not true in higher dimensions.^{ [8] }

The topological properties of 3–manifolds are sufficiently intricate that in many cases it is interesting to know that a property holds virtually for a class of manifolds, that is for any manifold in the class there exists a finite covering space of the manifold with the property. The virtual properties of hyperbolic 3–manifolds are the objects of a series of conjectures by Waldhausen and Thurston, which were recently all proven by Ian Agol following work of Jeremy Kahn, Vlad Markovic, Frédéric Haglund, Dani Wise and others. The first part of the conjectures were logically related to the virtually Haken conjecture. In order of strength they are:^{ [9] }

- (the surface subgroup conjecture) The fundamental group of any hyperbolic manifold of finite volume contains a (non-free) surface group (the fundamental group of a closed surface).
- (the Virtually Haken conjecture) Any hyperbolic 3–manifold of finite volume is virtually Haken; that is, it contains an embedded closed surface such that the embedding induces an injective map between fundamental groups.
- Any hyperbolic 3–manifold of finite volume has a finite cover with a nonzero first Betti number.
- Any hyperbolic 3–manifold of finite volume has a finite cover whose fundamental group surjects onto a non-abelian free group (such groups are usually called
*large*).

Another conjecture (also proven by Agol) which implies 1-3 above but a priori has no relation to 4 is the following :

- 5. (the virtually fibered conjecture) Any hyperbolic 3–manifold of finite volume has a finite cover which is a surface bundle over the circle.

A sequence of Kleinian groups is said to be *geometrically convergent* if it converges in the Chabauty topology. For the manifolds obtained as quotients this amounts to them being convergent in the pointed Gromov-Hausdorff metric.

The hyperbolic volume can be used to order the space of all hyperbolic manifold. The set of manifolds corresponding to a given volume is at most finite, and the set of volumes is well-ordered and of order type . More precisely, Thurston's hyperbolic Dehn surgery theorem implies that a manifold with cusps is a limit of a sequence of manifolds with cusps for any , so that the isolated points are volumes of compact manifolds, the manifolds with exactly one cusp are limits of compact manifolds, and so on. Together with results of Jørgensen the theorem also proves that any convergent sequence must be obtained by Dehn surgeries on the limit manifold.^{ [10] }

Sequences of quasi-fuchsian surface groups of given genus can converge to a doubly degenerate surface group, as in the double limit theorem.

- ↑ Aschenbrenner, Friedl & Wilton 2015, Chapter 9.
- ↑ Thurston 1982, Corollary 2.5.
- ↑ Maher 2010.
- ↑ Ratcliffe 2006, Theorem 12.7.2.
- ↑ Ratcliffe 2006, Theorems 10.1.2 and 10.1.3.
- ↑ Petronio & Porti 2000.
- ↑ Callahan, Hildebrand & Weeks 1999.
- ↑ Gromov & Thurston 1987.
- ↑ Aschenbrenner, Friedl & Wilton 2015.
- ↑ Gromov 1981.

**William Paul Thurston** was an American mathematician. He was 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. From 2003 until his death he was a professor of mathematics and computer science at Cornell University.

**Riemannian geometry** is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a *Riemannian metric*, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

In mathematics, **Thurston's geometrization conjecture** states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries . In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.

In mathematics, **low-dimensional topology** is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

**Geometric group theory** is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act.

In mathematics, a **3-manifold** is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

In mathematics, a **Kleinian group** is a discrete subgroup of PSL(2, **C**). The group PSL(2, **C**) 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 space **H**^{3}, and as orientation-preserving conformal maps of the open unit ball *B*^{3} in **R**^{3} to itself. Therefore, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.

In mathematics, **Mostow's rigidity theorem**, or **strong rigidity theorem**, or **Mostow–Prasad rigidity theorem**, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique. The theorem was proven for closed manifolds by Mostow (1968) and extended to finite volume manifolds by Marden (1974) in 3 dimensions, and by Prasad (1973) in all dimensions at least 3. Gromov (1981) gave an alternate proof using the Gromov norm. Besson, Courtois, and Gallot (1996) gave the simplest available proof.

In mathematics, a **pair of pants** is a surface which is homeomorphic to the three-holed sphere. The name comes from considering one of the removed disks as the waist and the two others as the cuffs of a pair of pants.

In group theory, more precisely in geometric group theory, a **hyperbolic group**, also known as a *word hyperbolic group* or *Gromov hyperbolic group*, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by Mikhail Gromov (1987). The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology, and combinatorial group theory. In a very influential chapter from 1987, Gromov proposed a wide-ranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others.

In mathematics, **hyperbolic Dehn surgery** is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes hyperbolic geometry in three dimensions from other dimensions.

In mathematics, the **geometric topology** is a topology one can put on the set *H* of hyperbolic 3-manifolds of finite volume.

In differential geometry, a subfield of mathematics, the **Margulis lemma** is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifolds. Roughly, it states that within a fixed radius, usually called the **Margulis constant**, the structure of the orbits of such a group cannot be too complicated. More precisely, within this radius around a point all points in its orbit are in fact in the orbit of a nilpotent subgroup.

In geometry, Thurston's **geometrization theorem** or **hyperbolization theorem** implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture.

In mathematics, the **2 π theorem** of Gromov and Thurston states a sufficient condition for Dehn filling on a cusped hyperbolic 3-manifold to result in a negatively curved 3-manifold.

In mathematics, more precisely in group theory and hyperbolic geometry, **Arithmetic Kleinian groups** are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An **arithmetic hyperbolic three-manifold** is the quotient of hyperbolic space by an arithmetic Kleinian group. These manifolds include some particularly beautiful or remarkable examples.

**James W. Cannon** is an American mathematician working in the areas of low-dimensional topology and geometric group theory. He was an Orson Pratt Professor of Mathematics at Brigham Young University.

In hyperbolic geometry, the **ending lamination theorem**, originally conjectured by William Thurston (1982), states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold.

In geometric group theory, the **Rips machine** is a method of studying the action of groups on **R**-trees. It was introduced in unpublished work of Eliyahu Rips in about 1991.

**Chern's conjecture for affinely flat manifolds** was proposed by Shiing-Shen Chern in 1955 in the field of affine geometry. As of 2018, it remains an unsolved mathematical problem.

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