In mathematics, a **hyperbolic manifold** is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman.

A **hyperbolic -manifold** is a complete Riemannian -manifold of constant sectional curvature .

Every complete, connected, simply-connected manifold of constant negative curvature is isometric to the real hyperbolic space . As a result, the universal cover of any closed manifold of constant negative curvature is . Thus, every such can be written as where is a torsion-free discrete group of isometries on . That is, is a discrete subgroup of . The manifold has finite volume if and only if is a lattice.

Its thick–thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and ends which are the product of a Euclidean ()-manifold and the closed half-ray. The manifold is of finite volume if and only if its thick part is compact.

The simplest example of a hyperbolic manifold is hyperbolic space, as each point in hyperbolic space has a neighborhood isometric to hyperbolic space.

A simple non-trivial example, however, is the once-punctured torus. This is an example of an (Isom(), )-manifold. This can be formed by taking an ideal rectangle in – that is, a rectangle where the vertices are on the boundary at infinity, and thus don't exist in the resulting manifold – and identifying opposite images.

In a similar fashion, we can construct the thrice-punctured sphere, shown below, by gluing two ideal triangles together. This also shows how to draw curves on the surface – the black line in the diagram becomes the closed curve when the green edges are glued together. As we are working with a punctured sphere, the colored circles in the surface – including their boundaries – are not part of the surface, and hence are represented in the diagram as ideal vertices.

Many knots and links, including some of the simpler knots such as the figure eight knot and the Borromean rings, are hyperbolic, and so the complement of the knot or link in is a hyperbolic 3-manifold of finite volume.

For the hyperbolic structure on a *finite volume* hyperbolic -manifold is unique by Mostow rigidity and so geometric invariants are in fact topological invariants. One of these geometric invariants used as a topological invariant is the hyperbolic volume of a knot or link complement, which can allow us to distinguish two knots from each other by studying the geometry of their respective manifolds.

In the mathematical disciplines of topology and geometry, an **orbifold** is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.

In mathematics, **hyperbolic space** of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, **H**^{2}, which was the first instance studied, is also called the hyperbolic plane.

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.

In mathematics, a **Kleinian group** is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space **H**^{3}. The latter, identifiable with PSL(2, **C**), is the quotient group of the 2 by 2 complex matrices of determinant 1 by their center, which consists of the identity matrix and its product by −1. PSL(2, **C**) has a natural representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open unit ball *B*^{3} in **R**^{3}. The group of Möbius transformations is also related as the non-orientation-preserving isometry group of **H**^{3}, PGL(2, **C**). So, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.

In mathematics, a **Fuchsian model** is a representation of a hyperbolic Riemann surface *R* as a quotient of the upper half-plane **H** by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation. The concept is named after Lazarus Fuchs.

In mathematics, the **Teichmüller space** of a (real) topological surface , is a space that parametrizes complex structures on up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller.

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 & Gallot (1996) gave the simplest available proof.

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 differential geometry, the **Margulis lemma** is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifold. 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 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.

In Lie theory and related areas of mathematics, a **lattice** in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of **R**^{n}, this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood.

In mathematics, the concept of a **relatively hyperbolic group** is an important generalization of the geometric group theory concept of a hyperbolic group. The motivating examples of relatively hyperbolic groups are the fundamental groups of complete noncompact hyperbolic manifolds of finite volume.

In mathematics, specifically in group theory, two groups are **commensurable** if they differ only by a finite amount, in a precise sense. The **commensurator** of a subgroup is another subgroup, related to the normalizer.

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.

**Arithmetic Fuchsian groups** are a special class of Fuchsian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. The prototypical example of an arithmetic Fuchsian group is the modular group . They, and the hyperbolic surface associated to their action on the hyperbolic plane often exhibit particularly regular behaviour among Fuchsian groups and hyperbolic surfaces.

In Lie theory, an area of mathematics, the **Kazhdan–Margulis theorem** is a statement asserting that a discrete subgroup in semisimple Lie groups cannot be too dense in the group. More precisely, in any such Lie group there is a uniform neighbourhood of the identity element such that every lattice in the group has a conjugate whose intersection with this neighbourhood contains only the identity. This result was proven in the sixties by David Kazhdan and Grigory Margulis.

In the mathematical subject of geometric group theory, the **Švarc–Milnor lemma** is a statement which says that a group , equipped with a "nice" discrete isometric action on a metric space , is quasi-isometric to .

In the mathematical subject of group theory, a **co-Hopfian group** is a group that is not isomorphic to any of its proper subgroups. The notion is dual to that of a Hopfian group, named after Heinz Hopf.

In mathematics, the **Cartan–Ambrose–Hicks theorem** is a theorem of Riemannian geometry, according to which the Riemannian metric is locally determined by the Riemann curvature tensor, or in other words, behavior of the curvature tensor under parallel translation determines the metric.

In mathematics, a **Cannon–Thurston map** is any of a number of continuous group-equivariant maps between the boundaries of two hyperbolic metric spaces extending a discrete isometric actions of the group on those spaces.

- Kapovich, Michael (2009) [2001],
*Hyperbolic manifolds and discrete groups*, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, doi:10.1007/978-0-8176-4913-5, ISBN 978-0-8176-4912-8, MR 1792613 - Maclachlan, Colin; Reid, Alan W. (2003),
*The arithmetic of hyperbolic 3-manifolds*, Graduate Texts in Mathematics, vol. 219, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98386-8, MR 1937957 - Ratcliffe, John G. (2006) [1994],
*Foundations of hyperbolic manifolds*, Graduate Texts in Mathematics, vol. 149 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-47322-2, ISBN 978-0-387-33197-3, MR 2249478

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