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.

We can also ask what the area of the boundary of the knot complement is. As there is a relationship between the volume of a knot complement and the volume of the complement under Dehn filling,^{ [1] } we can use the area of the boundary to inform us of how the volume might change under such a filling.

In Riemannian geometry, the **scalar curvature** is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.

In geometry, the **Dehn invariant** of a polyhedron is a value used to determine whether polyhedra can be dissected into each other or whether they can tile space. It is named after Max Dehn, who used it to solve Hilbert's third problem on whether all polyhedra with equal volume could be dissected into each other.

In mathematics, in the subfield of geometric topology, the **mapping class group** is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.

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, 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 topology, a branch of mathematics, a **Dehn surgery**, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: *drilling* then *filling*.

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, 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, **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. {{harvtxt|Besson, Courtois, and Gallot|1996} https://www.researchgate.net/profile/Gilles_Courtois2/publication/231902765_Minimal_entropy_and_Mostow's_rigidity_theorems/links/02e7e538a32469eacc000000.pdf} gave the simplest available proof.

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 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 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.

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 the mathematical subject of geometric group theory, a **Dehn function**, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the *area* of a *relation* in that group in terms of the length of that relation. The growth type of the Dehn function is a quasi-isometry invariant of a finitely presented group. The Dehn function of a finitely presented group is also closely connected with non-deterministic algorithmic complexity of the word problem in groups. In particular, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation of this group is recursive. The notion of a Dehn function is motivated by isoperimetric problems in geometry, such as the classic isoperimetric inequality for the Euclidean plane and, more generally, the notion of a filling area function that estimates the area of a minimal surface in a Riemannian manifold in terms of the length of the boundary curve of that surface.

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 mathematics, and more precisely in topology, the **mapping class group** of a surface, sometimes called the **modular group** or **Teichmüller modular group**, is the group of homeomorphisms of the surface viewed up to continuous deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves.

**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 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 mathematics, the notion of 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.

- ↑ Purcell, Jessica S.; Kalfagianni, Efstratia; Futer, David (2006-12-06). "Dehn filling, volume, and the Jones polynomial". arXiv: math/0612138 . Bibcode:2006math.....12138F.Cite journal requires
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- 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,**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,**149**(2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-47322-2, ISBN 978-0-387-33197-3, MR 2249478 - Hyperbolic Voronoi diagrams made easy, Frank Nielsen

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