topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric...

3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds...

arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space H 3 {\displaystyle \mathbb {H} ^{3}} by an arithmetic Kleinian...

Weeks manifold is an arithmetic hyperbolic 3-manifold, its volume can be computed using its arithmetic data and a formula due to Armand Borel: V w = 3 ⋅ 23...

(a 2-manifold) with a hyperbolic metric (a Riemannian metric of constant sectional curvature −1). If Γ {\displaystyle \Gamma } is an arithmetic Fuchsian...

at s = 2 in terms of the dilogarithm function, by studying arithmetic hyperbolic 3-manifolds. He later formulated a general conjecture giving formulas...

manifolds. A particularly active research topic has been arithmetic hyperbolic 3-manifolds, which as William Thurston wrote, "...often seem to have special...

\mathbf {R} )}}} is a line bundle over the hyperbolic plane. When imbued with a left-invariant metric, the 3-manifold SL ( 2 , R ) ¯ {\displaystyle {\overline...

In hyperbolic geometry, the Meyerhoff manifold is the arithmetic hyperbolic 3-manifold obtained by ( 5 , 1 ) {\displaystyle (5,1)} surgery on the figure-8...

a Scottish-American mathematician working primarily with arithmetic hyperbolic 3-manifolds. He is the Edgar Odell Lovett Chair of mathematics at Rice...

EMS Press Maclachlan, Colin; Reid, Alan W. (2003), The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics, vol. 219, Berlin, New York:...

Thurston in his influential 1982 paper Three-dimensional manifolds, Kleinian groups and hyperbolic geometry published in the Bulletin of the American Mathematical...

direct product of hyperbolic Riemann surfaces. Otherwise it is irreducible. The irreducible manifolds fundamental groups are arithmetic groups by Margulis'...

isospectral, non-isometric closed hyperbolic 2-manifolds and 3-manifolds as quotients of hyperbolic 2-space and 3-space by arithmetic subgroups, constructed using...

studying systolic invariants of manifolds and polyhedra. Systolic hyperbolic geometry the study of systoles in hyperbolic geometry. Contents:  Top A B C...

is. For this reason it is natural to think of the squeeze mapping as a hyperbolic rotation, as did Émile Borel in 1914, by analogy with circular rotations...

model Constructions in hyperbolic geometry Hjelmslev transformation Hyperbolic 3-manifold Hyperbolic manifold Hyperbolic set Hyperbolic tree Kleinian group...

rings is referred to as arithmetic geometry. Algebraic number theory is also used in the study of arithmetic hyperbolic 3-manifolds. Class field theory Kummer...

geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of single variable calculus, vector calculus,...

to hyperbolic groups and to higher rank linear groups[citation needed]. They have many applications in Thurston's theory of geometric three-manifolds (for...

complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables...

S<\infty .\end{aligned}}} Koch snowflake – Fractal curve Picard horn – Hyperbolic 3-manifold proposed as a model for the shape of the universe Pseudosphere –...

precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group...

theory of word-hyperbolic groups (also referred to as Gromov-hyperbolic or negatively curved groups)." Brian Bowditch, Hyperbolic 3-manifolds and the geometry...

prototypical example of a Riemann surface, and is one of the simplest complex manifolds. In projective geometry, the sphere is an example of a complex projective...

geometry (through the construction of locally homogeneous manifolds), in number theory (through arithmetic groups), in ergodic theory (through the study of homogeneous...

{\displaystyle 2^{I}} is the Cantor space. All points of a zero-dimensional manifold are isolated. Arhangel'skii, Alexander; Tkachenko, Mikhail (2008). Topological...

{\displaystyle \mathbb {R} ^{3}.} Gieseking manifold − A cusped hyperbolic 3-manifold of finite volume. Horosphere Horocycle Picard horn Seifert–Weber...

an ideal polyhedron forms a hyperbolic manifold, topologically equivalent to a punctured sphere, and every such manifold forms the surface of a unique...

Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem. Many knots were shown to be hyperbolic knots, enabling the...