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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.
Generalising the example of the modular group a Fuchsian group is a group admitting a properly discontinuous action on the hyperbolic plane (equivalently, a discrete subgroup of ()). The hyperbolic plane is a δ {\displaystyle \delta } -hyperbolic space and hence the Svarc—Milnor lemma tells us that cocompact Fuchsian groups are hyperbolic.
The Weeks manifold is the hyperbolic three-manifold of smallest volume [3] and the Meyerhoff manifold is the one of next smallest volume. The complement in the three-sphere of the figure-eight knot is an arithmetic hyperbolic three-manifold [4] and attains the smallest volume among all cusped hyperbolic three-manifolds. [5]
As a result, the universal cover of any closed manifold M of constant negative curvature −1, which is to say, a hyperbolic manifold, is H n. Thus, every such M can be written as H n / Γ, where Γ is a torsion-free discrete group of isometries on H n. That is, Γ is a lattice in SO + (n, 1).
In geometry, if X is a manifold with an action of a topological group G by analytical diffeomorphisms, the notion of a (G, X)-structure on a topological space is a way to formalise it being locally isomorphic to X with its G-invariant structure; spaces with a (G, X)-structure are always manifolds and are called (G, X)-manifolds.
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). Hyperbolic 3-manifolds of finite volume have a particular importance in 3-dimensional topology as follows from Thurston's geometrisation ...
In mathematics, relatively hyperbolic groups form an important class of groups of interest for geometric group theory.The main purpose in their study is to extend the theory of Gromov-hyperbolic groups to groups that may be regarded as hyperbolic assemblies of subgroups , called peripheral subgroups, in a way that enables "hyperbolic reduction" of problems for to problems for the s.
Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, which describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s, [2] while an early form is found in the 1856 icosian calculus of William Rowan Hamilton ...