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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.
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 fundamental group of any oriented hyperbolic 3-manifold is a Kleinian group. There are many examples of these, such as the complement of a figure 8 knot or the Seifert–Weber space. Conversely if a Kleinian group has no nontrivial torsion elements then it is the fundamental group of a hyperbolic 3-manifold.
An example of a noncompact, finite volume hyperbolic manifold obtained in this way is the Gieseking manifold which is constructed by gluing 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.
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).
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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 ...