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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 ...
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.
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 ...
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).
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 ...