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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 perspective projection of a dodecahedral tessellation in H 3. This is an example of what ...
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 ...
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).
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]
A hyperbolic manifold is called geometrically finite if it has a finite number of components, each of which is the quotient of hyperbolic space by a geometrically finite discrete group of isometries (Ratcliffe 1994, 12.7).
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 ...
Discrete normal subgroups play an important role in the theory of covering groups and locally isomorphic groups. A discrete normal subgroup of a connected group G necessarily lies in the center of G and is therefore abelian. Other properties: every discrete group is totally disconnected; every subgroup of a discrete group is discrete.