Kleinian Groups

The Isometric spheres of a Kleinian group bound the group's fundamental polyhedron. This fundamental polyhedron is analogous to the Ford region, which is bounded by isometric circles in the two dimensional case.

The construction of a canonical polygon described by Linda Keen yields intrinsic parameters for Teichmüller spaces.

Cutting and pasting a surface hyperbolic polygon can have the same affect as applying the Mapping Class group to the surface which the polygon represents.

Poincare described the conditions necessary and sufficient for a hyperbolic polygon to represent a Reimann surface.

There are four types of Möbius transformations, three of which can be isometries of the hyperbolic plane. Under the Poincare extension, all four are isometries of hyperbolic 3-space.

A Thurston pair of pants can be represented by a hyperbolic nonagon with appropriately identified sides. Two of the boundaries can be glued together to form a torus with a single hole.

Thus, two pairs of pants that share one boundary, represent a double torus.

Each of the four types of Möbius transformation has its own intrinsic Steiner net. These can be viewed in the plane and on the Riemann sphere.

The one- and two-dimensional manifolds that are stabalized by the extended Möbius transformations make spectacular introductions to the actions of hyperbolic isometries.

A closed Riemann surface of genus two can be topologically embedded into R^3 by wrapping up a hyperbolic octagon and gluing together its sides.