Hyperbolic Velocity Space is a non-euclidean geometry, i.e. the sum of the angles of a triangle is less than 180 degrees.

This particular form, due to Henre Poincare, is a conformal mapping onto a disc

• Angles are preserved.
• The straightest lines (geodesics) are arcs of circles which intersect the edge of the disc at a right angle.
• The distance from the center of the disc to the edge is infinite.

Here, one starts with 3 geodesics, which intersect at equal angles, and performs an iterative inversion of each of these circles with respect to the other, and to their products. This tesselates (non-repetatively covers) the plane.

One thus obtains the Fuchsian Groups.

Here, for the first time, this well known group is extended. The geodesics are pulled out slightly into a higher-dimensional space, quantizing the group.

© November 1980
Mario Giannella
Cornell University
Ithaca, NY

Now located at:
Spallation Neutron Source
Oak Ridge, TN 37830

Email: giannella@sns.gov