Modular Group - Relationship To Hyperbolic Geometry

Relationship To Hyperbolic Geometry

The modular group is important because it forms a subgroup of the group of isometries of the hyperbolic plane. If we consider the upper half-plane model H of hyperbolic plane geometry, then the group of all orientation-preserving isometries of H consists of all Möbius transformations of the form

where a, b, c, and d are real numbers and ad − bc = 1. Put differently, the group PSL(2,R) acts on the upper half-plane H according to the following formula:

This (left-)action is faithful. Since PSL(2,Z) is a subgroup of PSL(2,R), the modular group is a subgroup of the group of orientation-preserving isometries of H.