Relationship To Hyperbolic Geometry
See also: PSL2(R)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.
Read more about this topic: Modular Group
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