structural:summarization
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, d are real numbers. In terms of projective coordinates, the group PSL(2, R) acts on the upper half-plane H by projectivity:

This 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.