Modular Group Gamma

The modular group is the group of all Möbius transformations of the form

(1)

where , , , and are integers with . Each transformation can be represented by a matrix

(2)

where . The matrices and represent the same transformation. Every such matrix can be expressed in the form

(3)

where

			(4)
			(5)

although the representation is not unique (Apostol 1997, pp. 28-29).

Writing with , the transformation satisfies

(6)

so it preserves the upper half-plane. Since , it also preserves the hyperbolic metric with line element

(7)

Consequently, the images of a fundamental region are congruent in hyperbolic geometry, even though they can have different Euclidean shapes and sizes (McCreary et al. 2018). Applying the generators and repeatedly produces a hyperbolic tiling (Trott 2006).

On the upper half-plane, a standard fundamental region is bounded by and , with (Milne 2017, pp. 32-33). The Möbius transformation carries this region to the Poincaré hyperbolic disk. Cutting along gives two hyperbolic triangles with angles , , and 0. The pair, rather than a single such triangle, is a fundamental region for .