The Action of the Modular Group on the Fundamental Domain
Choose eight transformations from the set of controls on the left. Each transformation can be either an inversion or a translation. Choose the exponent of each of these transformations (-1, 0, or 1) from the set of controls on the right. This Demonstration shows the fundamental domain and its images under the consecutive compositions of the chosen transformations.
The modular group is the quotient of the group of integer matrices with by the subgroup , where denotes the identity matrix. This group acts on the upper half-plane by . It turns out that the modular group is generated by just two elements: inversion, given by the matrix , and translation, given by . The fundamental domain is the subset of the upper half-plane formed by all points such that and . The action of the modular group tiles the upper half-plane exactly with images of the fundamental domain. This tiling requires the use of the entire infinite fundamental domain, so it cannot be demonstrated by Mathematica. However, this tiling can be approximated by considering the images of a finite part of the fundamental domain under a finite number of compositions , where each and is either a translation by an integer or a reflection.