# The Action of the Modular Group on the Fundamental Domain

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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.

Contributed by: Andrzej Kozlowski (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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.

## Permanent Citation

"The Action of the Modular Group on the Fundamental Domain"

http://demonstrations.wolfram.com/TheActionOfTheModularGroupOnTheFundamentalDomain/

Wolfram Demonstrations Project

Published: March 7 2011