Fundamental Domain - Fundamental Domain For The Modular Group

Fundamental Domain For The Modular Group

The existence and description of a fundamental domain is in general something requiring painstaking work to establish. The diagram to the right shows part of the construction of the fundamental domain for the action of the modular group Γ on the upper half-plane H.

This famous diagram appears in all classical books on modular functions. (It was probably well known to C. F. Gauss, who dealt with fundamental domains in the guise of the reduction theory of quadratic forms.) Here, each triangular region (bounded by the blue lines) is a free regular set of the action of Γ on H. The boundaries (the blue lines) are not a part of the free regular sets. To construct a fundamental domain of H/Γ, one must also consider how to assign points on the boundary, being careful not to double-count such points. Thus, the free regular set in this example is

The fundamental domain is built by adding the boundary on the left plus half the arc on the bottom including the point in the middle:

The choice of which points of the boundary to include as a part of the fundamental domain is arbitrary, and varies from author to author.

The core difficulty of defining the fundamental domain lies not so much with the definition of the set per se, but rather with how to treat integrals over the fundamental domain, when integrating functions with poles and zeros on the boundary of the domain.