Editing Fundamental domain (section)

== Fundamental domain for the modular group ==

[[Image:ModularGroup-FundamentalDomain.svg|thumb|400px|Each triangular region is a free regular set of H/Γ; the grey one (with the third point of the triangle at infinity) is the canonical fundamental domain.]]
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 function]]s. (It was probably well known to [[C. F. Gauss]], who dealt with fundamental domains in the guise of the [[Binary_quadratic_form#Reduction_and_class_numbers|reduction theory]] of [[quadratic form]]s.) 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

:<math>U = \left\{ z \in H: \left| z \right| > 1,\, \left| \,\mbox{Re}(z) \,\right| < \frac{1}{2} \right\}.</math>

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:

:<math>D=U\cup\left\{ z \in H: \left| z \right| \geq 1,\, \mbox{Re}(z)=\frac{-1}{2} \right\} \cup \left\{ z \in H: \left| z \right| = 1,\, \frac{-1}{2}<\mbox{Re}(z)\leq 0 \right\}.</math>

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.