Cusp neighborhood

[1] The cusp neighborhood for a hyperbolic Riemann surface can be defined in terms of its Fuchsian model.

is understood to mean the group generated by g. That is, γ acts properly discontinuously on U.

Because of this, it can be seen that the projection of U onto H/G is thus Here, E is called the neighborhood of the cusp corresponding to g. Note that the hyperbolic area of E is exactly 1, when computed using the canonical Poincaré metric.

This is most easily seen by example: consider the intersection of U defined above with the fundamental domain of the modular group, as would be appropriate for the choice of T as the parabolic element.

When integrated over the volume element the result is trivially 1.