In mathematics, a cusp neighborhood is defined as a set of points near a cusp.

Cusp neighborhood for a Riemann surface

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

Suppose that the Fuchsian group G contains a parabolic element g. For example, the element where

is a parabolic element. Then the set

where is the upper half-plane has

for any where is understood to mean the group generated by g. 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.