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Cusp NeighborhoodIn 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 -
\frac{1\cdot z+1}{0 \cdot z + 1} = z+1 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 Poincar 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.
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