Cusp neighborhood

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 T \in SL(2,\mathbb{Z}) where

T(z)=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}:z = \frac{1\cdot z+1}{0 \cdot z + 1} = z+1

is a parabolic element. Then the set

U=\{ z \in \mathbb{H} : \Im z > 1 \}

where \mathbb{H} is the upper half-plane has

\gamma(U) \cap U = \emptyset

for any \gamma \in G - \langle g \rangle where \langle g \rangle 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

E = U/ \langle g \rangle.

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

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

of the modular group, as would be appropriate for the choice of T as the parabolic element. When integrated over the volume element

d\mu=\frac{dxdy}{y^2}

the result is trivially 1.

See also: Cusp neighborhood, Cusp, Fuchsian group, Fuchsian model, Fundamental domain, Group (mathematics), Mathematics, Modular group, Parabolic element, Poincaré metric