Cusp neighborhood

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

[edit] 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. Note that all parabolic elements of $SL(2,\mathbb{C})$ are conjugate to this element. That is, if $g \in SL(2,\mathbb{C})$ is parabolic, then $g = h - 1 t h$ for some $h \in SL(2,\mathbb{C})$ .

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

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