Cusp neighborhood

Short description: Neighborhood of a singularity of cusp type

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

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 ∈ SL(2,Z) where

[math]\displaystyle{ t(z)=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}:z = \frac{1\cdot z+1}{0 \cdot z + 1} = z+1 }[/math]

is a parabolic element. Note that all parabolic elements of SL(2,C) are conjugate to this element. That is, if g ∈ SL(2,Z) is parabolic, then [math]\displaystyle{ g=h^{-1}th }[/math] for some h ∈ SL(2,Z).

The set

[math]\displaystyle{ U=\{ z \in \mathbf{H} : \Im z \gt 1 \} }[/math]

where H is the upper half-plane has

[math]\displaystyle{ \gamma(U) \cap U = \emptyset }[/math]

for any [math]\displaystyle{ \gamma \in G - \langle g \rangle }[/math] where [math]\displaystyle{ \langle g \rangle }[/math] 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

[math]\displaystyle{ E = U/ \langle g \rangle }[/math].

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

[math]\displaystyle{ \left\{ z \in H: \left| z \right| \gt 1,\, \left| \,\mbox{Re}(z) \,\right| \lt \frac{1}{2} \right\} }[/math]

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

[math]\displaystyle{ d\mu=\frac{dxdy}{y^2} }[/math]

the result is trivially 1. Areas of all cusp neighborhoods are equal to this, by the invariance of the area under conjugation.