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

$t(z)=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right):z=\frac{1\cdot z+1}{0\cdot z+1}=z+1$

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 $g=h^{-1}th$ for some h ∈ SL(2,Z).

The set

$U=\{z\in \mathbf{𝐇}:\mathrm{\Im }z>1\}$

where H is the upper half-plane has

$\gamma (U)\cap U=\mathrm{\varnothing }$

for any $\gamma \in G-⟨g⟩$ where $⟨g⟩$ 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

$E=U/⟨g⟩$.

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

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

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

See also

• Cusp form