Cusp neighborhood

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

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

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

The set

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

where H is the upper half-plane has

${\displaystyle \gamma (U)\cap U=\emptyset }$

for any ${\displaystyle \gamma \in G-\langle g\rangle }$ where ${\displaystyle \langle g\rangle }$ 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

${\displaystyle E=U/\langle g\rangle }$.

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

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

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

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

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

References