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

This page was last edited on 29 October 2023, at 09:56