Cusp neighborhood explained

In mathematics, a cusp neighborhood is defined as a set of points near a cusp singularity.^[1]

Cusp neighborhood for a Riemann surface

The cusp neighborhood for a hyperbolic Riemann surface can be defined in terms of its Fuchsian model.^[2] Suppose that the Fuchsian group G contains a parabolic element g. For example, the element t ∈ SL(2,Z) where

t(z) = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}:z = (1 ⋅ z +1)/(0 ⋅ 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 H : \Im z > 1 \}

where 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 thusE = U/ \langle g \rangle .

Here, E is called the neighborhood of the cusp corresponding to g.

\left\{ z \in H: \left| z \right| > 1 , \left| Re (z) \right| < (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

d\mu = (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.

Notes and References

1. Fujikawa . Ege . Shiga . Hiroshige . Taniguchi . Masahiko . 2004 . On the action of the mapping class group for Riemann surfaces of infinite type . Journal of the Mathematical Society of Japan . 56 . 4 . 1069–1086 . 10.2969/jmsj/1190905449.
2. Basmajian . Ara . 1992 . Generalizing the hyperbolic collar lemma . Bulletin of the American Mathematical Society . en . 27 . 1 . 154–158 . 10.1090/S0273-0979-1992-00298-7 . 0273-0979. math/9207211 .