Cusp neighborhood for a Riemann surface
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 for some h ∈ SL(2,Z).
where H is the upper half-plane has
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
the result is trivially 1. Areas of all cusp neighborhoods are equal to this, by the invariance of the area under conjugation.