Cusp of a finite-volume hyperbolic three-orbifold

Let $M=\Gamma\backslash\mathbb H^3$ be a complete orientable finite-volume hyperbolic three-orbifold. A cusp is an end represented by an embedded quotient $P\backslash B$ , where:

$\xi$ is an ideal fixed point of a parabolic element of $\Gamma$ , and $P=\operatorname{Stab}_\Gamma(\xi)$ ;
$B$ is an open horoball centered at $\xi$ , preserved by $P$ , with $\gamma B\cap B=\varnothing$ for $\gamma\notin P$ ;
the horosphere quotient $P\backslash\partial B$ is compact.

Shrinking $B$ toward $\xi$ gives the same end. Boundary points in the same $\Gamma$ -orbit specify the same cusp.

Shape and finite volume

After moving $\xi$ to infinity, a cusp has a compact Euclidean two-orbifold cross-section. With $t=\log(r/r_0)$ , its metric is $dt^2+e^{-2t}g_0$ . The volume beyond height $t_0$ is proportional to $\int_{t_0}^{\infty}e^{-2t}\,dt$ , which is finite despite the unbounded length of the end.

Torsion matters

A torsion-free orientable three-dimensional cusp has a torus cross-section. An orbifold cusp may instead have finite isotropy; do not assume every Bianchi cusp cross-section is a torus.

References

F. Paulin, Regards croisés sur les séries de Poincaré et leurs applications, Theorem 6, p. 12. Author’s text See the cusp coordinates and stabilizer discussion preceding Theorem 6.