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Suppose $H^n$ is the complex space form of bisectional curvature $-1$, $\Gamma$ is a discrete subgroup of $PU(n, 1)$, the holomorphic isometry group of $H^n$. Let $M=H^n/\Gamma$, then $M$ is a hyperbolic orbifold. If $\Gamma$ is torsion free, we know that $M$ is a manifold. Suppose further that $M$ has finite volume and is non-compact, then $M$ must have cusp ends. What do we know about the cross sections of the cusp ends? Can we say that each cross section is a smooth manifold, or the cross section itself maybe a singular orbifold? What references can I see for this question? Thanks!

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Does this paper by Hummel and Schroeder help? – Alain Valette Aug 30 '11 at 17:00
This paper has some information on this question: – Ian Agol Aug 30 '11 at 18:45
See papers by Kamishima: and – Igor Belegradek Aug 30 '11 at 19:04
Thanks! These papers imply that it is restrictive to be a cross-section of a complex hyperbolic cusp. – user17314 Aug 31 '11 at 18:14

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