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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$\begingroup$ Does this paper by Hummel and Schroeder help? springerlink.com/content/v74834r21r82637x/fulltext.pdf $\endgroup$– Alain ValetteAug 30, 2011 at 17:00
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$\begingroup$ This paper has some information on this question: front.math.ucdavis.edu/0409.5278 $\endgroup$– Ian AgolAug 30, 2011 at 18:45
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$\begingroup$ See papers by Kamishima: springerlink.com/content/71003531454282n0 and arxiv.org/abs/math/0603726 $\endgroup$– Igor BelegradekAug 30, 2011 at 19:04
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$\begingroup$ Thanks! These papers imply that it is restrictive to be a cross-section of a complex hyperbolic cusp. $\endgroup$– user17314Aug 31, 2011 at 18:14
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$\begingroup$ The link in Ian's comment is broken, here's a replacement: arxiv.org/abs/math/0409278 $\endgroup$– David Roberts ♦Mar 29, 2022 at 7:00
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