The number of cusps of higher-dimensional hyperbolic manifolds Suppose $n$ is an integer greater than 3. Sometimes ago I heard somewhere that it is still not known if there exist complete finite-volume hyperbolic $n$-manifolds having exactly one cusp.  
Could someone either confirm that the problem of finding such examples in every dimension is still open, or, preferably, give me a reference for examples of one-cusped hyperbolic manifolds in arbitrary dimension?
 A: Dear all, today a paper by Kolpakov and Martelli on the arxiv appeared that shows that there exist lots of 4-dimensional cusped hyperbolic manifolds with one cusp. Here is the reference
http://arxiv.org/abs/1303.6122
The general case is still open, I think.
A: Indeed, the general case is open. In the case of dimension four, we used the Coxeter $24$-cell, which is an ideal right-angled polytope (ideal means that all of its vertices are on $\partial \mathbb{H}^4$). It is know (a little result of mine) that there are now ideal right angled polytopes in dimensions greater than or equal to $7$ (however, I do not know anything about dimensions $5$ and $6$). Thus, there is no hope to use ideal right-angled polytopes in higher dimensions.
A: Dear Roberto, to add information of Agol's comment, in Theorem 1.3 of this paper it is proved that there aren't one-cusped arithmetic hyperbolic $n$-orbifolds for $n\geq 30$. Moreover, Stover shows one-cusped arithmetic hyperbolic orbifolds in dimensions 10 and 11. 
