TITLE: The number of cusps of higher-dimensional hyperbolic manifolds
QUESTION [18 upvotes]: 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?

REPLY [6 votes]: 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.