MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## 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?

-
I think this is still open. This is mentioned as an open problem here: ams.org/mathscinet-getitem?mr=1917053 – Ian Agol Sep 7 2010 at 15:31
Dear Ian, thank you very much for your reply. I also checked the papers that cite Long-Reid's work, and it seems that the problem is still open. – Roberto Frigerio Sep 9 2010 at 10:41
There are known 1-cusped hyperbolic orbifolds up to dimension 9, I think. See e.g.: dx.doi.org/doi:10.1016/j.jalgebra.2006.12.024 So one could attempt to look for irregular covers of these orbifolds which are manifolds. I don't know what is known about this though. – Ian Agol Sep 10 2010 at 0:24
Dear Ian, thank you again! this reference sounds quite interesting! I will give a look at it soon. – Roberto Frigerio Sep 19 2010 at 20:49

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.

-
 Thanks for your answer. I saw Stover's paper on the arxiv some days ago, I hope I will have time to read it soon! – Roberto Frigerio Dec 27 2011 at 17:52 Are there known non-arithmetic methods for constructing finite-volume hyperbolic manifolds of arbitrary dimension? – Anton Lukyanenko Dec 27 2011 at 20:30 Anton, there are special constructions of non-arithmetic hyperbolic manifolds in small dimensions >3, coming e.g. from reflection groups or certain special moduli spaces. Otherwise, in high dimensions, the only construction of non-arithmetic lattices available seems to be the Gromov-Piatetskii-Shapiro method of interbreeding, or a slight variation I call inbreeding. It may be possible to take a multicusp arithmetic manifold and interbreed it to construct a one cusp manifold. – Ian Agol Dec 27 2011 at 20:47

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.

-