Is compact flat manifold cusp cross-sections of a complete finite volume hyperbolic manifold?

Question

Let $M^{n-1}$ be a closed flat manifold. Is it true that there exists a hyperbolic manifold $N^n$ with finite volume such that $M$ is a cusp cross-section of $N$?

It was proved in "On the geometric boundaries of hyperbolic 4-manifolds" by Long and Reid in Geom. Topol. 4 (2000), 171–178 that there are 3-manifolds are not cusp cross-sections of any complete finite volume one-cusped hyperbolic 4-manifold, due to the obstructions of eta-invariant. They constructed a 3-dim flat manifold with eta invariant = -4/3. if it was the cusp of a hyperbolic manifold then it has to be integer according to Aatiyah-Patodi-Singer's theorem.

My question is can it always a cross-section of several-cusped hyperbolic 4-manifold?

Answer 1

Yes. Every (compact) flat $n$-manifold is diffeomorphic to a cusp cross section of a hyperbolic $(n+1)$-manifold. This is a theorem of McReynolds, Controlling manifold covers of orbifolds, Math. Res. Lett. 16 (2009), 651-662.

In your case, $n=3$, this is a theorem of Nimershiem. B. E. Nimershiem, All flat three-manifolds appear as cusps of hyperbolic four-manifolds, Topology Appl. 90 (1998), 109–133.

Answer 2

Since there are countably many finite-volume $3$-manifolds, and uncountably many flat tori, the answer is NO for $n=3,$ at least if your question is geometric, as opposed to topological.