# Finitely generated nilpotent groups as cusp groups

I recently learned about the following question, asked by I. Kapovich :

Is there an example of a group $$G$$ which is hyperbolic relative to some parabolic subgroups that are nilpotent of class $$\geq 3$$, whose Bowditch boundary is homeomorphic to some $$n$$-sphere ?

One can find the question here, together with a list of questions about boundaries of groups.

However, I was told once that one could realize any finitely generated nilpotent group as the stabilizer of a cusp in a finite volume manifold of pinched negative curvature. This result would follow from

• Gromov-Ruh theorem that says that any infranilmanifold is almost flat (the theorem actually states that this is an if and only if condition).

• Corollary 6 in P. Ontaneda's paper Pinched smooth hyperbolization (arXiv link here). It is proved there that for any almost flat manifold $$Q$$, $$Q\bigsqcup Q$$ bounds geometrically a pinched negatively curved manifold

See also Theorem 1.1 in I. Belegradek and V. Kapovich's paper Classification of negatively pinched manifolds with amenable fundamental groups (Acta Math. Volume 196 (2006), 229-260, arXiv link here). It is proved there that any infranilmanifold can be realized as a horosphere quotient.

Now, if $$M$$ is a manifold with pinched negative curvature, then its fundamental group $$\pi_1(M)$$ is hyperbolic relative to the stabilizers of the cusps and the Bowditch boundary is the limit set of $$\pi_1(M)$$ in $$\tilde{M}$$, so if $$M$$ has finite volume, the Bowdtich boundary of $$\pi_1(M)$$ is a $$n$$-sphere.

Of course, it could be the case that the question was asked before those papers were written, but

• Could someone clarify this ?

• What is the current status of this conjecture ?

• You are correct: the conjecture was resolved by Ontaneda's Riemannian hyperbolization. Thus any finite collection of finitely generated torsion-free nilpotent groups can be realized as the set of peripheral subgroups of a relatively hyperbolic group with sphere boundary, provided the almost flat manifold corresponding to the collection bounds a smooth manifold. The almost flat manifold has the same number of connected components as the number of groups in the collection. Jul 24, 2019 at 12:16
• @IgorBelegradek Well thank you a lot. Do we know for which f.g. torsion free nilpotent group the (connected) almost flat manifold smoothly bounds ? That is do we know when we can realize those nipotent groups as cusp groups of a manifold with only one cusp ? This does not matter so much but the trick of doubling $Q$ to make it bound $Q\times I$ seems a bit artificial to get the nilpotent group as a cusp group. Jul 24, 2019 at 12:28
• Ontaneda reviews the literature (on which almost flat manifolds bound) in his "Riemannian hyperbolization" paper. Conjectually all connected almost flat manifolds bound. See front.math.ucdavis.edu/1501.00300 for more recent works. Jul 24, 2019 at 12:41