For X = R, C, or H, it is well known that
cusp cross-sections of finite volume X–hyperbolic (n+1)–orbifolds
are flat n–orbifolds or almost flat orbifolds modelled on the
(2n+1)–dimensional Heisenberg group N2n+1
or the (4n+3)–dimensional quaternionic Heisenberg group
N4n+3(H). We give a necessary and sufficient
condition for such manifolds to be diffeomorphic to a cusp cross-section
of an arithmetic X–hyperbolic (n+1)–orbifold.
A principal tool in the proof of this classification theorem is a subgroup
separability result which may be of independent interest.