Peripheral separability and cusps of arithmetic hyperbolic orbifolds

Abstract

For $X=ℝ$, $ℂ$, or $ℍ$, it is well known that cusp cross-sections of finite volume $X$–hyperbolic $\left(n+1\right)$–orbifolds are flat $n$–orbifolds or almost flat orbifolds modelled on the $\left(2n+1\right)$–dimensional Heisenberg group ${\mathfrak{N}}_{2n+1}$ or the $\left(4n+3\right)$–dimensional quaternionic Heisenberg group ${\mathfrak{N}}_{4n+3}\left(ℍ\right)$. We give a necessary and sufficient condition for such manifolds to be diffeomorphic to a cusp cross-section of an arithmetic $X$–hyperbolic $\left(n+1\right)$–orbifold.

A principal tool in the proof of this classification theorem is a subgroup separability result which may be of independent interest.