Volume 4, issue 2 (2004)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Peripheral separability and cusps of arithmetic hyperbolic orbifolds

D B McReynolds

Algebraic & Geometric Topology 4 (2004) 721–755
 arXiv: math.GT/0409278
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.

Keywords
Borel subgroup, cusp cross-section, hyperbolic space, nil manifold, subgroup separability.
Primary: 57M50
Secondary: 20G20