#### Algebraic and Geometric Topology 4 (2004),
paper no. 32, pages 721-755.

## Peripheral separability and cusps of arithmetic hyperbolic orbifolds

### D.B. McReynolds

**Abstract**.
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
N_{2n+1} or the (4n+3)-dimensional quaternionic Heisenberg group
N_{4n+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.
**Keywords**.
Borel subgroup, cusp cross-section, hyperbolic space, nil manifold, subgroup separability.

**AMS subject classification**.
Primary: 57M50.
Secondary: 20G20.

**DOI:** 10.2140/agt.2004.4.721

**E-print:** `arXiv:math.GT/0409278`

Submitted: 2 April 2004.
(Revised: 24 August 04.)
Accepted: 3 September 2004.
Published: 11 September 2004.

Notes on file formats
D.B. McReynolds

University of Texas, Austin, TX 78712, USA

Email: dmcreyn@math.utexas.edu

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