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.

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D.B. McReynolds
University of Texas, Austin, TX 78712, USA

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