Peripheral separability and cusps of arithmetic hyperbolic orbifolds

McReynolds, D B

doi:10.2140/agt.2004.4.721

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Peripheral separability and cusps of arithmetic hyperbolic orbifolds

D B McReynolds

Algebraic & Geometric Topology 4 (2004) 721–755

DOI: 10.2140/agt.2004.4.721

arXiv: math.GT/0409278

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 $(2 n + 1)$ –dimensional Heisenberg group $N_{2 n + 1}$ or the $(4 n + 3)$ –dimensional quaternionic Heisenberg group $N_{4 n + 3} (ℍ)$ . 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.

Mathematical Subject Classification 2000

Primary: 57M50

Secondary: 20G20

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Publication

Received: 2 April 2004

Accepted: 3 September 2004

Published: 11 September 2004

Authors

D B McReynolds
University of Texas Austin TX 78712 USA

Volume 4, issue 2 (2004)