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 (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.

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-32.abs.html

References in corpus (2)

• On the geometric boundaries of hyperbolic 4-manifolds
• All flat manifolds are cusps of hyperbolic orbifolds

Cited by in corpus (13)

• Convex projective manifolds with a cusp of any non-diagonalizable type
• Determining Fuchsian groups by their finite quotients
• Nonexistence of cusp cross-section of one-cusped complete complex hyperbolic manifolds II
• Primitive geodesic lengths and (almost) arithmetic progressions
• Cusps of Hilbert modular varieties
• Cusps and Commensurability Classes of Hyperbolic 4-Manifolds
• Arithmetic cusp shapes are dense
• Separable subgroups of mapping class groups
• Coarse models of homogeneous spaces and translations-like actions
• Some cusp-transitive hyperbolic 4-manifolds
• The fundamental group of a toroidal compactification of a Hermitian locally symmetric space
• Toroidal compactifications of torsion free local complex hyperbolic spaces
• Controlling manifold covers of orbifolds