Algebraic & Geometric Topology

Peripheral separability and cusps of arithmetic hyperbolic orbifolds

D B McReynolds

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For X=, , or , 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 N2n+1 or the (4n+3)–dimensional quaternionic Heisenberg group N4n+3(). 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.

Article information

Algebr. Geom. Topol., Volume 4, Number 2 (2004), 721-755.

Received: 2 April 2004
Accepted: 3 September 2004
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 20G20: Linear algebraic groups over the reals, the complexes, the quaternions

Borel subgroup cusp cross-section hyperbolic space nil manifold subgroup separability.


McReynolds, D B. Peripheral separability and cusps of arithmetic hyperbolic orbifolds. Algebr. Geom. Topol. 4 (2004), no. 2, 721--755. doi:10.2140/agt.2004.4.721.

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  • D Allcock, Reflection groups on the octave hyperbolic plane, J. Algebra 213 (1999) 467–498
  • N Bergeron, Premier nombre de Betti et spectre du laplacien de certaines variétés hyperboliques, Enseign. Math. 46 (2000) 109–137
  • A Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963) 111–122
  • A Borel, Linear algebraic groups, second edition, Graduate Texts in Mathematics 126, Springer–Verlag, New York (1991)
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften 319, Springer–Verlag, Berlin (1999)
  • J S Chahal, Solution of the congruence subgroup problem for solvable algebraic groups, Nagoya Math. J. 79 (1980) 141–144
  • K Corlette, Archimedean superrigidity and hyperbolic geometry, Ann. of Math. 135 (1992) 165–182
  • C W Curtis, I Reiner, Representation theory of finite groups and associative algebras, Reprint of the 1962 original, Wiley Classics Library, John Wiley & Sons Inc. New York (1988)
  • K Dekimpe, Almost-Bieberbach groups: affine and polynomial structures, Lecture Notes in Mathematics 1639, Springer–Verlag, Berlin (1996)
  • K Dekimpe, B Eick, Computational aspects of group extensions and their applications in topology, Experiment. Math. 11 (2002) 183–200
  • F T Farrell, S Zdravkovska, Do almost flat manifolds bound?, Michigan Math. J. 30 (1983) 199–208
  • W M Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York (1999)
  • M Gromov, Almost flat manifolds, J. Differential Geom. 13 (1978) 231–241
  • E Hamilton, Abelian subgroup separability of Haken 3-manifolds and closed hyperbolic $n$-orbifolds, Proc. London Math. Soc. 83 (2001) 626–646
  • G C Hamrick, D C Royster, Flat Riemannian manifolds are boundaries, Invent. Math. 66 (1982) 405–413
  • Y Kamishima, Cusp cross-sections of hyperbolic orbifolds by nilmanifolds, in preparation
  • I Kim, J R Parker, Geometry of quaternionic hyperbolic manifolds, Math. Proc. Cambridge Philos. Soc. 135 (2003) 291–320
  • D D Long, Immersions and embeddings of totally geodesic surfaces, Bull. London Math. Soc. 19 (1987) 481–484
  • D D Long, A W Reid, On the geometric boundaries of hyperbolic $4$–manifolds, \gtref420005171178
  • D D Long, A W Reid, All flat manifolds are cusps of hyperbolic orbifolds, \agtref2200213285296
  • C Maclachlan, A W Reid, The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics 219, Springer–Verlag, New York (2003)
  • A I Mal$'$cev, On homomorphisms onto finite groups, Ivanov. Gos. Ped. Inst. Ucen. Zap. 18 (1958) 49–60
  • G A Margulis, G A Soĭfer, Maximal subgroups of infinite index in finitely generated linear groups, J. Algebra 69 (1981) 1–23
  • D B McReynolds, Ph.D. Thesis (In preparation)
  • M Q Ouyang, Geometric invariants for Seifert fibred $3$–manifolds, Trans. Amer. Math. Soc. 346 (1994) 641–659
  • V Platonov, A Rapinchuk, Algebraic groups and number theory, translated from the 1991 Russian original by Rachel Rowen, Pure and Applied Mathematics 139, Academic Press, Boston, MA (1994)
  • M S Raghunathan, Discrete subgroups of Lie groups, Ergebnisse series 68, Springer–Verlag, New York (1972),
  • J G Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer–Verlag, New York (1994)
  • P Scott, The geometries of $3$–manifolds, Bull. London Math. Soc. 15 (1983) 401–487
  • J-P Serre, Linear representations of finite groups, translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics 42, Springer–Verlag, New York (1977)
  • W P Thurston, Three-dimensional geometry and topology. Vol. 1, edited by Silvio Levy, Princeton Mathematical Series 35, Princeton University Press, Princeton, NJ (1997)
  • S Upadhyay, A bounding question for almost flat manifolds, Trans. Amer. Math. Soc. 353 (2001) 963–972
  • M-F Vignéras, Arithmétique des algèbres de quaternions, volume 800 of Lecture Notes in Mathematics, Springer–Verlag, Berlin (1980)
  • B A F Wehrfritz, Infinite linear groups. An account of the group-theoretic properties of infinite groups of matrices, Ergebnisse series 76, Springer–Verlag, New York (1973)
  • A Weil, Algebras with involutions and the classical groups, J. Indian Math. Soc. (N.S.) 24 (1960) 589–623 (1961)
  • A Weil, Basic number theory, reprint of the second (1973) edition, Classics in Mathematics, Springer–Verlag, Berlin (1995)