Algebraic & Geometric Topology

All flat manifolds are cusps of hyperbolic orbifolds

Darren D Long and Alan W Reid

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Abstract

We show that all closed flat n–manifolds are diffeomorphic to a cusp cross-section in a finite volume hyperbolic (n+1)–orbifold.

Article information

Source
Algebr. Geom. Topol., Volume 2, Number 1 (2002), 285-296.

Dates
Received: 6 December 2001
Accepted: 10 April 2002
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882693

Digital Object Identifier
doi:10.2140/agt.2002.2.285

Mathematical Reviews number (MathSciNet)
MR1917053

Zentralblatt MATH identifier
0998.57038

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57R99: None of the above, but in this section

Keywords
flat manifolds hyperbolic orbifold cusp cross-sections

Citation

Long, Darren D; Reid, Alan W. All flat manifolds are cusps of hyperbolic orbifolds. Algebr. Geom. Topol. 2 (2002), no. 1, 285--296. doi:10.2140/agt.2002.2.285. https://projecteuclid.org/euclid.agt/1513882693


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References

  • I. Agol, D.D. Long and A.W. Reid, The Bianchi groups are separable on geometrically finite subgroups, Annals of Math. 153 (2001), pp. 599–621.
  • A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), pp. 111 – 122.
  • A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Annals of Math. 75 (1962), 485–535.
  • L.S. Charlap, Bieberbach Groups and Flat Manifolds, Universitext, Springer-Verlag (1986).
  • F.T. Farrell and S. Zdravkovska, Do almost flat manifolds bound, Michigan J. Math. 30 (1983), pp 199–208.
  • M.L. Gromov, Almost flat manifolds, J. Diff. Geom. 13 (1978), pp 231–241.
  • G. Hamrick and D. Royster, Flat Riemannian manifolds are boundaries, Invent. Math. 66 (1982), pp. 405 –413.
  • T.Y. Lam, The Algebraic Theory of Quadratic Forms, Benjamin (1973).
  • D.D. Long, Immersions and embeddings of totally geodesic surfaces, Bull. London Math. Soc. 19 (1987), pp. 481–484.
  • D.D. Long and A.W. Reid, On the geometric boundaries of hyperbolic 4-manifolds, Geometry and Topology, 4 (2000), pp. 171–178.
  • B.E. Nimershiem, All flat three-manifolds appear as cusps of hyperbolic four-manifolds, Topology and Its Appl. 90 (1998), pp. 109-133.
  • J.G. Ratcliffe, Foundations of Hyperbolic Manifolds, G.T.M. 149, Springer-Verlag, (1994).
  • J.G. Ratcliffe and S.T. Tschantz, The volume spectrum of hyperbolic 4-manifolds, Experimental Math 9 (2000), pp. 101–125.
  • W.P. Thurston, Three-Dimensional Geometry and Topology, Volume 1, Princeton University Press (1997).
  • E.B. Vinberg, On groups of unit elements of certain quadratic forms, Mat. Sb. 87 (1972) pp. 17–35.
  • E.B. Vinberg and O.V. Shvartsman, Discrete groups of motions of spaces of constant curvature, Geometry II, Encyc. Math. Sci. 29, pp 139–248, Springer-Verlag (1993).