Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Gluing copies of a 3-dimensional polyhedron to obtain a closed nonpositively curved pseudomanifold

Authors: D. Burago, S. Ferleger, B. Kleiner and A. Kononenko
Journal: Proc. Amer. Math. Soc. 129 (2001), 1493-1498
MSC (2000): Primary 51K10, 53C20; Secondary 52B10
Published electronically: January 8, 2001
MathSciNet review: 1707510
Full-text PDF

Abstract | References | Similar Articles | Additional Information


Let $S$ be a smooth 3-dimensional nonpositively curved Riemannian manifold with corners, whose boundary consists of a finite number of geodesically convex nonpositively curved faces (for example, a Euclidean or hyperbolic polyhedron). We show that it is always possible to glue together finitely many copies of $S$ so as to get a nonpositively curved pseudomanifold without boundary.

References [Enhancements On Off] (What's this?)

  • [AB] S. Alexander, R. Bishop.
    The Hadamard-Cartan theorem in locally convex metric spaces.
    Enseign. Math, (2) 36 (1990), no. 3-4, 309-320. MR 92c:53044
  • [Ba] W.Ballmann.
    Lectures on spaces of nonpositive curvature. With an appendix by Misha Brin. DMV Seminar, 25. Birkhauser Verlag, Basel, 1995. MR 97a:53053
  • [BFK1] D.Burago, S.Ferleger, A.Kononenko.
    Uniform estimates on the number of collisions in semi-dispersing billiards.
    Annals of Mathematics, (2) 147 (1998), 695-708. MR 99f:58120
  • [BFK2] D.Burago, S.Ferleger, A.Kononenko.
    Topological entropy of semi-dispersing billiards.
    Ergodic Theory and Dynamical Systems, 18 (1998), 791-805. MR 99m:58146
  • [BFK3] D.Burago, S.Ferleger, A.Kononenko.
    Unfoldings and global bounds on the number of collisions for generalized semi-dispersing billiards.
    Asian J. Math., 2 (1998), 141-152. MR 2000c:37039
  • [GH] E. Ghys, P. de la Harpe (eds.).
    Sur les groupes hyperboliques d'aprés Mikhael Gromov.
    Prog. Math., vol. 83, Birkhäuser, Boston, 1990. MR 92f:53050
  • [Gr] M.Gromov. ``Hyperbolic groups.'' in
    Essays in group theory, S.M.Gersten (ed.).
    M.S.R.I. Publ., Vol.8: 75-263, Springer 1987. MR 89e:20070
  • [He] J.Hempel.
    Residual finiteness for $3$-manifolds.
    Combinatorial group theory and topology, 379-396,
    Ann. of Math. Stud., 111, Princeton Univ. Press, Princeton, NJ, 1987. MR 89b:57002
  • [Ka] M. Kapovich.
    Hyperbolic manifolds and Discrete Groups. Notes on Thurston's Hyperbolization.
    University of Utah Lecture Notes, 1995. To appear in Birkhäuser, Progress in Mathematics.
  • [Ma] A.Malcev.
    On isomorphic matrix representations of infinite groups. (Russian)
    Rec. Math. [Mat.Sbornik] N.S., 8, no. 50, 405-422, 1940. MR 2:216d
  • [Mc] C.McMullen.
    Iteration on Teichmuller space.
    Invent. Math. 99 (1990), no. 2, 425-454. MR 91a:57008
  • [Ot1] J.-P. Otal.
    Le théorème d'hyperbolisation pour les variétès fibrées de dimension 3.
    Astérisque 235, Société mathématique de France, 1996. MR 97e:57013
  • [Ot2] J.-P. Otal.
    Thurston's hyperbolization of Haken manifolds. Surveys in differential geometry, Vol. III (Cambridge, MA, 1996), Int. Press, Boston, MA, 1998, pp. 77-194. MR 2000b:57025
  • [Re] Yu.G.Reshetnyak (ed.).
    Geometry 4, non-regular Riemannian geometry.
    Encyclopedia of Mathematical Sciences, Vol.70, 1993. MR 94i:53038
  • [Sp] E. Spanier.
    Algebraic Topology.
    Springer, 1996. MR 96a:55001
  • [St] J.Stillwell.
    Classical Topology and Combinatorial Group Theory.
    Springer-Verlag, New York, 1993. MR 94a:57001
  • [Sto] D.Stone.
    Geodesics in piecewise linear manifolds.
    Trans. Amer. Math. Soc., 215 (1976), 1-44. MR 53:6464
  • [Th] W.Thurston.
    Three-dimensional manifolds, Kleinian groups and hyperbolic geometry.
    Bull. Amer. Math. Soc. (N.S.), 6, no. 3, 357-381, 1982. MR 83h:57019

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 51K10, 53C20, 52B10

Retrieve articles in all journals with MSC (2000): 51K10, 53C20, 52B10

Additional Information

D. Burago
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802

S. Ferleger
Affiliation: Renaissance Technology Corporation, 600 Rt. 25-A, East Setanket, New York 11787

B. Kleiner
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112-0090

A. Kononenko
Affiliation: Renaissance Technology Corporation, 600 Rt. 25-A, East Setanket, New York 11787

Received by editor(s): August 31, 1998
Published electronically: January 8, 2001
Additional Notes: The first author was partially supported by a Sloan Foundation Fellowship and NSF grant DMS-9803129. The second author was partially supported by a Sloan Dissertation Fellowship. The third author was supported by a Sloan Foundation Fellowship, and NSF grants DMS-95-05175, DMS-96-26911. The fourth author was supported by NSF grant DMS-9803092
Communicated by: Christopher Croke
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society