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)



Non-separable surfaces in cubed manifolds

Author: Saburo Matsumoto
Journal: Proc. Amer. Math. Soc. 125 (1997), 3439-3446
MSC (1991): Primary 57Q35; Secondary 57M50
MathSciNet review: 1415352
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that there are 3-manifolds with cubings of non-positive curvature such that their fundamental groups are not subgroup separable (LERF). We also give explicit examples of non-separable surfaces in certain cubed manifolds.

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

  • [AR] I. R. Aitchison and J. H. Rubinstein, An introduction to polyhedral metrics of nonpositive curvature on 3-manifolds, Geometry of low-dimensional manifolds, 2 (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 151, Cambridge Univ. Press, Cambridge, 1990, pp. 127–161. MR 1171913
  • [BGS] Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 823981
  • [BKS] R. G. Burns, A. Karrass, and D. Solitar, A note on groups with separable finitely generated subgroups, Bull. Austral. Math. Soc. 36 (1987), no. 1, 153–160. MR 897431, 10.1017/S0004972700026393
  • [Gm] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, 10.1007/978-1-4613-9586-7_3
  • [HS] Joel Hass and Peter Scott, Homotopy equivalence and homeomorphism of 3-manifolds, Topology 31 (1992), no. 3, 493–517. MR 1174254, 10.1016/0040-9383(92)90046-K
  • [Lo] D. D. Long, Immersions and embeddings of totally geodesic surfaces, Bull. London Math. Soc. 19 (1987), no. 5, 481–484. MR 898729, 10.1112/blms/19.5.481
  • [LN] D. D. Long and G. A. Niblo, Subgroup separability and 3-manifold groups, Math. Z. 207 (1991), no. 2, 209–215. MR 1109662, 10.1007/BF02571384
  • [Ma] S. Matsumoto, Subgroup Separability of 3-Manifold Groups, Ph.D. thesis, University of Michigan, 1995.
  • [Ma1] -, A 3-manifold with a non-subgroup-separable fundamental group, to appear in Bull. Austral. Math. Soc. 55 (1997), 261-279. CMP 97:09
  • [Ma2] -, Separability criterion for graph-manifold groups, to appear in Topology Appl.
  • [Mo] L. Mosher, Geometry of cubulated 3-manifolds, Topology 34 (1995), 789-813. CMP 96:04
  • [Re] L. Reeves, Biautomatic structure of the fundamental groups of cubulated manifolds, preprint, 1992.
  • [RW] J. H. Rubinstein and S. Wang, On $\pi _{1}$-injective surfaces in graph manifolds, to appear in Comm. Math. Helv.
  • [Sc] Peter Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555–565. MR 0494062
    Peter Scott, Correction to: “Subgroups of surface groups are almost geometric” [J. London Math. Soc. (2) 17 (1978), no. 3, 555–565; MR0494062 (58 #12996)], J. London Math. Soc. (2) 32 (1985), no. 2, 217–220. MR 811778, 10.1112/jlms/s2-32.2.217
  • [Sk] A. Skinner, The word problem in the fundamental groups of a class of three-dimensional manifolds, Ph.D. thesis, University of Melbourne, 1991.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 57Q35, 57M50

Retrieve articles in all journals with MSC (1991): 57Q35, 57M50

Additional Information

Saburo Matsumoto
Affiliation: Department of Mathematics, University of Melbourne, Parkville, Victoria 3052 Australia
Address at time of publication: Department of Mathematical and Computing Sciences, Tokyo Institute of Technoloy, O-okayama, Meguro-ku, Tokyo 152, Japan

Keywords: Cubed manifolds, separable surfaces
Received by editor(s): February 5, 1996
Received by editor(s) in revised form: June 24, 1996
Additional Notes: The research partially supported by Australian Research Council.
The author would like to thank Prof. P. Scott for many helpful comments.
Communicated by: R. Fintushel
Article copyright: © Copyright 1997 American Mathematical Society