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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Non-separable surfaces in cubed manifolds
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by Saburo Matsumoto PDF
Proc. Amer. Math. Soc. 125 (1997), 3439-3446 Request permission


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.
  • 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
  • Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 823981, DOI 10.1007/978-1-4684-9159-3
  • 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, DOI 10.1017/S0004972700026393
  • M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
  • Joel Hass and Peter Scott, Homotopy equivalence and homeomorphism of $3$-manifolds, Topology 31 (1992), no. 3, 493–517. MR 1174254, DOI 10.1016/0040-9383(92)90046-K
  • D. D. Long, Immersions and embeddings of totally geodesic surfaces, Bull. London Math. Soc. 19 (1987), no. 5, 481–484. MR 898729, DOI 10.1112/blms/19.5.481
  • D. D. Long and G. A. Niblo, Subgroup separability and $3$-manifold groups, Math. Z. 207 (1991), no. 2, 209–215. MR 1109662, DOI 10.1007/BF02571384
  • S. Matsumoto, Subgroup Separability of 3-Manifold Groups, Ph.D. thesis, University of Michigan, 1995.
  • —, A 3-manifold with a non-subgroup-separable fundamental group, to appear in Bull. Austral. Math. Soc. 55 (1997), 261–279.
  • —, Separability criterion for graph-manifold groups, to appear in Topology Appl.
  • L. Mosher, Geometry of cubulated 3-manifolds, Topology 34 (1995), 789–813.
  • L. Reeves, Biautomatic structure of the fundamental groups of cubulated manifolds, preprint, 1992.
  • J. H. Rubinstein and S. Wang, On $\pi _{1}$-injective surfaces in graph manifolds, to appear in Comm. Math. Helv.
  • Peter Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555–565. MR 494062, DOI 10.1112/jlms/s2-17.3.555
  • A. Skinner, The word problem in the fundamental groups of a class of three-dimensional manifolds, Ph.D. thesis, University of Melbourne, 1991.
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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
  • Email:
  • 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
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 3439-3446
  • MSC (1991): Primary 57Q35; Secondary 57M50
  • DOI:
  • MathSciNet review: 1415352