Nonseparable surfaces in cubed manifolds
Author:
Saburo Matsumoto
Journal:
Proc. Amer. Math. Soc. 125 (1997), 34393446
MSC (1991):
Primary 57Q35; Secondary 57M50
MathSciNet review:
1415352
Fulltext PDF Free Access
Abstract 
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Abstract: We show that there are 3manifolds with cubings of nonpositive curvature such that their fundamental groups are not subgroup separable (LERF). We also give explicit examples of nonseparable surfaces in certain cubed manifolds.
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 W. Ballman, M. Gromov, and V. Schroeder, Manifolds of nonpositive curvature, Birkhäuser, 1985. MR 87h:53050
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 [Gm]
 M. Gromov, Hyperbolic groups, Essays in Group Theory (S. M. Gersten, ed.), SpringerVerlag, 1987, pp. 75264. MR 89e:20070
 [HS]
 J. Hass and P. Scott, Homotopy equivalence and homeomorphism of 3Manifolds, Topology 31 (1992), 493517. MR 94g:57021
 [Lo]
 D. Long, Immersions and embeddings of totally geodesic surfaces, Bull. London Math. Soc. 19 (1987), 481484. MR 89g:57014
 [LN]
 D. Long and G. A. Niblo, Subgroup separability and 3manifold groups, Math. Z. 207 (1991), 209215. MR 92g:20047
 [Ma]
 S. Matsumoto, Subgroup Separability of 3Manifold Groups, Ph.D. thesis, University of Michigan, 1995.
 [Ma1]
 , A 3manifold with a nonsubgroupseparable fundamental group, to appear in Bull. Austral. Math. Soc. 55 (1997), 261279. CMP 97:09
 [Ma2]
 , Separability criterion for graphmanifold groups, to appear in Topology Appl.
 [Mo]
 L. Mosher, Geometry of cubulated 3manifolds, Topology 34 (1995), 789813. 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 injective surfaces in graph manifolds, to appear in Comm. Math. Helv.
 [Sc]
 P. Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2), 17 (1978), 555565. MR 58:12996; MR 87k:57003
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 A. Skinner, The word problem in the fundamental groups of a class of threedimensional 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, Ookayama, Meguroku, Tokyo 152, Japan
Email:
saburo@is.titech.ac.jp
DOI:
http://dx.doi.org/10.1090/S000299399704015X
PII:
S 00029939(97)04015X
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
