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Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation

Authors: R. Roberts, J. Shareshian and M. Stein
Journal: J. Amer. Math. Soc. 16 (2003), 639-679
MSC (2000): Primary 57M25; Secondary 57R30
Published electronically: March 3, 2003
MathSciNet review: 1969207
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Abstract: We investigate group actions on simply-connected (second countable but not necessarily Hausdorff) 1-manifolds and describe an infinite family of closed hyperbolic 3-manifolds whose fundamental groups do not act nontrivially on such 1-manifolds. As a corollary we conclude that these 3-manifolds contain no Reebless foliation. In fact, these arguments extend to actions on oriented $\mathbb R$-order trees and hence these 3-manifolds contain no transversely oriented essential lamination; in particular, they are non-Haken.

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Additional Information

R. Roberts
Affiliation: Department of Mathematics, Washington University, St Louis, Missouri 63130

J. Shareshian
Affiliation: Department of Mathematics, Washington University, St Louis, Missouri 63130

M. Stein
Affiliation: Department of Mathematics, Trinity College, Hartford, Connecticut 06106

Keywords: Hyperbolic, 3-manifold, tree, order tree, 1-manifold, group action, punctured surface bundle, Reebless foliation, essential lamination
Received by editor(s): June 2, 2002
Published electronically: March 3, 2003
Additional Notes: The first author was partially supported by National Science Foundation grant DMS-9971333
The second author was partially supported by National Science Foundation grant DMS-0070757
The third author was partially supported by a Trinity College Faculty Research Grant
Article copyright: © Copyright 2003 American Mathematical Society