Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation

Roberts, R.; Shareshian, J.; Stein, M.

doi:10.1090/S0894-0347-03-00426-0

Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation
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by R. Roberts, J. Shareshian and M. Stein

J. Amer. Math. Soc. 16 (2003), 639-679

DOI: https://doi.org/10.1090/S0894-0347-03-00426-0

Published electronically: March 3, 2003

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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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