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Transactions of the American Mathematical Society

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Taut foliations in branched cyclic covers and left-orderable groups


Authors: Steven Boyer and Ying Hu
Journal: Trans. Amer. Math. Soc. 372 (2019), 7921-7957
MSC (2010): Primary 57M50, 57R30, 20F60; Secondary 57M25, 57M99, 20F36
DOI: https://doi.org/10.1090/tran/7833
Published electronically: June 10, 2019
MathSciNet review: 4029686
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Abstract: We study the left-orderability of the fundamental groups of cyclic branched covers of links which admit co-oriented taut foliations. In particular we do this for cyclic branched covers of fibred knots in integer homology $ 3$-spheres and cyclic branched covers of closed braids. The latter allows us to complete the proof of the L-space conjecture for closed, connected, orientable, irreducible $ 3$-manifolds containing a genus one fibred knot. We also prove that the universal abelian cover of a manifold obtained by generic Dehn surgery on a hyperbolic fibred knot in an integer homology $ 3$-sphere admits a co-oriented taut foliation and has left-orderable fundamental group, even if the surgered manifold does not, and that the same holds for many branched covers of satellite knots with braided patterns. A key fact used in our proofs is that the Euler class of a universal circle representation associated to a co-oriented taut foliation coincides with the Euler class of the foliation's tangent bundle. Though known to experts, no proof of this important result has appeared in the literature. We provide such a proof in the paper.


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Additional Information

Steven Boyer
Affiliation: Département de Mathématiques, Université du Québec à Montréal, 201 avenue du Président-Kennedy, Montréal, Quebec H2X 3Y7, Canada
Email: boyer.steven@uqam.ca

Ying Hu
Affiliation: Department of Mathematics, University of Nebraska Omaha, 6001 Dodge Street, Omaha, Nebraska 68182
Email: yinghu@unomaha.edu

DOI: https://doi.org/10.1090/tran/7833
Keywords: Cyclic branched covers, left-orderable groups, fractional Dehn twist coefficient, taut foliation, contact structure
Received by editor(s): June 19, 2018
Received by editor(s) in revised form: February 18, 2019, and February 22, 2019
Published electronically: June 10, 2019
Additional Notes: The first author was partially supported by NSERC grant RGPIN 9446-2013
The second author was partially supported by a CIRGET postdoctoral fellowship
Article copyright: © Copyright 2019 American Mathematical Society