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On L-spaces and left-orderable fundamental groups

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Abstract

Examples suggest that there is a correspondence between L-spaces and three-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric three-manifold that is non-hyperbolic, a family which includes all closed, connected, orientable Seifert fibred spaces. We also show that they are equivalent for the twofold branched covers of non-split alternating links. To do this we prove that the fundamental group of the twofold branched cover of an alternating link is left-orderable if and only if it is a trivial link with two or more components. We also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in \(\text{ Homeo}_+(S^1)\).

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Acknowledgments

The authors thank Adam Clay, Josh Greene, Tye Lidman and Ciprian Manolescu for their comments on and interest in this work, Alan Reid for mentioning Corollary 3, Michael Polyak for showing them the presentation described in Sect. 3.1, and Józef Przytycki for pointing out Wada’s paper [47]. They also thank Adam Levine, Robert Lipshitz, Peter Ozsváth and Dylan Thurston for patiently answering questions about bordered Heegaard Floer homology, which proved to be a key tool for establishing Corollary 1. Finally, they thank the referee for his or her careful reading of the manuscript and many helpful comments.

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Authors and Affiliations

  1. Département de Mathématiques, Université du Québec à Montréal, PO Box 8888, Centre-Ville, Montréal, QC,  H3C 3P8, Canada

    Steven Boyer

  2. Department of Mathematics, University of Texas at Austin, 1 University Station, Austin, TX,  78712, USA

    Cameron McA. Gordon

  3. Department of Mathematics, University of California at Los Angeles, 520 Portola Plaza, Los Angeles, CA, 90095, USA

    Liam Watson

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  1. Steven Boyer
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  2. Cameron McA. Gordon
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  3. Liam Watson
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Correspondence to Steven Boyer.

Additional information

S. Boyer partially supported by NSERC grant OGP 9446-2008; C. McA. Gordon partially supported by NSF grant DMS-0906276; L. Watson partially supported by an NSERC postdoctoral fellowship.

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Boyer, S., Gordon, C.M. & Watson, L. On L-spaces and left-orderable fundamental groups. Math. Ann. 356, 1213–1245 (2013). https://doi.org/10.1007/s00208-012-0852-7

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