Mathematische Annalen

, Volume 356, Issue 4, pp 1213–1245 | Cite as

On L-spaces and left-orderable fundamental groups

Article

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 \(\text{ Homeo}_+(S^1)\).

Mathematics Subject Classification (2000)

57M25 57M50 57M99 20F60 06F15 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Steven Boyer
    • 1
  • Cameron McA. Gordon
    • 2
  • Liam Watson
    • 3
  1. 1.Département de MathématiquesUniversité du Québec à MontréalCentre-Ville, MontréalCanada
  2. 2.Department of MathematicsUniversity of Texas at AustinAustinUSA
  3. 3.Department of MathematicsUniversity of California at Los AngelesLos AngelesUSA

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