Algebraic & Geometric Topology

Left-orderability and cyclic branched coverings

Ying Hu

Full-text: Open access


We provide an alternative proof of a sufficient condition for the fundamental group of the nth cyclic branched cover of S3 along a prime knot K to be left-orderable, which is originally due to Boyer, Gordon and Watson. As an application of this sufficient condition, we show that for any (p,q) two-bridge knot, with p 3  mod 4, there are only finitely many cyclic branched covers whose fundamental groups are not left-orderable. This answers a question posed by Da̧bkowski, Przytycki and Togha.

Article information

Algebr. Geom. Topol., Volume 15, Number 1 (2015), 399-413.

Received: 3 February 2014
Revised: 25 June 2014
Accepted: 30 June 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M05: Fundamental group, presentations, free differential calculus
Secondary: 57M12: Special coverings, e.g. branched 57M27: Invariants of knots and 3-manifolds

left-orderable groups cyclic branched coverings group representations two-bridge knots Riley's polynomial


Hu, Ying. Left-orderability and cyclic branched coverings. Algebr. Geom. Topol. 15 (2015), no. 1, 399--413. doi:10.2140/agt.2015.15.399.

Export citation


  • G,M Bergman, Right orderable groups that are not locally indicable, Pacific J. Math. 147 (1991) 243–248
  • H,U Boden, S Friedl, Metabelian ${\rm SL}(n,\mathbb{C})$ representations of knot groups, Pacific J. Math. 238 (2008) 7–25
  • S Boyer, C,M Gordon, L Watson, On L–spaces and left-orderable fundamental groups, Math. Ann. 356 (2013) 1213–1245
  • S Boyer, D Rolfsen, B Wiest, Orderable $3$–manifold groups, Ann. Inst. Fourier $($Grenoble$)$ 55 (2005) 243–288
  • D Calegari, N,M Dunfield, Laminations and groups of homeomorphisms of the circle, Invent. Math. 152 (2003) 149–204
  • A Clay, T Lidman, L Watson, Graph manifolds, left-orderability and amalgamation, Algebr. Geom. Topol. 13 (2013) 2347–2368
  • M,K Dąbkowski, J,H Przytycki, A,A Togha, Non-left-orderable $3$–manifold groups, Canad. Math. Bull. 48 (2005) 32–40
  • É Ghys, Groups acting on the circle, Enseign. Math. 47 (2001) 329–407
  • C Gordon, T Lidman, Taut foliations, left-orderability, and cyclic branched covers, Acta Math. Vietnam. 39 (2014) 599–635
  • J,E Greene, Alternating links and left-orderability
  • J Hilgert, K-H Neeb, Structure and geometry of Lie groups, Springer, New York (2012)
  • J Howie, H Short, The band-sum problem, J. London Math. Soc. 31 (1985) 571–576
  • T Ito, Non-left-orderable double branched coverings, Algebr. Geom. Topol. 13 (2013) 1937–1965
  • A Kawauchi, Survey on knot theory, Birkhäuser, Basel (1996)
  • V,T Khoi, A cut-and-paste method for computing the Seifert volumes, Math. Ann. 326 (2003) 759–801
  • X,S Lin, Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. $($Engl. Ser.$)$ 17 (2001) 361–380
  • R,C Lyndon, P,E Schupp, Combinatorial group theory, Springer, Berlin (2001)
  • F Nagasato, Finiteness of a section of the ${\rm SL}(2,\mathbb{C})$–character variety of the knot group, Kobe J. Math. 24 (2007) 125–136
  • P Ozsváth, Z Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005) 1281–1300
  • P Ozsváth, Z Szabó, On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005) 1–33
  • T Peters, On L–spaces and non left-orderable $3$–manifold groups
  • S,P Plotnick, Finite group actions and nonseparating $2$–spheres, Proc. Amer. Math. Soc. 90 (1984) 430–432
  • R Riley, Parabolic representations of knot groups, I, Proc. London Math. Soc. 24 (1972) 217–242
  • R Riley, Nonabelian representations of $2$–bridge knot groups, Quart. J. Math. Oxford Ser. 35 (1984) 191–208
  • M Teragaito, Fourfold cyclic branched covers of genus one two-bridge knots are $L$–spaces, Bol. Soc. Mat. Mex. 20 (2014) 391–403
  • A Tran, On left-orderablility and cyclic branched coverings to appear in J. Math. Soc. Japan
  • C,A Weibel, An introduction to homological algebra, Cambridge Studies Adv. Math. 38, Cambridge Univ. Press, Cambridge (1994)