## Taut foliations in branched cyclic covers and left-orderable groups

HTML articles powered by AMS MathViewer

- by Steven Boyer and Ying Hu PDF
- Trans. Amer. Math. Soc.
**372**(2019), 7921-7957 Request permission

## 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.## References

- J. W. Alexander,
*A lemma on system of knotted curves*, Proc. Nat. Acad. Sci. USA**9**(1923), 93–95. - K. Baker and K. Motegi,
*Seifert vs slice genera of knots in twist families and a characterization of braid axes*, preprint arXiv:1705.10373, 2017. - John A. Baldwin,
*Heegaard Floer homology and genus one, one-boundary component open books*, J. Topol.**1**(2008), no. 4, 963–992. MR**2461862**, DOI 10.1112/jtopol/jtn029 - M. Boileau, S. Boyer, and C. McA. Gordon,
*Branched covers of quasipositive links and L-spaces*, preprint arXiv:1710.07658, 2017. - Steven Boyer and Adam Clay,
*Foliations, orders, representations, L-spaces and graph manifolds*, Adv. Math.**310**(2017), 159–234. MR**3620687**, DOI 10.1016/j.aim.2017.01.026 - Steven Boyer, Cameron McA. Gordon, and Liam Watson,
*On L-spaces and left-orderable fundamental groups*, Math. Ann.**356**(2013), no. 4, 1213–1245. MR**3072799**, DOI 10.1007/s00208-012-0852-7 - Michel Boileau and Joan Porti,
*Geometrization of 3-orbifolds of cyclic type*, Astérisque**272**(2001), 208 (English, with English and French summaries). Appendix A by Michael Heusener and Porti. MR**1844891** - Steven Boyer, Dale Rolfsen, and Bert Wiest,
*Orderable 3-manifold groups*, Ann. Inst. Fourier (Grenoble)**55**(2005), no. 1, 243–288 (English, with English and French summaries). MR**2141698**, DOI 10.5802/aif.2098 - Jonathan Bowden,
*Approximating $C^0$-foliations by contact structures*, Geom. Funct. Anal.**26**(2016), no. 5, 1255–1296. MR**3568032**, DOI 10.1007/s00039-016-0387-2 - Gerhard Burde and Heiner Zieschang,
*Knots*, 2nd ed., De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 2003. MR**1959408** - Danny Calegari,
*Leafwise smoothing laminations*, Algebr. Geom. Topol.**1**(2001), 579–585. MR**1875608**, DOI 10.2140/agt.2001.1.579 - Danny Calegari,
*Foliations and the geometry of 3-manifolds*, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007. MR**2327361** - Alberto Candel,
*Uniformization of surface laminations*, Ann. Sci. École Norm. Sup. (4)**26**(1993), no. 4, 489–516. MR**1235439**, DOI 10.24033/asens.1678 - Alberto Candel and Lawrence Conlon,
*Foliations. II*, Graduate Studies in Mathematics, vol. 60, American Mathematical Society, Providence, RI, 2003. MR**1994394**, DOI 10.1090/gsm/060 - Danny Calegari and Nathan M. Dunfield,
*Laminations and groups of homeomorphisms of the circle*, Invent. Math.**152**(2003), no. 1, 149–204. MR**1965363**, DOI 10.1007/s00222-002-0271-6 - Adam Clay, Tye Lidman, and Liam Watson,
*Graph manifolds, left-orderability and amalgamation*, Algebr. Geom. Topol.**13**(2013), no. 4, 2347–2368. MR**3073920**, DOI 10.2140/agt.2013.13.2347 - Mieczysław K. Da̧bkowski, Józef H. Przytycki, and Amir A. Togha,
*Non-left-orderable 3-manifold groups*, Canad. Math. Bull.**48**(2005), no. 1, 32–40. MR**2118761**, DOI 10.4153/CMB-2005-003-6 - William D. Dunbar,
*Geometric orbifolds*, Rev. Mat. Univ. Complut. Madrid**1**(1988), no. 1-3, 67–99. MR**977042** - Albert Fathi, François Laudenbach, and Valentin Poénaru,
*Thurston’s work on surfaces*, Mathematical Notes, vol. 48, Princeton University Press, Princeton, NJ, 2012. Translated from the 1979 French original by Djun M. Kim and Dan Margalit. MR**3053012** - Benson Farb and Dan Margalit,
*A primer on mapping class groups*, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR**2850125** - David Gabai,
*Foliations and the topology of $3$-manifolds*, J. Differential Geom.**18**(1983), no. 3, 445–503. MR**723813** - David Gabai,
*Foliations and $3$-manifolds*, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 609–619. MR**1159248** - David Gabai and Ulrich Oertel,
*Essential laminations in $3$-manifolds*, Ann. of Math. (2)**130**(1989), no. 1, 41–73. MR**1005607**, DOI 10.2307/1971476 - Hansjörg Geiges,
*An introduction to contact topology*, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR**2397738**, DOI 10.1017/CBO9780511611438 - Étienne Ghys,
*Groups acting on the circle*, Enseign. Math. (2)**47**(2001), no. 3-4, 329–407. MR**1876932** - Paolo Ghiggini,
*Knot Floer homology detects genus-one fibred knots*, Amer. J. Math.**130**(2008), no. 5, 1151–1169. MR**2450204**, DOI 10.1353/ajm.0.0016 - C. McA. Gordon,
*Riley’s conjecture on $\textrm {SL}(2,\Bbb R)$ representations of 2-bridge knots*, J. Knot Theory Ramifications**26**(2017), no. 2, 1740003, 6. MR**3604485**, DOI 10.1142/S021821651740003X - Cameron Gordon and Tye Lidman,
*Taut foliations, left-orderability, and cyclic branched covers*, Acta Math. Vietnam.**39**(2014), no. 4, 599–635. MR**3292587**, DOI 10.1007/s40306-014-0091-y - Cameron Gordon and Tye Lidman,
*Corrigendum to “Taut foliations, left-orderability, and cyclic branched covers” [ MR3292587]*, Acta Math. Vietnam.**42**(2017), no. 4, 775–776. MR**3708042**, DOI 10.1007/s40306-017-0216-1 - J. Hanselman, J. Rasmussen, and L. Watson,
*Bordered Floer homology for manifolds with torus boundary via immersed curves*, preprint arXiv:1604.03466, 2016. - J. Hanselman, J. Rasmussen, S. Rasmussen, and L. Watson,
*Taut foliations on graph manifolds*, preprint arXiv:1508.05911v1, 2015. - Shelly Harvey, Keiko Kawamuro, and Olga Plamenevskaya,
*On transverse knots and branched covers*, Int. Math. Res. Not. IMRN**3**(2009), 512–546. MR**2482123**, DOI 10.1093/imrn/rnn138 - Matthew Hedden,
*Notions of positivity and the Ozsváth-Szabó concordance invariant*, J. Knot Theory Ramifications**19**(2010), no. 5, 617–629. MR**2646650**, DOI 10.1142/S0218216510008017 - Ko Honda, William H. Kazez, and Gordana Matić,
*Right-veering diffeomorphisms of compact surfaces with boundary*, Invent. Math.**169**(2007), no. 2, 427–449. MR**2318562**, DOI 10.1007/s00222-007-0051-4 - Ko Honda, William H. Kazez, and Gordana Matić,
*Right-veering diffeomorphisms of compact surfaces with boundary. II*, Geom. Topol.**12**(2008), no. 4, 2057–2094. MR**2431016**, DOI 10.2140/gt.2008.12.2057 - Jennifer Hom,
*Satellite knots and L-space surgeries*, Bull. Lond. Math. Soc.**48**(2016), no. 5, 771–778. MR**3556360**, DOI 10.1112/blms/bdw044 - James Howie and Hamish Short,
*The band-sum problem*, J. London Math. Soc. (2)**31**(1985), no. 3, 571–576. MR**812788**, DOI 10.1112/jlms/s2-31.3.571 - Ying Hu,
*Left-orderability and cyclic branched coverings*, Algebr. Geom. Topol.**15**(2015), no. 1, 399–413. MR**3325741**, DOI 10.2140/agt.2015.15.399 - Tetsuya Ito,
*Braid ordering and knot genus*, J. Knot Theory Ramifications**20**(2011), no. 9, 1311–1323. MR**2844810**, DOI 10.1142/S0218216511009169 - Tetsuya Ito,
*Braid ordering and the geometry of closed braid*, Geom. Topol.**15**(2011), no. 1, 473–498. MR**2788641**, DOI 10.2140/gt.2011.15.473 - Tetsuya Ito and Keiko Kawamuro,
*Essential open book foliations and fractional Dehn twist coefficient*, Geom. Dedicata**187**(2017), 17–67. MR**3622682**, DOI 10.1007/s10711-016-0188-7 - András Juhász,
*A survey of Heegaard Floer homology*, New ideas in low dimensional topology, Ser. Knots Everything, vol. 56, World Sci. Publ., Hackensack, NJ, 2015, pp. 237–296. MR**3381327**, DOI 10.1142/9789814630627_{0}007 - William H. Kazez and Rachel Roberts,
*Fractional Dehn twists in knot theory and contact topology*, Algebr. Geom. Topol.**13**(2013), no. 6, 3603–3637. MR**3248742**, DOI 10.2140/agt.2013.13.3603 - William H. Kazez and Rachel Roberts,
*Approximating $C^{1,0}$-foliations*, Interactions between low-dimensional topology and mapping class groups, Geom. Topol. Monogr., vol. 19, Geom. Topol. Publ., Coventry, 2015, pp. 21–72. MR**3609903**, DOI 10.2140/gtm.2015.19.21 - Paolo Lisca and András I. Stipsicz,
*Ozsváth-Szabó invariants and tight contact 3-manifolds. III*, J. Symplectic Geom.**5**(2007), no. 4, 357–384. MR**2413308**, DOI 10.4310/JSG.2007.v5.n4.a1 - Yu Li and Liam Watson,
*Genus one open books with non-left-orderable fundamental group*, Proc. Amer. Math. Soc.**142**(2014), no. 4, 1425–1435. MR**3162262**, DOI 10.1090/S0002-9939-2014-11847-8 - A. V. Malyutin,
*Writhe of (closed) braids*, Algebra i Analiz**16**(2004), no. 5, 59–91 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**16**(2005), no. 5, 791–813. MR**2106667**, DOI 10.1090/S1061-0022-05-00879-4 - John W. Morgan,
*The Smith conjecture*, The Smith conjecture (New York, 1979) Pure Appl. Math., vol. 112, Academic Press, Orlando, FL, 1984, pp. 3–6. MR**758460**, DOI 10.1016/S0079-8169(08)61632-3 - John Milnor,
*On the existence of a connection with curvature zero*, Comment. Math. Helv.**32**(1958), 215–223. MR**95518**, DOI 10.1007/BF02564579 - Shigenori Matsumoto and Shigeyuki Morita,
*Bounded cohomology of certain groups of homeomorphisms*, Proc. Amer. Math. Soc.**94**(1985), no. 3, 539–544. MR**787909**, DOI 10.1090/S0002-9939-1985-0787909-6 - Shigeyuki Morita,
*Geometry of differential forms*, Translations of Mathematical Monographs, vol. 201, American Mathematical Society, Providence, RI, 2001. Translated from the two-volume Japanese original (1997, 1998) by Teruko Nagase and Katsumi Nomizu; Iwanami Series in Modern Mathematics. MR**1851352**, DOI 10.1090/mmono/201 - John W. Milnor and James D. Stasheff,
*Characteristic classes*, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR**0440554**, DOI 10.1515/9781400881826 - William Meeks III, Leon Simon, and Shing Tung Yau,
*Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature*, Ann. of Math. (2)**116**(1982), no. 3, 621–659. MR**678484**, DOI 10.2307/2007026 - Kunio Murasugi,
*On closed $3$-braids*, Memoirs of the American Mathematical Society, No. 151, American Mathematical Society, Providence, R.I., 1974. MR**0356023** - Yi Ni,
*Knot Floer homology detects fibred knots*, Invent. Math.**170**(2007), no. 3, 577–608. MR**2357503**, DOI 10.1007/s00222-007-0075-9 - S. P. Novikov,
*The topology of foliations*, Trudy Moskov. Mat. Obšč.**14**(1965), 248–278 (Russian). MR**0200938** - Peter Ozsváth and Zoltán Szabó,
*Holomorphic disks and genus bounds*, Geom. Topol.**8**(2004), 311–334. MR**2023281**, DOI 10.2140/gt.2004.8.311 - Peter Ozsváth and Zoltán Szabó,
*On knot Floer homology and lens space surgeries*, Topology**44**(2005), no. 6, 1281–1300. MR**2168576**, DOI 10.1016/j.top.2005.05.001 - T. Peters,
*On L-spaces and non-left-orderable 3-manifold groups*, preprint arXiv:0903.4495, 2009. - J. F. Plante,
*Foliations with measure preserving holonomy*, Ann. of Math. (2)**102**(1975), no. 2, 327–361. MR**391125**, DOI 10.2307/1971034 - Rachel Roberts,
*Taut foliations in punctured surface bundles. II*, Proc. London Math. Soc. (3)**83**(2001), no. 2, 443–471. MR**1839461**, DOI 10.1112/plms/83.2.443 - D. Rolfsen,
**Knots and links**, American Mathematical Soc.**346**, 2003. - Harold Rosenberg,
*Foliations by planes*, Topology**7**(1968), 131–138. MR**228011**, DOI 10.1016/0040-9383(68)90021-9 - Edwin H. Spanier,
*Algebraic topology*, Springer-Verlag, New York, [1995?]. Corrected reprint of the 1966 original. MR**1325242** - W. Thurston,
*Hyperbolic structures on $3$-manifolds, II: Surface groups and $3$-manifolds which fiber over the circle*, preprint (1986), arXiv:math.GT/9801045. - William P. Thurston,
*On the geometry and dynamics of diffeomorphisms of surfaces*, Bull. Amer. Math. Soc. (N.S.)**19**(1988), no. 2, 417–431. MR**956596**, DOI 10.1090/S0273-0979-1988-15685-6

## Additional Information

**Steven Boyer**- Affiliation: Département de Mathématiques, Université du Québec à Montréal, 201 avenue du Président-Kennedy, Montréal, Quebec H2X 3Y7, Canada
- MR Author ID: 219677
- 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
- 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 - © Copyright 2019 American Mathematical Society
- 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
- MathSciNet review: 4029686