Annales mathématiques du Québec

, Volume 42, Issue 2, pp 191–214 | Cite as

On Sandon-type metrics for contactomorphism groups

  • Maia FraserEmail author
  • Leonid Polterovich
  • Daniel Rosen


For certain contact manifolds admitting a 1-periodic Reeb flow we construct a conjugation-invariant norm on the universal cover of the contactomorphism group. With respect to this norm the group admits a quasi-isometric monomorphism of the real line. The construction involves the partial order on contactomorphisms and symplectic intersections. This norm descends to a conjugation-invariant norm on the contactomorphism group. As a counterpoint, we discuss conditions under which conjugation-invariant norms for contactomorphisms are necessarily bounded.


Contact manifold Contactomorphism Conjugation Invariant norm 


On construit une norme invariante par conjugaison sur le revêtement universel du groupe des contactomorphismes associé à certaines variétés de contact admettant un flot de Reeb 1-périodique. Par rapport à cette norme, le groupe admet un monomorphisme quasi-isométrique des réels. La construction utilise l’ordre partiel sur les contactomorphismes et des propriétés des intersections symplectiques. Cette norme induit une norme invariante par conjugaison sur le groupe des contactomorphismes. Par contraste avec cette construction, nous discutons de conditions sous lesquelles des normes invariantes par conjugaison sur des groupes des contactomorphismes sont nécessairement bornées.

Mathematics Subject Classification




The results of the present paper have been presented at the AIM workshop ‘Contact topology in higher dimensions’ in May, 2012. We thank the organizers, J. Etnyre, E. Giroux and K. Niederkrueger, for the invitation and AIM for the excellent research atmosphere. We are grateful to V. Colin, Y. Karshon, and S. Sandon for very helpful discussions during the workshop. We also thank V. Colin, S. Sandon, F. Zapolsky, P. Albers and W. Merry for communicating early versions of [3, 9, 39], respectively with us. We thank M. Usher for useful discussions. Finally we thank an anonymous referee for extremely helpful comments and corrections. L.P. and D.R. were partially supported by the Israel Science Foundation Grants 509/07 and 178/13, by the National Science Foundation grant DMS-1006610 and by the European Research Council advanced grant 338809.


  1. 1.
    Albers, P., Frauenfelder, U.: A nondisplaceable Lagrangian torus in \(T^\ast S^2\). Comm. Pure Appl. Math. 61, 1046–1051 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Albers, P., Frauenfelder, U.: Leaf-wise intersections and Rabinowitz Floer homology. J. Topol. Anal. 2(1), 77–98 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Albers, P., Merry, W.: Translated points and Rabinowitz Floer homology. J. Fixed Point Theory Appl. 13(1), 201–214 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Banyaga, A.: The structure of classical diffeomorphism groups. Mathematics and its Applications, 400. Kluwer Academic Publishers Group, Dordrecht (1997)Google Scholar
  5. 5.
    Biran, P., Cieliebak, K.: Lagrangian embeddings into subcritical Stein manifolds. Israel J. Math. 127, 221–244 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Biran, P., Cieliebak, K.: Symplectic topology on subcritical manifolds. Comm. Math. Helv. 76, 712–753 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Borman, S., Zapolsky, F.: Quasi-morphisms on contactomorphism groups and contact rigidity. Geom. Topol. 19(1), 365–411 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Burago, S., Ivanov, D., Polterovich, L.: Conjugation-invariant norms on groups of geometric origin. Groups of diffeomorphisms. Adv. Stud. Pure Math. 52, 221–250 (2008)zbMATHGoogle Scholar
  9. 9.
    Colin, V., Sandon, S.: The discriminant length for contact and Legendrian isotopies. J. Eur. Math. Soc. (JEMS) 17(7), 1657–1685 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Eliashberg, Y.: Contact 3-manifolds twenty years since. J. Martinet’s work. Ann. Inst. Fourier (Grenoble) 42, 165–192 (1992)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Eliashberg, Y.: Topological characterization of Stein manifolds of dimension \(>2\). Internat. J. Math. 1(1), 29–46 (1990)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Eliashberg, Y., Gromov, M., Convex symplectic manifolds. In: Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, : Proc. Sympos. Pure Math. 52, Part 2, Amer. Math. Soc. Providence, R I 1991, 135–162 (1989)Google Scholar
  13. 13.
    Eliashberg, Y., Hofer, H., Salamon, D.: Lagrangian intersections in contact geometry. Geom. Funct. Anal. 5, 244–269 (1995)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Eliashberg, Y., Polterovich, L.: Partially ordered groups and geometry of contact transformations. Geom. Funct. Anal. 10, 1448–1476 (2000)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Eliashberg, Y., Kim, S.S., Polterovich, L.: Geometry of contact transformations and domains: orderability versus squeezing. Geom. Topol. 10, 1635–1747 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Floer, A.: Morse theory for Lagrangian intersections. J. Differ. Geom. 28(3), 513–547 (1988)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ginzburg, V.: On Maslov class rigidity for coisotropic submanifolds. Pacific J. Math. 250, 139–161 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Givental, A.: The nonlinear Maslov index. Geometry of low-dimensional manifolds, 2. Durham : London Math. Soc. Lecture Note Ser. 151(1990), 35–43 (1989)Google Scholar
  19. 19.
    Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Granja, G., Karshon, Y., Pabiniak, M., Sandon, S.: Givental’s non-linear Maslov index on lens spaces. arXiv:1704.05827 (2017)
  21. 21.
    Laudenbach, F., Sikorav, J.-C.: Persistence of intersection with the zero section during a Hamiltonian isotopy into a cotangent bundle. Invent. Math. 82(2), 349–357 (1985)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Laudenbach, F., Sikorav, J.-C.: Hamiltonian disjunction and limits of Lagrangian submanifolds. Internat. Math. Res. Notices 4, 161–168 (1994)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Milin, I.: Orderability and Non-squeezing in Contact Geometry. Ph.D. thesis, Stanford University (2008)Google Scholar
  24. 24.
    Moser, J.: A fixed point theorem in symplectic geometry. Acta Math. 141(1–2), 17–34 (1978)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Müller, S., Spaeth, P.: Topological contact dynamics I: symplectization and applications of the energy-capacity inequality. Adv. Geom. 15(3), 349–380 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Paternain, G.P., Polterovich, L., Siburg, K.F.: Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry–Mather theory. Mosc. Math. J. 3(2), 593–619 (2003)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Polterovich, L.: The geometry of the group of symplectic diffeomorphism. Lectures in Mathematics. ETH-Zurich, Birkhauser, Basel (2001)CrossRefGoogle Scholar
  28. 28.
    Rybicki, T.: Commutators of contactomorphisms. Adv. Math. 225(6), 329–336 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Rybicki, T.: Bi-invariant metric on the strict contactomorphism group. arXiv:1202.5897 (2012) (Now withdrawn)
  30. 30.
    Rybicki, T.: Hofer metric from the contact point of view. arXiv:1304.1971 (2013) (Now withdrawn)
  31. 31.
    Sandon, S.: An integer valued bi-invariant metric on the group of contactomorphisms of \(R^{2n}\times S^1\). J. Topol. Anal. 2, 327–339 (2010)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Sandon, S.: Contact homology, capacity and non-squeezing in \(\mathbb{R}^{2n} \times S^1\) via generating functions. Ann. Inst. Fourier (Grenoble) 61, 145–185 (2011)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sandon, S.: Equivariant homology for generating functions and orderability of lens spaces. J. Symplectic Geom. 9(2), 123–146 (2011)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Sandon, S.: On iterated translated points for contactomorphisms of \(\mathbb{R }^{2n+1}\) and \(\mathbb{R }^{2n} \times S^1\). Internat. J. Math. 23(2) (2012)Google Scholar
  35. 35.
    Shelukhin, E.: The Hofer norm of a contactomorphism. J. Symplectic. Geom. arXiv:1411.1457 (2014) (to appear)
  36. 36.
    Sullivan, D.: Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math. 36, 225–255 (1976)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Tsuboi, T.: On the simplicity of the group of contactomorphisms. Groups of diffeomorphisms, Adv. Stud. Pure Math. 52, 491–504. Soc. Japan, Tokyo (2008)Google Scholar
  38. 38.
    Usher, M.: Hofer geometry and cotangent fibers. J. Symplectic Geom. 12(3), 619–656 (2014)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Zapolsky, F.: Geometric structures on contactomorphism groups and contact rigidity in jet spaces. Internat. Math. Res. Notices 2013(20), 4687–4711 (2013)CrossRefGoogle Scholar

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© Fondation Carl-Herz and Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  2. 2.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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