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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
Article
  • 94 Downloads

Abstract

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.

Keywords

Contact manifold Contactomorphism Conjugation Invariant norm 

Resume

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

53Dxx 

Notes

Acknowledgements

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.

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Copyright information

© 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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