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On Sandon-type metrics for contactomorphism groups

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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.

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.

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Notes

  1. In fact, although the study of \(\succeq \) on \(\widetilde{\mathcal {G}}(V)\) was initiated in [14], the terminology orderable was introduced later in [15] and was applied to V itself; to avoid confusion we specify instead the relevant group.

  2. See the argument in Remark 1.23 of that paper and the proof of Theorem 1.3 on the same page.

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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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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada

    Maia Fraser

  2. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Leonid Polterovich & Daniel Rosen

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  1. Maia Fraser
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  2. Leonid Polterovich
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Correspondence to Maia Fraser.

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Fraser, M., Polterovich, L. & Rosen, D. On Sandon-type metrics for contactomorphism groups. Ann. Math. Québec 42, 191–214 (2018). https://doi.org/10.1007/s40316-017-0092-z

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