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Ĝ-invariant quasimorphisms and symplectic geometry of surfaces

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Abstract

In this paper, for a group Ĝ and its normal subgroup G, we study Ĝ-invariant quasimorphisms on G which appear in symplectic geometry and low dimensional topology. We will show a Bavard-type duality for Ĝ-invariant quasimorphisms and a variant of commutator length and obtain some comparison result on that length. As their application, we prove the non-existence of a section of the flux homomorphism on closed surfaces of higher genus.

We also prove that Py’s Calabi quasimorphism and Entov—Polterovich’s partial Calabi quasimorphism are non-extendable to the group of symplec-tomorphisms.

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Acknowledgment

The authors would like to thank Takahiro Matsushita for fruitful discussions on Propositions 3.1 and 3.2. They also thank Kazuhiko Fukui, Tomohiko Ishida, Jarek Kȩdra, Shuhei Maruyama, Atsuhide Mori, Ryuma Orita, Kaoru Ono and Leonid Polterovich for nice advice and comments. This work has been supported by JSPS KAKENHI Grant Number JP18J00765.

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Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Morimichi Kawasaki

  2. Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, 153-8914, Japan

    Mitsuaki Kimura

Authors
  1. Morimichi Kawasaki
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  2. Mitsuaki Kimura
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Correspondence to Morimichi Kawasaki.

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Kawasaki, M., Kimura, M. Ĝ-invariant quasimorphisms and symplectic geometry of surfaces. Isr. J. Math. 247, 845–871 (2022). https://doi.org/10.1007/s11856-021-2283-1

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  • DOI: https://doi.org/10.1007/s11856-021-2283-1

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