Bavard’s duality theorem for mixed commutator length

  • Morimichi Kawasaki

    Aoyama Gakuin University, Kanagawa, Japan
  • Mitsuaki Kimura

    Kyoto University, Japan
  • Takahiro Matsushita

    University of the Ryukyus, Okinawa, Japan
  • Masato Mimura

    Tohoku University, Sendai, Japan
Bavard’s duality theorem for mixed commutator length cover
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Let be a normal subgroup of a group . A quasimorphism on is -invariant if for every and every . The goal of this paper is to establish Bavard’s duality theorem of -invariant quasimorphisms, which was previously proved by Kawasaki and Kimura for the case

Our duality theorem provides a connection between -invariant quasimorphisms and -commutator lengths. Here, for , the -commutator length of is the minimum number such that is a product of commutators, which are written as with and . In the proof, we give a geometric interpretation of -commutator lengths. As an application of our Bavard duality, we obtain a sufficient condition on a pair under which and are bi-Lipschitz equivalent on .

Cite this article

Morimichi Kawasaki, Mitsuaki Kimura, Takahiro Matsushita, Masato Mimura, Bavard’s duality theorem for mixed commutator length. Enseign. Math. 68 (2022), no. 3/4, pp. 441–481

DOI 10.4171/LEM/1037