Bavard’s duality theorem for mixed commutator length
Morimichi Kawasaki
Aoyama Gakuin University, Kanagawa, JapanMitsuaki Kimura
Kyoto University, JapanTakahiro Matsushita
University of the Ryukyus, Okinawa, JapanMasato Mimura
Tohoku University, Sendai, Japan
Abstract
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