# 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

## Abstract

Let $N$ be a normal subgroup of a group $G$. A quasimorphism $f$ on $N$ is $G$-invariant if $f(gxg_{−1})=f(x)$ for every $g∈G$ and every $x∈N$. The goal of this paper is to establish Bavard’s duality theorem of $G$-invariant quasimorphisms, which was previously proved by Kawasaki and Kimura for the case $N=[G,N]$

Our duality theorem provides a connection between $G$-invariant quasimorphisms and $(G,N)$-commutator lengths. Here, for $x∈[G,N]$, the $(G,N)$-commutator length $cl_{G,N}(x)$ of $x$ is the minimum number $n$ such that $x$ is a product of $n$ commutators, which are written as $[g,h]$ with $g∈G$ and $h∈N$. In the proof, we give a geometric interpretation of $(G,N)$-commutator lengths. As an application of our Bavard duality, we obtain a sufficient condition on a pair $(G,N)$ under which $scl_{G}$ and $scl_{G,N}$ are bi-Lipschitz equivalent on $[G,N]$.

## 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