Bavard’s duality theorem for mixed commutator length

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 $g \in G$ and every $x \in N$. 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 \in \[G,N]$, the $(G,N)$-commutator length $\operatorname{cl}\_{G,N}(x)$ of $x$ is the minimum number $n$ such that $x$ is a product of $n$ commutators, which are written as $\[g,h]$ with $g \in G$ and $h \in N$. 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 $\operatorname{scl}G$ and $\operatorname{scl}{G,N}$ are bi-Lipschitz equivalent on $\[G,N]$.