Bavard's duality theorem for mixed commutator length.

Let $N$ be a normal subgroup of a group $G$. A quasimorphism $f$ on $N$ is $G$-quasi-invariant if there exists a non-negative number $D$ satisfying $|f(g x g^{-1}) - f(x)| \le D$ for every $g \in G$ and every $x \in N$. The goal in this paper is to establish Bavard's duality theorem of $G$-quasi-invariant quasimorphisms, which was previously proved by Kawasaki and Kimura in the case $N = [G,N]$. Our duality theorem provides a connection between $G$-quasi-invariant quasimorphisms and $(G, N)$-commutator lengths. Here for $x \in [G, N]$, the $(G, N)$-commutator length ${\rm cl}_{G, N}(x)$ 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 ${\rm scl}_G$ and ${\rm scl}_{G, N}$ are bi-Lipschitzly equivalent on $[G, N]$.