Bavard's duality theorem of invariant quasimorphisms

Let $H$ be a normal subgroup of a group $G$. A quasimorphism $f$ on $H$ is $G$-invariant if there is a non-negative number $D$ satisfying $|f(gxg^{-1}) - f(x)| \le D$ for every $g \in G$ and every $x \in H$. The purpose in this paper is to prove Bavard's duality theorem of $G$-invariant quasimorphisms, which was previously proved by Kawasaki and Kimura in the case $H = [G,H]$. Our duality theorem gives a connection between $G$-invariant quasimorphisms and $(G,H)$-commutator lengths. Here for $x \in [G,H]$, the $(G,H)$-commutator length $\cl_{G,H}(x)$ of $x$ is the minimum number $n$ such that $x$ is a product of $n$ commutators which are written by $[g,h]$ with $g \in G$ and $h \in H$. In the proof, we give a geometric interpretation of $(G,H)$-commutator lengths.