Bavard’s duality theorem for mixed commutator length

Let $N$ be a normal subgroup of group $G$. A quasimorphism $f$ on is $G$-invariant if $f(gxg^{-1}) = f(x)$ for every $g \in G$ and $x N$. The goal this paper to establish Bavard’s duality theorem quasimorphisms, which was previously proved by Kawasaki Kimura the case $N \[G,N]$ Our provides connection between quasimorphisms $(G,N)$-commutator lengths. Here, \[G,N]$, length $\operatorname{cl}\_{G,N}(x)$ $x$ minimum number $n$ such that product commutators, are written as $\[g,h]$ with $h In proof, we give geometric interpretation As an application our Bavard duality, obtain sufficient condition pair $(G,N)$ under $\operatorname{scl}G$ $\operatorname{scl}{G,N}$ bi-Lipschitz equivalent $\[G,N]$.