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 $f(gx{g}^{-1})=f(x)$ 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 ${\mathrm{cl}}_{G,N}(x)$ 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 ${\mathrm{scl}}_G$ and ${\mathrm{scl}}_{G,N}$ are bi-Lipschitz equivalent on $[G,N]$.