Bavard's duality theorem for mixed commutator length (Q2674719)

Summary: Let \(N\) be a normal subgroup of a group \(G\). A quasimorphism \(f\) on \(N\) is \(G\)-invariant if \(f (g x g^{- 1}) = f (x)\) 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 \(\operatorname{cl}_{G, N} (x)\) 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 \([G, N]\).