Coarse geometry of stable mixed commutator length I: duality and functional analysis on chains

Let $G$ be a group and $N$ its normal subgroup. On the mixed commutator subgroup $[G,N]$, the mixed stable commutator length ${\mathrm{s}\mathrm{c}\mathrm{l}}_{G,N}$ and the restriction of the ordinary stable commutator length ${\mathrm{s}\mathrm{c}\mathrm{l}}_G$ are defined. We characterize when they are bi-Lipschitz equivalent by the vanishing of a certain $\mathbb{R}$-linear space $\mathrm{W}(G,N)$ related to invariant quasimorphisms. For the proof, we obtain a refined version of the generalized mixed Bavard duality theorem, and perform functional analysis on the completion of a certain space of $1$-chains.