Invariant quasimorphisms for groups acting on the circle and non-equivalence of SCL

We construct invariant quasimorphisms for groups acting on the circle. Furthermore, we provide a criterion for the non-extendablity of the resulting quasimorphisms and an explicit formula which relates the values of our quasimorphisms to those of the Poincaré translation number. By using them, we show that the stable commutator length $\mathrm{scl}_G$ and the stable mixed commutator length $\mathrm{scl}_{G,N}$ are not bi-Lipschitzly equivalent for the surface group $G=\pi_1(\Sigma_{\ell})$ of genus at least $2$ and its commutator subgroup $N = [\pi_1(\Sigma_{\ell}), \pi_1(\Sigma_{\ell})]$. We also show the non-equivalence for a pair $(G,N)$ such that $G$ is the fundamental group of a $3$-dimensional closed hyperbolic mapping torus. These pairs serve as the first family of examples of such $(G,N)$ in which $G$ is finitely generated.