Commuting symplectomorphisms on a surface and the flux homomorphism.

Let $(S,\omega)$ be a closed orientable surface whose genus $l$ is at least two. Then we provide an obstruction for commuting symplectomorphisms in terms of the flux homomorphism. More precisely, we show that for every non-negative integer $n$ and for every homomorphism $\alpha \colon \mathbb{Z}^n \to \mathrm{Symp}_0(S, \omega)$, the image of $\mathrm{Flux}_\omega \circ \alpha \colon \mathbb{Z}^n \to H^1(S ; \mathbb{R})$ is contained in an $l$-dimensional real linear subspace of $H^1(S; \mathbb{R})$. For the proof, we show the following two keys: a refined version of the non-extendability of Py's Calabi quasimorphism $\mu_P$ on $\mathrm{Ham}(S, \omega)$, and an extension theorem of $\hat{G}$-invariant quasimorphisms on $G$ for a group $\hat{G}$ and a normal subgroup $G$ with certain conditions. We also pose the conjecture that the cup product of the fluxes of commuting symplectomorphisms is trivial.