SL2R action on differentials (${\mathrm{SL}}_2(\mathbb{R})\curvearrowright \Omega {\mathcal{M}}_g,\mathcal{Q}{\mathcal{M}}_g$)

For nonzero holomorphic one-forms and holomorphic quadratic differentials, apply $A\in {\mathrm{SL}}_2(\mathbb{R})$ to every flat coordinate. Translation and sign transition maps remain of the same type, defining a new complex structure and differential. Composition gives a group action, zero orders persist and area is preserved. The zero section has no such atlas; fixing it gives only a set-theoretic extension that is generally discontinuous.