Epstein-Poincaré surfaces for $G-$opers

Given a complex, simple Lie group $G$ of adjoint type, we introduce the notion of an Epstein-Poincaré surface associated to a $G$-oper. These surfaces generalize Epstein’s classical construction for $G=PG{L}_2(\mathbb{C})$. As an application, we provide a criterion that ensures that the holonomy of the oper is $\mathrm{\Delta }-$Anosov. Finally, we discuss how the developing map of the oper interacts with domains of discontinuity of the holonomy (whenever Anosov) and the transversality properties it satisfies. Along the way, we provide a quick review of opers that we hope serves as a self-contained introduction.