Epstein-Poincar\'e surfaces for $G-$opers

TL;DR

This paper introduces Epstein-Poincaré surfaces for complex Lie group opers, generalizing classical constructions, and explores their implications for holonomy properties and transversality in geometric structures.

Contribution

It defines Epstein-Poincaré surfaces for $G$-opers, extends Epstein's classical construction, and links these surfaces to holonomy and transversality properties in complex Lie group geometry.

Findings

Provides a criterion for $ riangle$-Anosov holonomy.

Analyzes the interaction of developing maps with domains of discontinuity.

Offers a self-contained review of opers.

Abstract

Given a complex, simple Lie group $G$ of adjoint type, we introduce the notion of an Epstein-Poincar\'e surface associated to a $G$-oper. These surfaces generalize Epstein's classical construction for $G=PGL_2 (\mathbb{C})$. As an application, we provide a criterion that ensures that the holonomy of the oper is $\Delta-$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.