Intersection of holonomy varieties of $CP^1$-structures

TL;DR

This paper investigates the intersection properties of holonomy varieties associated with ${ m CP}^1$-structures on a closed surface, revealing countably infinite pairs sharing the same holonomy between distinct structures.

Contribution

It establishes the existence of countably infinite pairs of ${ m CP}^1$-structures on different Riemann surfaces with identical holonomy representations.

Findings

Countably infinite pairs of structures share the same holonomy.

Holonomy varieties intersect in a countably infinite set.

Results deepen understanding of ${ m CP}^1$-structure moduli spaces.

Abstract

Let $\Sigma$ be a closed orientable surface of genus at least two, and let $X, Y$ be distinct marked Riemann surface structures on $\Sigma$, possibly with opposite orientations. In this paper, we show that there are (exactly) countably infinite pairs of ${\rm CP}^1$-structures on $X$ and on $Y$ sharing holonomy $\pi_1(\Sigma) \to {\rm PSL}_2 {\rm C}$.