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

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 $\pi_1(\Sigma) \to {\rm PSL}_2 {\rm C}$.