Intersection of holonomy varieties of $C{P}^1$-structures

Abstract

Let $\hat{I}\pounds$ be a closed orientable surface of genus at least two, and let $X,Y$ be distinct marked Riemann surface structures on $\hat{I}\pounds$, possibly with opposite orientations. In this paper, we show that there are (exactly) countably infinite pairs of ${\mathrm{CP}}^1$-structures on $X$ and on $Y$ sharing holonomy $\ddot{I}{€}_1(\hat{I}\pounds )\to {\mathrm{PSL}}_2\mathrm{C}$.

54 pages, 20 figures. An alternative proof is provided for the case in which the orientations of $X$ and $Y$ are opposite