Holonomy map fibers of $\mathbb{C}{\rm P}^1$-structures in moduli space

Abstract

Let $S$ be a closed oriented surface of genus $g\geq 2$. Fix an arbitrary non-elementary representation $ρ\colonπ_1(S)\to {\rm SL}_2(\mathbb{C})$ and consider all marked (complex) projective structures on $S$ with holonomy $ρ$. We show that their underlying conformal structures are dense in the moduli space of $S$.

26 pages, 6 figures. To appear in Journal of Topology