Geometrization of almost extremal representations in ${\text{PSL}}_2\mathbb{R}$

Abstract

Let $S$ be a closed surface of genus $g$. In this paper, we investigate the relationship between hyperbolic cone-structure on $S$ and representations of the fundamental group into ${\text{PSL}}_2\mathbb{R}$. We consider surfaces of genus greater than $g$ and we show that, under suitable conditions, every representation $\rho :{\pi }_{1}S⟶{\text{PSL}}_2\mathbb{R}$ with Euler number $\mathcal{E}(\rho )=\pm (\chi (S)+1)$ arises as holonomy of a hyperbolic cone-structure $\sigma$ on $S$ with a single cone point of angle $4\pi$. From this result, we derive that for surfaces of genus $2$ every representation with $\mathcal{E}(\rho )=\pm 1$ arises as the holonomy of some hyperbolic cone-structure.

Improved version. 24 pages and 5 figures. Comments are welcome!