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

TL;DR

This paper explores the connection between hyperbolic cone-structures on surfaces and certain representations of their fundamental groups into PSL(2,R), showing that almost extremal representations correspond to specific hyperbolic cone-structures with a single cone point.

Contribution

It establishes a correspondence between almost extremal representations and hyperbolic cone-structures with one cone point, extending known results to surfaces of genus greater than one.

Findings

Representations with Euler number ±(χ(S)+1) correspond to hyperbolic cone-structures with one cone point.

For genus 2 surfaces, all representations with Euler number ±1 arise from hyperbolic cone-structures.

The results connect geometric structures on surfaces with algebraic properties of their fundamental group representations.

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 $\text{PSL}_2\Bbb R$. We consider surfaces of genus greater than $g$ and we show that, under suitable conditions, every representation $\rho:\pi_1 S\longrightarrow \text{PSL}_2\Bbb R$ with Euler number $\mathcal{E}(\rho)=\pm\big(\chi(S)+1\big)$ 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 $\mathcal{E}(\rho)=\pm1$ arises as the holonomy of some hyperbolic cone-structure.