Geometrization of almost extremal representations in PSL₂\Bbb R

2018/02/02 by Gianluca Faraco, Faraco, Gianluca
Mathematics · #57M50 20H10 #Advanced Algebra and Geometry #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1802.00755

openalex publication_date 2018/02/02 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

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 PSL₂\Bbb R. We consider surfaces of genus greater than g and we show that, under suitable conditions, every representation ρ:π₁ S\longrightarrow PSL₂\Bbb R with Euler number E(ρ)=±(χ(S)+1) arises as holonomy of a hyperbolic cone-structure σ on S with a single cone point of angle 4π. From this result, we derive that for surfaces of genus 2 every representation with E(ρ)=±1 arises as the holonomy of some hyperbolic cone-structure.

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