On totally hyperbolic non-Fuchsian type-preserving representations

2025/11/17 by Ryu, Inyoung
Mathematics · #57K20 #57M05 #57M50 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology

paper · doi:10.48550/arxiv.2511.13989

openalex publication_date 2025/11/17 · openalex created_date 2025/11/20 · openalex updated_date 2026/07/28

Abstract

We identify type-preserving representations ϕ: π₁(Σ)→ PSL(2,ℝ) of the fundamental group of every punctured surface Σ= Σ_g,p that are not Fuchsian yet send all non-peripheral simple closed curves to hyperbolic elements, which give a negative answer to a question of Bowditch. These representations have relative Euler class e(ϕ) = ± (χ(Σ) + 1), and their PSL(2,ℝ)-conjugacy classes form a full-measure subset of 2p connected components of the relative character variety. We further show that, while these representations are not Fuchsian, their restrictions to certain subsurfaces of Σ are Fuchsian.

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