Bending Fuchsian representations of fundamental groups of cusped surfaces in PU(2,1)

2011/01/05 by Pierre Will, Will, Pierre
Mathematics · #22E40 #32M15 #51M10 #Advanced Combinatorial Mathematics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1101.0925

openalex publication_date 2011/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe a family of representations of π₁(Σ) in PU(2,1), where Σ is a hyperbolic Riemann surface with at least one deleted point. This family is obtained by a bending process associated to an ideal triangulation of Σ. We give an explicit description of this family by describing a coordinates system in the spirit of shear coordinates on the Teichmüller space. We identify within this family new examples of discrete, faithful and type-preserving representations of π₁(Σ). In turn, we obtain a 1-parameter family of embeddings of the Teichmüller space of Σ in the PU(2,1)-representation variety of π₁(Σ). These results generalise to arbitrary Σ the results obtained in a previous paper for the 1-punctured torus.

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