Bending fuchsian representations of fundamental groups of cusped surfaces in pu(2,1)

Abstract

We describe a new 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 [42] for the 1-punctured torus. © 2012 J. Differential Geometry.