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

Abstract

We describe a family of representations of $π_1(Σ)$ 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 $π_1(Σ)$. In turn, we obtain a 1-parameter family of embeddings of the Teichmüller space of $Σ$ in the PU(2,1)-representation variety of $π_1(Σ)$. These results generalise to arbitrary $Σ$ the results obtained in a previous paper for the 1-punctured torus.