Bending Fuschsian representations of fundamental groups of cusped surfaces in $\mathrm{PU}(2,1)$

Abstract

We describe a new family of representations of $\pi_1(\Sigma)$ in $\mathrm{PU}(2,1)$, where $\Sigma$ 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 $\Sigma$. 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 $\pi_1(\Sigma)$. In turn, we obtain a 1-parameter family of embeddings of the Teichmüller space of $\Sigma$ in the $\mathrm{PU}(2,1)$-representation variety of $\pi_1(\Sigma)$. These results generalise to arbitrary $\Sigma$ the results obtained in "The punctured torus and Lagrangian triangle groups in $\mathrm{PU}(2,1)$," J. reine angew. Math., 602 (2007), 95–121, for the 1-punctured torus.