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

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

2011-01-05

arxiv.org/abs/1101.0925v1

Mathematics

Geometric Topology

46 pages, 11 figures

Scientific paper

We describe a family of representations of $\pi_1(\Sigma)$ in 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 Teichm\"uller 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 Teichm\"uller space of $\Sigma$ in the PU(2,1)-representation variety of $\pi_1(\Sigma)$. These results generalise to arbitrary $\Sigma$ the results obtained in a previous paper for the 1-punctured torus.

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