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March 2012 Bending Fuschsian representations of fundamental groups of cusped surfaces in $\mathrm{PU}(2,1)$
Pierre Will
J. Differential Geom. 90(3): 473-520 (March 2012). DOI: 10.4310/jdg/1335273392

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.

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Pierre Will. "Bending Fuschsian representations of fundamental groups of cusped surfaces in $\mathrm{PU}(2,1)$." J. Differential Geom. 90 (3) 473 - 520, March 2012. https://doi.org/10.4310/jdg/1335273392

Information

Published: March 2012
First available in Project Euclid: 24 April 2012

zbMATH: 1255.30043
MathSciNet: MR2916044
Digital Object Identifier: 10.4310/jdg/1335273392

Rights: Copyright © 2012 Lehigh University

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Vol.90 • No. 3 • March 2012
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