## Journal of Differential Geometry

- J. Differential Geom.
- Volume 90, Number 3 (2012), 473-520.

### 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.

#### Article information

**Source**

J. Differential Geom., Volume 90, Number 3 (2012), 473-520.

**Dates**

First available in Project Euclid: 24 April 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1335273392

**Digital Object Identifier**

doi:10.4310/jdg/1335273392

**Mathematical Reviews number (MathSciNet)**

MR2916044

**Zentralblatt MATH identifier**

1255.30043

#### Citation

Will, Pierre. Bending Fuschsian representations of fundamental groups of cusped surfaces in $\mathrm{PU}(2,1)$. J. Differential Geom. 90 (2012), no. 3, 473--520. doi:10.4310/jdg/1335273392. https://projecteuclid.org/euclid.jdg/1335273392