Abstract
In this work, we describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented punctured surface Σ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of a triple of flags in the complex hyperbolic plane \({\bf H^2_{\mathbb {C}}}\) . We establish a bijection between a set of decorations of an ideal triangulation of Σ and a subset of the PU(2,1)-representation variety of π 1(Σ).
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Marché, J., Will, P. Configurations of flags and representations of surface groups in complex hyperbolic geometry. Geom Dedicata 156, 49–70 (2012). https://doi.org/10.1007/s10711-011-9589-9
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DOI: https://doi.org/10.1007/s10711-011-9589-9