A la Fock-Goncharov coordinates for PU(2,1)

TL;DR

This paper introduces a new coordinate system for the PU(2,1)-representation variety of punctured surfaces, utilizing geometric invariants of flag triples in complex hyperbolic space, enabling a bijective correspondence with ideal triangulation decorations.

Contribution

It develops a novel set of coordinates based on flag invariants for PU(2,1) representations, linking geometric structures to surface triangulations.

Findings

Established a bijection between decorations and a subset of the representation variety.

Defined geometric invariants of flag triples in complex hyperbolic space.

Provided a coordinate system for PU(2,1)-representations of punctured surfaces.

Abstract

We describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented punctured surface $S$ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of a triple of flags in the complex hyperpolic plane. We establish a bijection between a set of decorations of an ideal triangulation of $S$ and a subset of the PU(2,1)-representation variety of $\pi_1(S)$.