Moduli spaces of local systems and higher Teichmüller theory

Let G be a split semisimple algebraic group over 𝐐 with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G ( 𝐑 ) , construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant…