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Moduli spaces of local systems and higher Teichmüller theory. (English) Zbl 1099.14025
Authors’ abstract: Let \(G\) be a split semisimple algebraic group over \(\mathbb{Q}\) 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 \(G(\mathbb{R})\), 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 \(S\) has holes, we define 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 under the action of the mapping class group of \(S\). We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmüller spaces related to \(G\) and \(S\), while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil-Petersson form for one of these spaces. It is related to the motivic dilogarithm.

MSC:
14H60 Vector bundles on curves and their moduli
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14D20 Algebraic moduli problems, moduli of vector bundles
14L24 Geometric invariant theory
14L35 Classical groups (algebro-geometric aspects)
22E46 Semisimple Lie groups and their representations
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
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