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      Moduli spaces of local systems and higher Teichmuller theory

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          Abstract

          Let G be a split semisimple algebraic group with trivial center. Let S be a compact oriented surface, with or without boundary. We define {\it 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 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 Teichmuller 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.

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          Journal
          10 November 2003
          2006-04-28
          Article
          math/0311149
          1e6b9525-1008-49e1-be78-9cc91dac3fe1
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          Custom metadata
          Version 4: hopefully the final version. reformatted, so became 220 pages. Section 6.8, 6.8, 7.9, 13 are new
          math.AG math.DG

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