Moduli spaces of local systems and higher Teichmüller theory

Vladimir Fock; Alexander Goncharov

Publications Mathématiques de l'IHÉS (2006)

  • Volume: 103, page 1-211
  • ISSN: 0073-8301

Abstract

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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 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.

How to cite

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Fock, Vladimir, and Goncharov, Alexander. "Moduli spaces of local systems and higher Teichmüller theory." Publications Mathématiques de l'IHÉS 103 (2006): 1-211. <http://eudml.org/doc/104216>.

@article{Fock2006,
abstract = {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 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.},
author = {Fock, Vladimir, Goncharov, Alexander},
journal = {Publications Mathématiques de l'IHÉS},
language = {eng},
pages = {1-211},
publisher = {Springer},
title = {Moduli spaces of local systems and higher Teichmüller theory},
url = {http://eudml.org/doc/104216},
volume = {103},
year = {2006},
}

TY - JOUR
AU - Fock, Vladimir
AU - Goncharov, Alexander
TI - Moduli spaces of local systems and higher Teichmüller theory
JO - Publications Mathématiques de l'IHÉS
PY - 2006
PB - Springer
VL - 103
SP - 1
EP - 211
AB - 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 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.
LA - eng
UR - http://eudml.org/doc/104216
ER -

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Citations in EuDML Documents

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  1. Vladimir V. Fock, Alexander B. Goncharov, Cluster ensembles, quantization and the dilogarithm
  2. Sophie Morier-Genoud, Valentin Ovsienko, Serge Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons
  3. François Labourie, Cross ratios, surface groups, P S L ( n , 𝐑 ) and diffeomorphisms of the circle
  4. Alexander B. Goncharov, Richard Kenyon, Dimers and cluster integrable systems
  5. François Labourie, Cross ratios, Anosov representations and the energy functional on Teichmüller space

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