Moduli spaces of local systems and higher Teichmüller theory

  title={Moduli spaces of local systems and higher Teichm{\"u}ller theory},
  author={Vladimir V. Fock and Alexander B. Goncharov},
  journal={Publications Math{\'e}matiques de l'Institut des Hautes {\'E}tudes Scientifiques},
  • V. FockA. Goncharov
  • Published 10 November 2003
  • Mathematics
  • Publications Mathématiques de l'Institut des Hautes Études Scientifiques
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… 

Quantum geometry of moduli spaces of local systems and representation theory

Let G be a split semi-simple adjoint group, and S an oriented surface with punctures and special boundary points. We introduce a moduli space P(G,S) parametrizing G-local system on S with some

Diophantine Analysis on Moduli of Local Systems

We develop a Diophantine analysis on moduli of special linear rank two local systems over surfaces with prescribed boundary traces. We first show that such a moduli space is a log Calabi-Yau variety

Parametrizing spaces of positive representations

. Using Lusztig’s total positivity in split real Lie groups V. Fock and A. Goncharov have introduced spaces of positive (framed) representations. For general semisimple Lie groups a generalization of

Maximal Representations of Surface Groups: Symplectic Anosov Structures

Let G be a connected semisimple Lie group such that the associ- ated symmetric space X is Hermitian and let Γg be the fundamental group of a compact orientable surface of genus g ≥ 2. We survey the

Coordinates on the augmented moduli space of convex RP2 structures

Let S be an orientable, finite‐type surface with negative Euler characteristic. The augmented moduli space of convex real projective structures on S was first defined and topologized by the first

Algebras of quantum monodromy data and decorated character varieties

The Riemann-Hilbert correspondence is an isomorphism between the de Rham moduli space and the Betti moduli space, defined by associating to each Fuchsian system its monodromy representation class. In

ON FOCK-GONCHAROV COORDINATES OF THE ONCE-PUNCTURED TORUS GROUP (Representation spaces, twisted topological invariants and geometric structures of 3-manifolds)

In their seminal paper [3], Fock and Goncharov defined positive representations of the fundamental group of a surface S into a split semi-simple real Lie group G (e.g. PSL(n, R)). They showed that

Higgs bundles and higher Teichmüller spaces

This paper is a survey on the role of Higgs bundle theory in the study of higher Teichmuller spaces. Recall that the Teichmuller space of a compact surface can be identified with a certain connected



The decorated Teichmüller space of punctured surfaces

A principal ℝ+5-bundle over the usual Teichmüller space of ans times punctured surface is introduced. The bundle is mapping class group equivariant and admits an invariant foliation. Several

Universal Constructions in Teichmüller Theory

We study a new model Tess of a universal Teichmuller space, which is defined to be the collection Tess′ of all ideal tesselations of the Poincare disk (together with a distinguished oriented edge)

Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization

The fundamental group is one of the most basic topological invariants of a space. The aim of this paper is to present a method of constructing representations of fundamental groups in complex

Anosov flows, surface groups and curves in projective space

Note that in [10], W. Goldman gives a complete description of these connected components in the case of finite covers of PSL(2,R). In the case of PSL(2,R), two homeomorphic components, called

Poisson structure on moduli of flat connections on Riemann surfaces and $r$-matrix

We consider the space of graph connections (lattice gauge fields) which can be endowed with a Poisson structure in terms of a ciliated fat graph. (A ciliated fat graph is a graph with a fixed linear

Parabolic Higgs bundles and Teichm\"uller spaces for punctured surfaces

In this paper we study the relation between parabolic Higgs bundles and irreducible representations of the fundamental group of punctured Riemann surfaces established by Simpson. We generalize a

Explicit construction of characteristic classes

Let E be a vector bundle over an algebraic manifold X. An explicit Iocal construction of characteristic classes cn(E) with values in Bigrassmannian cohomology that are defined in § 1 is given. In the

Lie groups and Lie algebras

From the reviews of the French edition "This is a rich and useful volume. The material it treats has relevance well beyond the theory of Lie groups and algebras, ranging from the geometry of regular

Cluster χ-varieties, amalgamation, and Poisson—Lie groups

In this paper, starting from a split semisimple real Lie group G with trivial center, we define a family of varieties with additional structures. We describe them as cluster χ-varieties, as defined