# Dual Teichmuller and lamination spaces

@article{Fock2007DualTA, title={Dual Teichmuller and lamination spaces}, author={Vladimir V. Fock and Alexander B. Goncharov}, journal={arXiv: Differential Geometry}, year={2007}, pages={647-684} }

We survey explicit coordinate descriptions for two (A and X) versions of Teichmuller and lamination spaces for open 2D surfaces, and extend them to the more general set-up of surfaces with distinguished collections of points on the boundary. Main features, such as mapping class group action, Poisson and symplectic structures and others, are described in these terms. The lamination spaces are interpreted as the tropical limits of the Teichmuller ones. Canonical pairings between lamination and… Expand

#### 77 Citations

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#### References

SHOWING 1-10 OF 10 REFERENCES

Dual Teichm\" uller spaces

- Mathematics
- 1997

We describe in elementary geometrical terms Teichm\" uller spaces of decorated and holed surfaces. We construct explicit global coordinates on them as well as on the spaces of measured laminations… Expand

Moduli spaces of local systems and higher Teichmüller theory

- Mathematics
- 2003

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… Expand

Cluster algebras and Weil-Petersson forms

- Mathematics
- 2003

In our previous paper we have discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper we consider the case of… Expand

ON THE RELATION BETWEEN QUANTUM LIOUVILLE THEORY AND THE QUANTIZED TEICHMÜLLER SPACES

- Physics, Mathematics
- 2003

We review both the construction of conformal blocks in quantum Liouville theory and the quantization of Teichmuller spaces as developed by Kashaev, Checkov and Fock. In both cases one assigns to a… Expand

THE DEFORMATION SPACES OF CONVEX RP²-STRUCTURES ON 2-ORBIFOLDS

- Mathematics
- 2001

We determine that the deformation space of convex real projective structures, that is, projectively flat torsion-free connections with the geodesic convexity property on a compact 2-orbifold of… Expand

The Symplectic Nature of Fundamental Groups of Surfaces

- Mathematics
- 1984

Soit #7B-G la categorie des groupes de Lie G avec une forme bilineaire symetrique non singuliere B sur l'algebre de Lie. Soit π le groupe fondamental d'une surface orientee close de genre 1 et soit… Expand

Quantum Teichm\"uller space

- Mathematics
- 1999

We describe explicitly a noncommutative deformation of the *-algebra of functions on the Teichm\"uller space of Riemann surfaces with holes equivariant w.r.t. the mapping class group action.

Cluster algebras: Notes for the CDM-03 conference

- Mathematics
- 2003

This is an expanded version of the notes of our lectures given at the conference Current Developments in Mathematics 2003 held at Harvard University on November 21–22, 2003. We present an overview of… Expand

Cluster algebras I: Foundations

- Mathematics
- 2001

In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.