Tetrahedra of flags, volume and homology of SL(3)

In the paper we define a "volume" for simplicial complexes of flag tetrahedra. This generalizes and unifies the classical volume of hyperbolic manifolds and the volume of CR tetrahedra complexes. We describe when this volume belongs to the Bloch group. In doing so, we recover and generalize results of Neumann-Zagier, Neumann, and Kabaya. Our approach is very related to the work of Fock and Goncharov.Comment: 45 pages, 14 figures. The first version of the paper contained a mistake which is correct here. Hopefully the relation between the works of Neumann-Zagier on one side and Fock-Goncharov on the other side is now much cleare

Infinitesimal Liouville currents, cross-ratios and intersection numbers

Many classical objects on a surface S can be interpreted as cross-ratio functions on the circle at infinity of the universal covering. This includes closed curves considered up to homotopy, metrics of negative curvature considered up to isotopy and, in the case of interest here, tangent vectors to the Teichm\"uller space of complex structures on S. When two cross-ratio functions are sufficiently regular, they have a geometric intersection number, which generalizes the intersection number of two closed curves. In the case of the cross-ratio functions associated to tangent vectors to the Teichm\"uller space, we show that two such cross-ratio functions have a well-defined geometric intersection number, and that this intersection number is equal to the Weil-Petersson scalar product of the corresponding vectors.Comment: 17 page

Hitchin characters and geodesic laminations

For a closed surface S, the Hitchin component Hit_n(S) is a preferred component of the character variety consisting of group homomorphisms from the fundamental group pi_1(S) to the Lie group PSL_n(R). We construct a parametrization of the Hitchin component that is well-adapted to a maximal geodesic lamination on the surface. This is a natural extension of Thurston's parametrization of the Teichmueller space of S by shear coordinates associated to a maximal geodesic lamination, corresponding to the case n=2. However, significantly new ideas are needed in this higher dimensional case. The article concludes with a few applications.Comment: 67 pages, 9 figures. Version 2: Minor polish (misprints, etc.) prior to submissio