233 research outputs found
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
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