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

2014; Mathematical Sciences Publishers; Volume: 18; Issue: 4 Linguagem: Inglês

10.2140/gt.2014.18.1911

1465-3060

Nicolas Bergeron, Elisha Falbel, Antonin Guilloux,

Advanced Algebra and Geometry

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 considered in [4,6].We describe when this volume belongs to the Bloch group and more generally describe a variation formula in terms of boundary data.In doing so, we recover and generalize results of Neumann-Zagier [13], Neumann [11], and Kabaya [10].Our approach is very related to the work of Fock and Goncharov [7, 8].Contents 1. Introduction 1 2. Configurations of flags and cross-ratios 4 3. Tetrahedra of flags and volume 9 4. Decoration of a tetrahedron and the pre-Bloch group 14 5. Decoration of a tetrahedra complex and its holonomy 19 6.Some linear algebra and the unipotent case 26 7. Neumann-Zagier bilinear relations for PGL(3, C) 28 8. Homologies and symplectic forms 36 9. Extension to the general case 38 10.Examples 41 11.