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

Abstract

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 tetrahedral complexes considered in Falbel [6], and Falbel and Wang [8]. 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 and Zagier [18], Neumann [16] and Kabaya [15]. Our approach is very related to the work of Fock and Goncharov [9; 10].