Tetrahedra of flags, volume and homology of $\mathrm{SL}(3)$

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 [Q. J. Math. 62 (2011) 397–415], and Falbel and Wang [Asian J. Math. 17 (2013) 391–422]. 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 [Topology 24 (1985) 307–332], Neumann [Topology ’90 (1992) 243–271] and Kabaya [Topology Appl. 154 (2007) 2656–2671]. Our approach is very related to the work of Fock and Goncharov [Publ. Math. Inst. Hautes Études Sci. 103 (2006) 1–211; Ann. Sci. Éc. Norm. Supér. 42 (2009) 865–930].