Tetrahedra of flags, volume and homology of SL(3) (Q463140)

Let \(M\) be a complete hyperbolic 3-manifold. By using tetrahedral decompositions of \(M\), Thurston developed a method to calculate representations \(\pi_1(M) \to \mathrm{PGL}(2, \mathbb{C})\), and the hyperbolic volume of the representation.NEWLINENEWLINEIn the paper under review, the authors develop a method to calculate representations \(\pi_1(M) \to \mathrm{PGL}(3, \mathbb{C})\) by using decorated tetrahedra of flags. Also, the notion of volume of this representation is introduced. This is a nice generalization of the volume of the hyperbolic structure and the Cauchy-Riemann structure of \(M\). This volume can be calculated explicitly and depends only on the boundary data of \(M\).