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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 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].

We describe a general geometrical construction of spherical CR structures. We construct then spherical CR structures on the complement of the figure eight knot and the Whitehead link. They have discrete holonomies contained in P U (2, 1, Z[ω]) and P U (2, 1, Z[i]) respectively. These are the same ring of integers appearing in the real hyperbolic geometry of the corresponding links.

Abstract. We show that the figure eight knot complement admits a uniformizable spherical CR structure, i.e. it occurs as the manifold at infinity of a complex hyperbolic orbifold. The uniformization is unique provided we require the peripheral subgroups to have unipotent holonomy.