Extended Bloch group and the Cheeger-Chern-Simons class

Abstract

We define an extended Bloch group and show it is naturally isomorphic to H3(PSL(2, C) δ; Z). Using the Rogers dilogarithm function this leads to an exact simplicial formula for the universal Cheeger-Chern-Simons class on this homology group. It also leads to an independent proof of the analytic relationship between volume and Chern-Simons invariant of hyperbolic 3-manifolds conjectured in [13] and proved in [21], as well as an effective formula for the Chern-Simons invariant of a hyperbolic 3-manifold.