Extended Bloch group and the Cheeger–Chern–Simons class

Abstract

We define an extended Bloch group and show it is naturally isomorphic to $H_3\left(PSL{\left(2,ℂ\right)}^{\delta };\mathbb{Z}\right)$. 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 by Neumann and Zagier and proved by Yoshida, as well as effective formulae for the Chern–Simons invariant of a hyperbolic $3$–manifold.