The complex volume of SL(n,C)-representations of 3-manifolds

For a compact 3-manifold M with arbitrary (possibly empty) boundary, we give a parameterization of the set of conjugacy classes of boundary-unipotent representations of π1(M) into SL(n,C). Our parameterization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmüller spaces due to Fock and Goncharov. We show that a boundary-unipotent representation determines an element in Neumann’s extended Bloch group Bˆ(C), and we use this to obtain an efficient formula for the Cheeger–Chern–Simons invariant, and, in particular, for the volume. Computations for the census manifolds show that boundary-unipotent representations are abundant, and numerical comparisons with census volumes suggest that the volume of a representation is an integral linear combination of volumes of hyperbolic 3-manifolds. This is in agreement with a conjecture of Walter Neumann, stating that the Bloch group is generated by hyperbolic manifolds.