ABSTRACT.
Two important invariants of hyperbolic three-manifolds are Riemannian volume and Chern-Simons invariant. The complex volume is defined from these as real and imaginary parts. We prove that the complex volume of any hyperbolic three-manifold is the Beilinson regulator value of a mixed Tate motive, and further that in one-cupsed case, under some assumption, this complex volume can be also expressed "directly" in terms of its fundamental group, via the mixed Hodge structure on a suitable fundamental groupoid of the augmented character variety of the given three-manifold.
CMC Algebra Seminar