Hyperbolic Volume and Chern-Simons

The first reference known to me is

Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (1982). ZBL0496.57005.

However, I highly recommend various papers by Walter Neumann for much more.

This is a Theorem of Yoshida, the reference is

• Yoshida, Tomoyoshi: ''The η-invariant of hyperbolic 3-manifolds.'' Invent. Math. 81, 473-514 (1985). http://mathlab.snu.ac.kr/~top/articles/Yoshida.pdf

The proof is by explicit computation and comparison of the Chern-Simons form and the volume form.

There is an alternative proof using the Extended Bloch Group, the reference is

• Neumann, Walter D.: ''Extended Bloch group and the Cheeger-Chern-Simons class.'' Geom. Topol. 8, 413-474 (2004). http://arxiv.org/abs/math/0307092

with some more details in

• Goette, Sebastian; Zickert, Christian K.: ''The extended Bloch group and the Cheeger-Chern-Simons class.'' Geom. Topol. 11, 1623-1635 (2007). http://www2.math.umd.edu/~zickert/The%20extended%20Bloch%20group%20and%20the%20Cheeger-Chern-Simons%20class%20(Goette,%20Zickert).pdf