Bloch invariants of hyperbolic 3-manifolds

Abstract

We define an invariant β(M) of a finite volume hyperbolic 3-manifold M in the Bloch group B(C) and show it is determined by the simplex parameters of any degree one ideal triangulation of M. β(M) lies in a subgroup of \B(\C) of finite \Q-rank determined by the invariant trace field of M. Moreover, the Chern-Simons invariant of M is determined modulo rationals by β(M). This leads to a simplicial formula and rationality results for the Chern Simons invariant which appear elsewhere. Generalizations of β(M) are also described, as well as several interesting examples. An appendix describes a scissors congruence interpretation of B(C).

25 pages. A slightly revised version will appear in Duke Math. J.