1. Volumes of (2n − 1)-dimensional hyperbolic manifolds and the Borel regulator on K2n−1(Q). Let M be an n-dimensional hyperbolic manifold with finite volume vol(M). If n = 2m is an even number, then by the Gauss-Bonnet theorem ([Ch]) vol(M) = −c2m · χ(M) where c2m = 1/2×(volume of sphere S of radius 1) and χ(M) is the Euler characteristic of M. This is straightforward for compact manifolds and a bit more delicate for noncompact ones. According to Mostow’s rigidity theorem (see [Th], Ch. 5… CONTINUE READING