Volumes of hyperbolic manifolds and mixed Tate motives

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) volumes of hyperbolic manifolds are homotopy invariants. For even-dimensional ones this is clear from the formula above. The volumes of odd-dimensional hyperbolic manifolds form a very interesting set of numbers. If the dimension is ≥ 5 it is discrete and, moreover, for any given c ∈ R there is only a finite number of hyperbolic manifolds of volume ≤ c ([W]). Thanks to Jorgensen and Thurston we know that the volumes of hyperbolic 3-folds form a nondiscrete well-odered set of ordinal type ω (see [Th]). We will denote by Q̄ the subfield of all algebraic numbers in C. There is the Borel regulator [Bo2] rm : K2m−1(C)→ R.