Volumes of hyperbolic manifolds and mixed Tate motives

Abstract

Two different constructions of an invariant of an odd-dimensional hyperbolic manifold with values in K 2 n − 1 ( Q ¯ ) ⊗ Q K_{2n-1}(\overline {\mathbb Q})\otimes \mathbb Q are given. We prove that the volume of the manifold equals the value of the Borel regulator on this invariant. The scissors congruence groups in noneuclidean geometries are studied and related to mixed Tate motives and algebraic K-theory of C \mathbb C . We contribute to the general theory of mixed Hodge structures by introducing for Hodge-Tate structures the big period map with values in C ⊗ C ∗ ( n − 2 ) \mathbb C \otimes \mathbb C^*(n-2) .