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Volumes of hyperbolic manifolds and mixed Tate motives. (English) Zbl 0919.11080

Let \(M^n\) be an \(n\)-dimensional hyperbolic manifold of finite volume. For odd \(n\), the author relates the geometry of \(M^n\) to the algebraic \(K\)-theory of fields. In particular, the volume of \(M^n\) is proved to be equal to \(r_n(\gamma(M^n))\), where \(\gamma(M^n)\) is a certain element of \(K_n(\overline\mathbb{Q}^n)\otimes\mathbb{Q}\), and \(r_n:K_n(\mathbb{C})\otimes\mathbb{Q}\to\mathbb{R}\) is the Borel regulator; as usual, \(\overline\mathbb{Q}\) stands for the algebraic closure of \(\mathbb{Q}\) (in \(\mathbb{C})\). Thereby the volume of an odd-dimensional hyperbolic manifold \(M^{2n-1}\) may be conjecturally expressed as a sum of some special values of the classical polylogarithm \(L_n\) (this conjecture has been proved for \(n=1,2)\). Furthermore, the author gives a motivic interpretation of the Dehn invariant, defined as a certain homomorphism of the scissors congruence group generated by the geodesic simplices of \(M^n\). In the course of these investigations, the author constructs an abelian category of mixed Tate motives over a number field, having all the expected properties. According to the author, “this result should have quite a lot of different applications”.
Reviewer: B.Z.Moroz (Bonn)

MSC:

11R70 \(K\)-theory of global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
51M10 Hyperbolic and elliptic geometries (general) and generalizations
51M25 Length, area and volume in real or complex geometry
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