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Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Volumes of hyperbolic manifolds and mixed Tate motives

Author: Alexander Goncharov
Journal: J. Amer. Math. Soc. 12 (1999), 569-618
MSC (1991): Primary 11Gxx; Secondary 19Fxx, 57-XX
MathSciNet review: 1649192
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Abstract: Two different constructions of an invariant of an odd-dimensional hyperbolic manifold with values in $K_{2n-1}(\overline {\mathbb Q})\otimes \mathbb Q$ 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 $\mathbb C$. We contribute to the general theory of mixed Hodge structures by introducing for Hodge-Tate structures the big period map with values in $\mathbb C \otimes \mathbb C^*(n-2)$.

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Alexander Goncharov
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912

Received by editor(s): October 16, 1996
Received by editor(s) in revised form: October 11, 1998
Article copyright: © Copyright 1999 American Mathematical Society