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Journal of the American Mathematical Society

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

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Volumes of hyperbolic manifolds and mixed Tate motives
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by Alexander Goncharov
J. Amer. Math. Soc. 12 (1999), 569-618


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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Bibliographic Information
  • Alexander Goncharov
  • Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
  • Email:
  • Received by editor(s): October 16, 1996
  • Received by editor(s) in revised form: October 11, 1998
  • © Copyright 1999 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 12 (1999), 569-618
  • MSC (1991): Primary 11Gxx; Secondary 19Fxx, 57-XX
  • DOI:
  • MathSciNet review: 1649192