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Journal of the American Mathematical Society

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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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  • [Ao] Aomoto K.: Analytic structure of Schlafli function, Nagoya Math. J. 68 (1977), 1-16. MR 58:27972
  • [B1] Beilinson A.A.: Height pairings between algebraic cycles, Lecture Notes in Math. N. 1289, (1987), p. 1-26. MR 89h:11027
  • [B2] Beilinson A.A.:Higher regulators and special values of $L$-functions VINITI, 24 (1984), 181-238. English translation: J.Soviet Math., 30 (1985), 2036-2070. MR 86h:11103
  • [BBD] Beilinson A.A., Bernstein J., Deligne P.: Faisceaux pervers, Asterisque 100, 1982. MR 86g:32015
  • [BD] Beilinson A.A., Deligne P.: Motivic polylogarithms and Zagier's conjecture. Preprint to appear.
  • [BD1] Beilinson A.A., Deligne P.: Interprétation motivique de la conjecture de Zagier. Proc. of AMS Research Summer Conference ``Motives'', Symposium in Pure Mathematics, vol 55, part 2. MR 95a:19008
  • [BMS] Beilinson A.A., Macpherson R.D. Schechtman V.V: Notes on motivic cohomology, Duke Math. J., 1987 vol 55 p. 679-710. MR 88f:14021
  • [BGSV] Beilinson A.A., Goncharov A.B., Schechtman V.V, Varchenko A.N.: Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of a pair of triangles in the plane, The Grothendieck Feschtrift, Birkhauser, 1990, p 131-172. MR 92h:19007
  • [Bl1] Bloch S.: Algebraic cycles and higher K-theory, Advances in Math. 61, n 3, (1985), 65-79. MR 88f:18010
  • [Bl2] Bloch S.: Moving lemma for Higher Chow groups, Journal of Algebraic geometry (1995). MR 96c:14007
  • [Bl3] Bloch S.: Applications of the dilogarithm function in Algebraic K - theory and algebraic geometry, Symp. on Alg. Geom., Kyoto, 1977, Kinokuniya Bookstore, Tokyo, 1978. MR 82f:14009
  • [Bl4] Bloch S.: Function theory of polylogarithms, Chapter 12 in ``Structural properties of Polylogarithms", L. Lewin, Editor, Math. Surveys and monographs, vol. 37; AMS, 1991. CMP 92:07
  • [Bo] Böhm J.: Inhaltsmessung in ${\mathbb R}^5$ constanter Krümmung , Arch.Math. 11 (1960), 298-309. MR 22:12448
  • [Bo1] Borel A.: Cohomologie des espaces fibrés principaux, Ann. Math. 57 (1953), 115-207. MR 14:490e
  • [Bo2] Borel A.: Cohomologie de $SL_{n}$ et valeurs de fonctions zêta aux points entiers, Annali Scuola Normale Superiore Pisa 4 (1977), 613-636. MR 58:22016
  • [Ca] Cathelineau J.-L.: $\lambda$-operations structure in algebraic $K$-theory and cyclic homology, $K$-theory 4, N6, 591-606 (1991). MR 92k:19003
  • [Ch] Chern S.-S.: A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math., vol 45, 1944, N 4. MR 6:106a
  • [Du1] Dupont J.: Algebra of polytops and homology of flag complexes, Osaka J. Math. 19 (1982), 599-641. MR 85f:52014
  • [Du2] Dupont J.: The dilogarithm as a characteristic class for a flat bundle, J. Pure and Appl. Algebra, 44 (1987), 137-164. MR 88k:57032
  • [DPS] Dupont J., Parry W., Sah H.: Homology of classical groups made discrete.II.$ H_{2}, H_{3}$, and relations with scissors congruences, J. Pure and Appl. Algebra, 113, 1988, p.215-260. MR 89h:22019
  • [DS1] Dupont J., Sah H.: Scissors congruences II, J. Pure and Appl. Algebra, v. 25, 1982, p. 159-195. MR 84b:53062b; MR 84m:53060
  • [DS2] Dupont J., Sah H.: Homology of Euclidean groups of motion made discrete and Euclidean scissors congruences, Acta Math. 164 (1990) 1-24. MR 91b:22016
  • [EP] Epstein D.B.A., Penner R.: Euclidean decomposition of non-compact hyperbolic manifolds, J.Diff. Geom. 27 (1988), 67-80. MR 89a:57020
  • [FT] Feigin B, Tsygan B.: Additive K-theory, Lect. Notes in Math. 1289. MR 89a:18017
  • [G1] Goncharov A.B.: Polylogarithms and motivic Galois groups, Proc. of AMS Research Summer Conference ``Motives'', Symposium in Pure Mathematics, vol 55, part 2, 43-97. MR 94m:19003
  • [G2] Goncharov A.B.: Geometry of configurations, polylogarithms and motivic cohomology, Advances in Mathematics, vol. 144, N2, 1995. 197-318. MR 96g:19005
  • [G3] Goncharov A.B.: Hyperlogarithms, multiple zeta numbers and mixed Tate motives, Preprint MSRI 058-93, June 1993.
  • [G4] Goncharov A.B.: Mixed elliptic motives. Proceedings of the $60$-th Durham Symposium ``Galois groups in arithmetic and algebraic geometry". (1998).
  • [G5] Goncharov A.B.: Geometry of polylogarithms and regulators, To appear.
  • [G6] Goncharov A.B.: Analytic motivic complexes and characteristic classes, To appear.
  • [Goo] Goodwillie T.G.: Algebraic K-theory and cyclic homology, Annals of Math. 24 (1986) 344-399. MR 88b:18008
  • [H] Hanamura M.: Preprint MPI 1996.
  • [K1] Kellerhals, R.: On the volume of hyperbolic 5-orthoschemes and the trilogarithm , Comm. Math. Helv., 67 (1992), 648-663 MR 93g:52006
  • [K2] Kellerhals, R.: Volumes in hyperbolic 5-space, GAFA, 5 (1995), 640-667. MR 96g:52011
  • [L] Levine, M.: Motivic cohomology and algebraic cycles: a categorical construction, I, Preprint 1993.
  • [ML] MacLane, S.: Homology, Springer-Verlag, 1963. MR 28:122
  • [M1] Milnor J.: Computation of volume, Chapter 7 in [Th].
  • [Mu] Müller P.: Über Simplexinhalte in nichteuklidischen Räumen, Dissertation Universität Bonn, 1954.
  • [NZ] Neumann W., Zagier D.: Volumes of hyperbolic three-manifolds, Topology 24 (1985), 307-331. MR 87j:57008
  • [P] Positselsky L.: Some remarks on triangulated categories, Preprint 1995.
  • [Ra] Raghunatan M.S. :Discrete subgroups of Lie groups, Springer Verlag, 1972.
  • [R] Ramakrishnan D.: Regulators, algebraic cycles and values of $L$-functions, Contemporary Math., vol 83, 183-310. MR 90e:11094
  • [Sah1] Sah H.: Hilbert's third problem: scissors congruences, Research notes in mathematics 33, Pitman Publishing Ltd., San-Francisco-London-Melbourne, 1979. MR 81g:51011
  • [Sah2] Sah H.: Scissor congruences, I: The Gauss-Bonnet map, Math. Scand. 49 (1981), 181-210. MR 84b:53062a; MR 85b:53059
  • [Sah3] Sah H.: Homology of classical groups made discrete. III, J.pure appl. algebra 56 (1989) 269-312. MR 91k:19002
  • [SW] Sah H., Wagoner J.B: Second homology of Lie groups made discrete, Comm. Algebra 5(1977), 611-642. MR 58:31128
  • [Sch] Schläfli: On the multiple integral $\int^{n}$, whose limits are $p_{1} = a_{1}x + b_{1}y + ... + h_{1}z > 0, p_{2} > 0, ... , p_{n} > 0$,and $ x^{2} + y^{2} + ... + z^{2} < 1$, in Gesammelte Mathematische Abhandlungen II, pp. 219-270, Verlag Birkhauser, Basel, 1953. MR 14:833p
  • [S] Suslin A.A.: $K_{3}$ of a field and Bloch's group, Proceedings of the Steklov Institute of Mathematics 1991, Issue 4. MR 91k:19003
  • [Th] Thurston W.:Geometry and topology of 3-manifolds, Princeton University, 1978.
  • [V] Voevodsky V.: Triangulated category of motives over a field, Preprint 1994.
  • [Z1] Zagier D.: Polylogarithms, Dedekind zeta functions and the algebraic $K$-theory of fields. Arithmetic Algebraic Geometry (G.v.d.Geer, F.Oort, J.Steenbrink, eds.), Prog. Math., Vol 89, Birkhauser, Boston, 1991, pp. 391-430. Proceedings of the Texel conference on Arithmetical Algebraic Geometry, 1990. MR 92f:11161
  • [Z2] Zagier D.: The remarkable Dilogarithm, J.Math. and Phys. Soc. 22 (1988), 131-145. MR 89f:33024
  • [W] Wang H.C.: Topics in totally discontinuous groups, in Symmetric spaces, Boothby-Weiss, Editors, New York, 1972.

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Additional Information

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

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