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Transactions of the American Mathematical Society

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The Farrell-Jones Isomorphism Conjecture for finite covolume hyperbolic actions and the algebraic $K$-theory of Bianchi groups

Authors: E. Berkove, F. T. Farrell, D. Juan-Pineda and K. Pearson
Journal: Trans. Amer. Math. Soc. 352 (2000), 5689-5702
MSC (1991): Primary 19A31, 19B28, 19D35
Published electronically: June 28, 2000
MathSciNet review: 1694279
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Abstract | References | Similar Articles | Additional Information


We prove the Farrell-Jones Isomorphism Conjecture for groups acting properly discontinuously via isometries on (real) hyperbolic $n$-space $\mathbb{H} ^n$ with finite volume orbit space. We then apply this result to show that, for any Bianchi group $\Gamma$, $Wh(\Gamma)$, $\tilde K_0(\mathbb{Z}\Gamma)$, and $K_i(\mathbb{Z}\Gamma)$ vanish for $i\leq -1$.

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  • 1. D. R. Anderson, and W.-C. Hsiang, The functors $K_{-i}$and pseudo-isotopies of polyhedra, Ann. of Math., 105, (1977), 210-223. MR 55:13447
  • 2. H. Bass, Algebraic K-Theory, W.A. Benjamin, Inc., New York, 1968. MR 40:2736
  • 3. W. Ballmann, M. Gromov, and V. Schroeder, Manifolds of Nonpositive Curvature, Birk- häuser, Progress in Mathematics, vol. 61, 1985. MR 87h:53050
  • 4. R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry, Springer-Verlag, Universitext, 1992. MR 94e:57015
  • 5. E. Berkove and D. Juan-Pineda, On the $K$-theory of Bianchi groups, Bol. Soc. Mat. Mexicana, 3 (1996), 15-29. MR 97d:19003
  • 6. K. Brown, Cohomology of Groups, Springer-Verlag Graduate Texts in Mathematics 87, 1982. MR 83k:20002
  • 7. D. Carter, Lower K-theory of finite groups, Comm. in Algebra, 8, (1980), 1927-1937. MR 81m:16027
  • 8. J. F. Davis and W. Lück, Spaces over a category and assembly maps in isomorphism conjectures in $K$- and $L$-Theory, $K$-Theory 15 (1998), 201-252. CMP 99:05
  • 9. F. T. Farrell and W.-C. Hsiang, A formula for $K_1R_{\alpha}[T]$, Proc. Symp. Pure Math., vol. 17, Applications of Categorical Algebra, American Mathematical Society, Providence (1970), 192-218. MR 41:5457
  • 10. F. T. Farrell and L. E. Jones, Isomorphism conjectures in algebraic K-theory, Journal of the American Mathematical Society, 6 (1993), 249-297. MR 93h:57032
  • 11. F. T. Farrell and L. E. Jones, Lower algebraic $K$-theory of virtually infinite cyclic groups, $K$-Theory, 9 (1995), 13-30. MR 90e:19003
  • 12. S. Gersten, On the spectrum of algebraic $K$-theory, Bull. of the AMS, 78, 216-219, 1972. MR 45:8705
  • 13. F. Grunewald and J. Schwermer, Subgroups of Bianchi groups and arithmetic quotients of hyperbolic $3$-space, Trans. Amer. Math. Soc. 335, (1993), 47-77. MR 43c:11024
  • 14. B. Maskit, Kleinian Groups, (Grundlehren der mathematischen Wissenschaften, 287) Springer-Verlag, 1988. MR 90a:30132
  • 15. J. Milnor, Introduction to Algebraic K-Theory, Princeton University Press, Princeton, 1971. MR 50:2304
  • 16. H. J. Munkholm and S. Prassidis, On the vanishing of certain $K$-theory Nil-groups, to appear.
  • 17. R. Oliver, Whitehead Groups of Finite Groups, London Mathematical Society, Lecture Notes Series 132, 1989. MR 89h:18014
  • 18. B. O'Neill, Semi-Riemannian Geometry, Academic Press, 1983. MR 85f:53002
  • 19. K. Pearson, Lower algebraic K-theory of two-dimensional crystallographic groups, to appear in $K$-Theory.
  • 20. E. Pedersen and C. Weibel, A non-connective delooping of algebraic $K$-theory, Topology, Lecture Notes in Math., vol. 1126, Springer-Verlag, Berlin-Heidelberg-New York, (1985), 166-181. MR 87b:18012
  • 21. F. Quinn, Ends of Maps II, Invent. Math., 68, (1982), 353-424. MR 84j:52011
  • 22. M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer-Verlag, Berlin-Heidelberg-New York, 1972. MR 58:22394
  • 23. I. Reiner, Class Groups and Picard Groups of Group Rings and Orders, CBMS Notes #26, American Mathematical Society, Providence, 1976. MR 53:8212
  • 24. P. Scott and T. Wall, Topological methods in group theory, Homological Group Theory, Proceedings held in Durham in September 1977, Ed. C. T. C. Wall, London Mathematical Society Lecture Note Series, 36, 1979, pp. 137-203. MR 81m:57002
  • 25. J. H. C. Whitehead, Simple homotopy types, American Journal of Math., 72, (1950), 1-57. MR 11:735c

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Additional Information

E. Berkove
Affiliation: Department of Mathematics, Lafayette College, Easton, Pennsylvania 18042-1781

F. T. Farrell
Affiliation: Department of Mathematics, Binghamton University, Binghamton, New York 13902

D. Juan-Pineda
Affiliation: Instituto de Matemáticas, UNAM Campus Morelia, Apartado Postal 61-3 (Xangari), Morelia, Michoacán, Mexico 58089

K. Pearson
Affiliation: Department of Mathematics and Computer Science, Valparaiso University, Valpa- raiso, Indiana 46383

Keywords: $K$-theory, discrete groups
Received by editor(s): July 2, 1998
Published electronically: June 28, 2000
Additional Notes: Research partially supported by NSF grant DMS-9701746 (the second author) and a DGAPA-UNAM research grant and CONACyT # 25314E (the third author)
Article copyright: © Copyright 2000 American Mathematical Society