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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
DOI: https://doi.org/10.1090/S0002-9947-00-02529-0
Published electronically: June 28, 2000
MathSciNet review: 1694279
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Abstract | References | Similar Articles | Additional Information

Abstract:

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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Additional Information

E. Berkove
Affiliation: Department of Mathematics, Lafayette College, Easton, Pennsylvania 18042-1781
Email: berkovee@lafayette.edu

F. T. Farrell
Affiliation: Department of Mathematics, Binghamton University, Binghamton, New York 13902
Email: farrell@math.binghamton.edu

D. Juan-Pineda
Affiliation: Instituto de Matemáticas, UNAM Campus Morelia, Apartado Postal 61-3 (Xangari), Morelia, Michoacán, Mexico 58089
Email: djuan@zeus.ccu.umich.mx

K. Pearson
Affiliation: Department of Mathematics and Computer Science, Valparaiso University, Valpa- raiso, Indiana 46383
Email: kimberly.pearson@valpo.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02529-0
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

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