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Cohomology of buildings and finiteness properties of $\widetilde{A}_n$-groups


Authors: Jacqui Ramagge and Wayne W. Wheeler
Journal: Trans. Amer. Math. Soc. 354 (2002), 47-61
MSC (2000): Primary 13D25, 20J06
DOI: https://doi.org/10.1090/S0002-9947-01-02818-5
Published electronically: August 21, 2001
MathSciNet review: 1859024
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Abstract: Borel and Serre calculated the cohomology of the building associated to a reductive group and used the result to deduce that torsion-free $S$-arithmetic groups are duality groups. By replacing their group-theoretic arguments with proofs relying only upon the geometry of buildings, we show that Borel and Serre's approach can be modified to calculate the cohomology of any locally finite affine building. As an application we show that any finitely presented $\widetilde{A}_n$-group is a virtual duality group. A number of other finiteness conditions for $\widetilde{A}_n$-groups are also established.


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Additional Information

Jacqui Ramagge
Affiliation: Department of Mathematics, University of Newcastle, NSW 2308 Callaghan, Australia
Email: jacqui@maths.newcastle.edu.au

Wayne W. Wheeler
Affiliation: Center for Communications Research, 4320 Westerra Court, San Diego, California 92117
Email: wheeler@member.ams.org

DOI: https://doi.org/10.1090/S0002-9947-01-02818-5
Received by editor(s): March 29, 2000
Published electronically: August 21, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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