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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Geometric cycles, arithmetic groups and their cohomology
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by Joachim Schwermer PDF
Bull. Amer. Math. Soc. 47 (2010), 187-279 Request permission


It is the aim of this article to give a reasonably detailed account of a specific bundle of geometric investigations and results pertaining to arithmetic groups, the geometry of the corresponding locally symmetric space $X/\Gamma$ attached to a given arithmetic subgroup $\Gamma \subset G$ of a reductive algebraic group $G$ and its cohomology groups $H^{\ast }(X/\Gamma , \mathbb C)$. We focus on constructing totally geodesic cycles in $X/\Gamma$ which originate with reductive subgroups $H \subset G$. In many cases, it can be shown that these cycles, to be called geometric cycles, yield non-vanishing (co)homology classes. Since the cohomology of an arithmetic group $\Gamma$ is strongly related to the automorphic spectrum of $\Gamma$, this geometric construction of non-vanishing classes leads to results concerning, for example, the existence of specific automorphic forms.
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Additional Information
  • Joachim Schwermer
  • Affiliation: Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria, and Erwin-Schrödinger International Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria
  • Email:
  • Received by editor(s): September 12, 2008
  • Received by editor(s) in revised form: June 8, 2009
  • Published electronically: February 2, 2010
  • Additional Notes: This work was supported in part by FWF Austrian Science Fund, grant number P 16762-N04.
  • © Copyright 2010 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 47 (2010), 187-279
  • MSC (2000): Primary 11F75, 22E40; Secondary 11F70, 57R95
  • DOI:
  • MathSciNet review: 2594629