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ISSN 1088-9485(e) ISSN 0273-0979(p)

Geometric cycles, arithmetic groups and their cohomology

Author(s): Joachim Schwermer
Journal: Bull. Amer. Math. Soc. 47 (2010), 187-279.
MSC (2000): Primary 11F75, 22E40; Secondary 11F70, 57R95
Posted: February 2, 2010
MathSciNet review: 2594629
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: 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 and its cohomology groups $H^{\ast}(X/\Gamma, \C)$ . 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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