Geometric cycles, arithmetic groups and their cohomology

Schwermer, Joachim

doi:10.1090/S0273-0979-10-01292-9

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

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 $G$ 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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