Geometric cycles, arithmetic groups and their cohomology
Author:
Joachim Schwermer
Journal:
Bull. Amer. Math. Soc. 47 (2010), 187-279
MSC (2000):
Primary 11F75, 22E40; Secondary 11F70, 57R95
DOI:
https://doi.org/10.1090/S0273-0979-10-01292-9
Published electronically:
February 2, 2010
MathSciNet review:
2594629
Full-text PDF Free Access
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 $G$ 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:
Joachim.Schwermer@univie.ac.at
Keywords:
Arithmetic groups,
geometric cycles,
cohomology,
automorphic forms
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.
Article copyright:
© Copyright 2010
American Mathematical Society