Geometric cycles, arithmetic groups and their cohomology

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 / Γ X/\Gamma attached to a given arithmetic subgroup Γ ⊂ G \Gamma \subset G of a reductive algebraic group G G and its cohomology groups H ∗ ( X / Γ , C ) H^{\ast }(X/\Gamma , \mathbb C) . We focus on constructing totally geodesic cycles in X / Γ X/\Gamma which originate with reductive subgroups H ⊂ G 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 Γ \Gamma is strongly related to the automorphic spectrum of Γ \Gamma , this geometric construction of non-vanishing classes leads to results concerning, for example, the existence of specific automorphic forms.