GEOMETRIC CYCLES, ARITHMETIC GROUPS AND THEIR COHOMOLOGY

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 of G of a reductive algebraic group G and its cohomology groups H*(X/Gamma,C) We focus on constructing totally geodesic cycles in X/Gamma which originate with reductive subgroups H c 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