Geometric cycles, Albert algebras and related cohomology classes for arithmetic groups

Abstract

We discuss the construction of totally geodesic cycles in locally symmetric spaces attached to arithmetic subgroups in algebraic groups $G$ of type $F_4$ which originate with reductive subgroups of the group $G$. 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 is related to the automorphic spectrum of the group, this geometric construction of non-vanishing classes leads to results concerning the existence of specific automorphic forms.