Lefschetz properties for arithmetic real and complex hyperbolic manifolds

We state new conjectures, of “Lefschetz type,” on the relationships between the (co)homology of arithmetic real and complex hyperbolic manifolds and the (co)homology of their totally geodesic subspaces. We prove particular cases of these conjectures by relating them to conjectures jointly made with Clozel on the automorphic spectrum of arithmetic real and complex hyperbolic manifolds, which appear to be particular cases of Arthur's conjectures of which some new approximations were obtained by Bergeron and Clozel.