Geometric construction of homology classes in Riemannian manifolds covered by products of hyperbolic planes

We study the homology of Riemannian manifolds of finite volume that are covered by a product of r copies  of the hyperbolic plane. Using a variation of a method developed by Avramidi and Nyguen-Phan, we show that any such manifold  M possesses, up to finite coverings, an arbitrarily large number of compact oriented flat totally geodesic r-dimensional submanifolds whose fundamental classes are linearly independent in the r-th homology group of M.