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 an r-fold product ($\mathbb{H}$$^{2}$)$^{r}$=$\mathbb{H}$$^{2}$×⋯×$\mathbb{H}$$^{2}$ of hyperbolic planes. Using a variation of a method developed by Avramidi and Nguyen-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 homology group H$_{r}$(M;$\mathbb{Z}$).