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 $$({\mathbb {H}}^2)^r = {\mathbb {H}}^2 \times \cdots \times {\mathbb {H}}^2$$ ( H 2 ) r = H 2 × ⋯ × 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 $$H_r(M;{\mathbb {Z}})$$ H r ( M ; Z ) .