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We study the homology of Riemannian manifolds of finite volume that arecovered by an $r$-fold product $(\mathbb{H}^2)^r = \mathbb{H}^2 \times \ldots\times \mathbb{H}^2$ of hyperbolic planes. Using a variation of a methoddeveloped by Avramidi and Nguyen-Phan, we show that any such manifold $M$possesses, up to finite coverings, an arbitrarily large number of compactoriented flat totally geodesic $r$-dimensional submanifolds whose fundamentalclasses are linearly independent in the homology group $H_r(M;\mathbb{Z})$.

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