  This is an exposition of results on the existence problem of
$\pi_1$-injective immersed and embedded surfaces in graph-manifolds, and also
of nonpositively curved metrics on graph-manifolds, obtained by different
authors. The results are represented from a unified point of view based on the
notion of compatible cohomological classes and some difference equation on the
graph of a graph-manifold (BKN-equation). Criteria for seven different
properties of graph-manifolds at three levels are given: at the level of
compatible cohomological classes; at the level of solutions to the
BKN-equation; in terms of spectral properties of operator invariants of a
graph-manifold.
