\(\pi_1\)-injective surfaces in graph manifolds (Q1281747)

Let \(f:S\to M\) be a \(\pi_1\)-injective, least area, proper immersion of a compact orientable surface with negative Euler characteristic into a compact orientable irreducible 3-manifold with infinite fundamental group. The authors find an \(f\) as above such that the preimage of \(f(S)\) in the universal cover of \(M\) contains no disjoint pair of planes; thus \(f(S)\) does not have the \(k\)-plane property for any \(k\), and no finite cover \(P:\widetilde M\to M\) of \(M\) contains a finite cover \(\widetilde S\) embedded by \(\widetilde f: \widetilde S\to \widetilde M\) and covering \(f:S\to M\). Other results on graph manifolds are obtained as well.