## $$\pi_1$$-injective surfaces in graph manifolds.(English)Zbl 0916.57001

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.

### MSC:

 57M10 Covering spaces and low-dimensional topology 57N10 Topology of general $$3$$-manifolds (MSC2010)

### Keywords:

graph manifold; immersion; 3-manifold; covering
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