## Essential surfaces and tameness of covers.(English)Zbl 0959.57002

A $$\pi_1$$-injective map from a closed surface $$S$$ into a closed orientable $$3$$-manifold $$M$$ is called an essential surface. It is conjectured that every irreducible $$M$$ with infinite fundamental group has an essential surface, in fact that $$M$$ has a finite cover containing an imbedded incompressible surface. In this paper, the authors examine two classes of essential surfaces, and develop some properties of $$M$$ that admit such surfaces.
The first class of essential surface is called topologically finite. Their defining criteria are properties known to be enjoyed by quasi-Fuchsian surfaces in hyperbolic $$3$$-manifolds. The authors prove that the covering space $$M_S$$ corresponding to the image of $$\pi_1(S)$$ is topologically tame, i. e. is the product of $$S$$ with the real line. For the case of Haken manifolds, this is a well-known theorem of J. Simon.
The second class of essential surface is called strongly filling. An essential surface $$S$$ is said to have the $$k$$-plane property if when $$S$$ is represented as a least area map, then any collection of $$k$$ planes in the universal cover $$\widetilde{M}$$ of $$M$$ that lie over $$S$$ must contain a pair of disjoint planes. To define strongly filling, one starts by fixing a generating set for $$\pi_1(M)$$. This defines a metric $$d$$ on the set of preimages of the basepoint $$x_0$$ of $$M$$ in $$\widetilde{M}$$, by taking the minimum distance between them in the Cayley graph of $$\pi_1(M)$$. An essential surface is called strongly filling when it has the $$k$$-plane property for some $$k$$, and for any $$n$$ there is a constant $$\alpha$$ so that for any two points $$x_i$$ and $$x_j$$ in the preimage of $$x_0$$ with $$d(x_i,x_j)>\alpha$$, there are at least $$n$$ pairwise disjoint planes in $$\widetilde{M}$$ lying over $$S$$ and separating $$x_i$$ from $$x_j$$.
The authors prove that a strongly filling surface fills $$M$$, which means that the image of every map in its homotopy class meets every homotopically nontrivial loop in $$M$$. As partial converses, a totally geodesic immersed surface that fills a hyperbolic $$3$$-manifold must be strongly filling, and a least area quasi-Fuchsian surface in a closed hyperbolic $$3$$-manifold is strongly filling if and only if every geodesic line in the universal cover has endpoints on both sides of some plane lying over $$S$$. They prove that if $$S$$ is strongly filling, then $$M_S$$ is topologically tame. Finally, if any essential surface in $$M$$ is either topologically finite or strongly filling, then any cover of $$M$$ with finitely generated freely indecomposable fundamental group is the interior of a compact $$3$$-manifold. Elsewhere, they prove that $$M$$ has a strongly filling surface if and only if its fundamental group acts freely on a finite-dimensional cubed complex with a metric of nonpositive curvature.

### MSC:

 57M10 Covering spaces and low-dimensional topology 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M99 General low-dimensional topology
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