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**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.

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.

Reviewer: Darryl McCullough (Norman)