Immersed essential surfaces in hyperbolic 3-manifolds (Q1866512)

Many of the central questions regarding 3-manifolds are known for the class of Haken manifolds (e.g. results of Waldhausen, Haken and Thurston). A 3-manifold is Haken if it is irreducible and contains an embedded \(\pi_1\)-injective surface. Since most questions of interest have been reduced to irreducible manifolds, this assumption is not in general problematic; however, there are known constructions, which are quite general, of manifolds not containing a \(\pi_1\)-injective surface. It is therefore of great interest to find structures on irreducible-3-manifolds that generalize embedded \(\pi_1\)-injective surfaces. The current paper is concerned with one such possibility: \(\pi_1\)-injective surfaces (but not necessarily embedded). \textit{B. Freedman} and \textit{M. H. Freedman} [Topology 37, 133-147 (1998; Zbl 0896.57012)] constructed immersed \(\pi_1\)-injective surfaces in a manifold with boundary torus (say \(X)\) by first considering a \(\pi_1\)-injective surface with boundary (say \(F)\). They proved that either \(F\) is a fiber in a fibration over \(S^1\) or, after tubing \(F\) to itself along an annulus that runs along the boundary many times, one obtains a closed surface that compresses down to a \(\pi_1\)-injective surface. According to \textit{M. Culler} and \textit{P. B. Shalen} [Invest. Math. 75, 537-545 (1984; Zbl 0542.57011)] (see also [\textit{M. Culler}, \textit{Mc A. Gordon}, \textit{J. Luecke} and \textit{P. B. Shalen}, Ann. Math. (2) 127, 663 (1988; Zbl 0645.57008)]) any manifold \(X\) contains bounded \(\pi_1\)-injective surfaces that are not fibers over \(S^1\) for at least 2 different boundary slopes. Their construction was used and improved in many papers, including the paper under review. Given a manifold with boundary torus one can obtain a closed manifold by attaching a solid torus to the boundary; this is called Dehn filling. It is well known that whenever \(X\) is hyperbolic infinitely many distinct manifolds are obtained. If, in addition, \(X\) contains no closed, non boundary parallel, \(\pi_1\)-injective surface then \textit{A. E. Hatcher} [Pac. J. Math. 99, 373-377 (1982; Zbl 0502.57005)] proved that only finitely many of the manifolds obtained by filling \(X\) are Haken. By contrast, the paper under review proves that for any manifold \(X\), all but finitely many manifolds obtained by Dehn filling \(X\) contain a \(\pi_1\)-injective surface. This result was also obtained by \textit{D. Cooper} and \textit{D. D. Long} [Geom. Topol. 5, 347-367 (2001; Zbl 1009.57017)] but the techniques used there are completely different. The paper under review gives informations about the slopes for which the existence of a \(\pi_1\)-injective surface is guaranteed: they are all the slopes of sufficiently large distance (in the sense of homological intersection number) from a boundary of a bounded surface \(F\) used in the Freedman-Freedman construction. The distance required is a linear function of the genus and number of punctures of \(F\).