Virtually Haken surgeries on once-punctured torus bundles (Q937825)

Waldhausen's conjecture states that every hyperbolic three-manifold is virtually Haken, namely it has a finite covering which is Haken. A special case is considered here: the author conjectures that, for a hyperbolic manifold with one cusp, almost all Dehn fillings are virtually Haken. This conjecture is shown to be true for an infinite family of hyperbolic punctured torus bundles over the circle. The family is constructed by considering bundles whose monodromy lies in certain finite index subgroups of the mapping class group. The subgroups are not normal. A complexity for punctured torus bundles over the circle is defined in terms of the complexity of the monodromy. With computer assistance, the author has checked that every manifold with complexity at most five has a cover in this family. According to the author, the proof is inspired by an argument of \textit{D. Cooper} and \textit{G. S. Walsh} [Geom. Topol. 10, 2237--2245 (2006; Zbl 1128.57018)]. Given a punctured torus bundle \(M\) in this family, the author works in a finite covering of \(M\), whose fiber has several boundary components, and such that it contains a nonseparating closed surface which is not a fiber. This allows to find an essential surface in a further finite covering. Since this three manifold covering \(M\) may have three or four boundary components, one has to control that the boundary slopes of the nonseparating surface project to a given slope of \(M\). The author develops techniques for constructing surfaces and computing slopes explicitly.