Heegaard splittings and virtually Haken Dehn filling (Q1774335)

In the paper under review, examples of virtually Haken \(3\)-manifolds are given, and they are obtained by Dehn surgery on two special families of hyperbolic \(2\)-bridge knots. The idea of the construction is as follows. Take a cyclic cover of the knot exterior and take the lift to the cover of the genus two Heegaard splitting of the knot exterior (note that non trivial \(2\)-bridge knots have tunnel number \(1\)). For a specific choice of the order of the covers, the obtained Heegaard splitting is weakly reducible, and the authors show that the Heegaard surface can be compressed to an essential surface (with some additional properties). Using a result of \textit{M. Culler, C. Gordon, J. Luecke} and \textit{P. Shalen} [Ann. Math. (2) 125, 237-300 (1987; Zbl 0633.57006)] they prove that there are infinitely many Dehn fillings on the cover which are Haken. It suffices then to consider the manifolds obtained as Dehn fillings of the knot exterior and which admit the manifolds just constructed as covers. Indeed, using again a result of [loc. cit.], one has that only a finite number of Dehn surgeries on these knot exteriors can be Haken, so that most of the manifolds constructed are virtually Haken but not Haken. Note that \textit{D. Cooper} and \textit{D. Long} [J. Differ. Geom. 52, No.1, 173-187 (1999; Zbl 1025.57020)] proved that, under certain assumptions, the Dehn-fillings of a non-fibred, atoroidal, Haken \(3\)-manifold with torus boundary are virtually Haken. Although the knot exteriors considered in the paper under review are non-fibred, the Dehn fillings providing virtually Haken manifolds do not always satisfy the aforementioned assumptions.