Heegaard splittings and virtually Haken Dehn filling. II. (Q1019587)

This is the second of a series of papers aimed at using Heegaard splittings in order to show that certain manifolds are virtually Haken. The idea is to obtain an incompressible closed surface in a finite cover of the manifold by compressing, along appropriately chosen discs, the lift of a Heegaard surface. The authors focus on a special class of \(3\)-manifolds, i.e. \textsl{knot manifolds with tunnel number \(1\)}: these are connected, compact, orientable \(3\)-manifolds with a unique boundary component which is a torus, and Heegaard genus equal to \(2\). Note that these manifolds are obtained by attaching a unique \(2\)-handle to a genus \(2\) handlebody. Assuming that the knot manifold has first Betti number equal to \(1\), the authors establish a criterion for it not to be fibred. For non fibred knot manifolds with first Betti number equal to \(1\) a condition is given on the attached \(2\)-handle which ensures that in cyclic covers of large enough degree the lift of the Heegaard surface can be compressed to obtain an essential closed surface. Using this latter result the authors also obtain a sufficient condition (in terms of the parameters of the surgery) for a manifold obtained by Dehn filling one of these knot manifolds to be virtually Haken. Finally, the authors verify via computational methods that their condition is fulfilled by all hyperbolic manifolds satisfying their hypotheses in the SnapPea census. This way they obtain bounds on the degree of large cyclic covers which improve other known theoretical bounds. Indeed, it was proved by \textit{B. Freedman} and \textit{M. H. Freedman} [Topology 37, No. 1, 133--147 (1998; Zbl 0896.57012)] that only finitely many cyclic covers of a(n arbitrary) non fibred knot manifold may not be large and a bound on the excluded covers was obtained by \textit{D. Cooper} and \textit{D. D. Long} [J. Differ. Geom. 52, No. 1, 173--187 (1999; Zbl 1025.57020)].