Virtual Haken 3-manifolds and Dehn filling
Let \(M\) be a compact, connected, orientable irreducible \(3\)-manifold whose boundary is a torus. A slope on \(\partial M\) is the isotopy class of an oriented, simple, essential loop in \(\partial M\) and gives rise to a closed \(3\)-manifold \(M(r)\) obtained by Dehn filling \(M\) with a solid torus so that a meridian of the torus has slope \(r\). Suppose \(M\) is hyperbolic in the sense that the interior of \(M\) admits a complete hyperbolic metric of finite volume. The distance between two slopes is their minimal geometric intersection number on \(\partial M\). The authors address the question, when is \(M(r)\) virtually Haken, i.e., finitely covered by a Haken manifold? The main results give conditions on one Dehn filling \(M(r_0)\) such that any other Dehn filling \(M(r)\) with the distance between \(r\) and \(r_0\) sufficiently large is virtually Haken. The virtual \(\mathbb Z\)-representability and virtual \(\mathbb Z\ast\mathbb Z\)-representability of \(\pi_1(M(r))\) is also discussed. Examples are given of \(M\) for which all but finitely many Dehn fillings are virtually Haken but not Haken.
scientific article; zbMATH DE number 1356313