Virtual Haken 3-manifolds and Dehn "lling

A stronger conjecture is that any closed, connected, orientable, irreducible 3-manifold= with in"nite fundamental group has a virtually positive "rst Betti number, i.e.= has a "nite cover which has positive "rst Betti number. Conjecture 0.1 becomes more compelling due to the recent work of Gabai et al. [16]. In fact it follows from [16] (as well as [9, 12, 13, 32]) that if a closed 3-manifold= is virtually Haken, then = is topologically rigid and admits a geometric decomposition in Thurston's sense [32]. In this paper we consider the conjecture through the Dehn "lling construction. Let M be a compact, connected, orientable, irreducible 3-manifold M such that LM is a torus. Recall that a slope on LM is the isotopy class of an unoriented, simple, essential loop in LM. We use *(r 1 , r 2 ) to denote the distance (i.e. the minimal geometric intersection number) between two slopes r 1 and r 2 on LM and use M(r) to denote the closed 3-manifold obtained by Dehn "lling M along LM with slope r. Call a slope r on LM a virtually Haken ,lling slope if M(r) is a virtually Haken 3-manifold. According to Thurston [32], either M is a Seifert "bred manifold, or it contains an incompressible,