We'll discuss the relations between various
group theoretic properties and properties of finite-sheeted
covers of 3-manifolds. It is a conjecture of Waldhausen
that every aspherical 3-manifold has a finite-sheeted cover
which is Haken. With the geometrization theorem, the
remaining open case is for closed hyperbolic 3-manifolds.
There are various properties of groups: LERF, large, RFRS, virtual betti number, etc. We will discuss how work of Wise and Kahn-Markovic implies the equivalence between these conditions for closed hyperbolic 3-manifold groups and the virtual Haken conjecture, and what further implications these properties have, in particular for the virtual fibering conjecture and the structure of the Thurston norm. We will also consider the prospects of extending these results to aspherical 3-manifolds with non-zero Gromov norm.