Covering Spaces of Arithmetic 3-orbifolds

This paper investigates properties of finite sheeted covering spaces of arithmetic hyperbolic 3-orbifolds (see §2). The main motivation is a central unresolved question in the theory of closed hyperbolic 3-manifolds; namely whether a closed hyperbolic 3-manifold is virtually Haken. Various strengthenings of this have also been widely studied. Of specific to interest to us is the question of whether the fundamental group of a given hyperbolic 3-manifold M is large; that is to say, some finite index subgroup of π1(M) admits a surjective homomorphism onto a non-abelian free group. This implies that M is virtually Haken, and indeed that M has infinite virtual first Betti number (see §2.4 for a definition). Of course, a weaker formulation is to only ask whether the virtual first Betti number of a closed hyperbolic 3-manifold M is positive. This has been verified in many cases, see [8] for some recent work on this. However, in general, passage from positive virtual first Betti number to infinite virtual first Betti number is difficult, as is passage from infinite virtual first Betti number to large. This paper makes some progress on the latter in certain settings.