On the virtual Betti numbers of arithmetic hyperbolic 3--manifolds

An interesting feature of our argument is that although it uses arithmetic in an essential way, it is largely geometric; in particular there is no use of Borel’s theorem [1]. This makes Theorem 1.1 strictly stronger than [1] in this setting, since no congruence assumptions are made. We recall that a group is said to be large if it has a subgroup of finite index which maps onto a free group of rank two. Using entirely different ideas, it is shown in [5] Theorem 6.1, that once an arithmetic hyperbolic 3-manifold has Betti number at least four, it is large, so that taken in conjunction with this result, Theorem 1.1 implies the somewhat stronger: