  Let G be an arithmetic Kleinian group, and let O be the associated hyperbolic
3-orbifold or 3-manifold. In this paper, we prove that, in many cases, G is
large, which means that some finite index subgroup admits a surjective
homomorphism onto a non-abelian free group. This has many consequences,
including that O has infinite virtual first Betti number and G has
super-exponential subgroup growth.
  Our first result, which forms the basis for the entire paper, is that G is
commensurable with a lattice containing the Klein group of order 4. Our second
result is that if G has a finite index subgroup with first Betti number at
least 4, then G is large. This is known to hold for many arithmetic lattices.
In particular, we show that it is always the case when G contains A_4, S_4 or
A_5. Our third main result is that the Lubotzky-Sarnak conjecture and the
geometrisation conjecture together imply that any arithmetic Kleinian group G
is large. We also give a new 'elementary' proof that arithmetic Kleinian groups
do not have the congruence subgroup property, which avoids the use of the
Golod-Shafarevich inequality.
