Quantifying virtual properties of Bianchi groups

This dissertation is concerned with quantifying virtual properties of hyperbolic 3-manifold groups. We determine C-special subgroups of the Bianchi groups with index bounded above by 120 by effectivising the arguments of Agol-Long-Reid. These subgroups are congruence subgroups of small level and retract to the free group on two generators. As a consequence, we find a C-special 20-sheeted cover of the figure-eight knot complement. We also determine C-special congruence subgroups for a family of cocompact arithmetic Kleinian groups and a family of non-cocompact arithmetic groups of hyperbolic dimension 4.