Special subgroups of Bianchi groups

Abstract

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 of level 2 or 4 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.