Special subgroups of Bianchi groups

TL;DR

This paper classifies certain special subgroups of Bianchi groups and related Kleinian groups, providing explicit examples of covers of the figure-eight knot complement and analyzing their properties.

Contribution

It explicitly determines C-special subgroups of Bianchi groups with bounded index and constructs a 20-sheeted cover of the figure-eight knot complement.

Findings

C-special subgroups are congruence of level 2 or 4

Constructed a 20-sheeted cover of the figure-eight knot complement

Identified C-special congruence subgroups for specific Kleinian groups

Abstract

We determine C-special subgroups of the Bianchi groups of 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.