Presentations for singly-cusped Bianchi groups

Abstract

We produce a complete list of group presentations for singly-cusped Bianchi groups, PSL2(Od) where Od is the ring of integers for Q(d) and d is −1, −2, −3, −7, −11, −19, −43, −67, or −163. To do this, we apply a theorem due to Macbeath to a sufficiently large horoball, treating the Bianchi groups as discrete subgroups of isometries for H3. As far as we know, explicit presentations were not previously known when d is −43, −67, or −163.