2-Generator arithmetic Kleinian groups III

Abstract

This paper forms part of the program to identify all the 2-generator arithmetic Kleinian groups. Here we identify all conjugacy classes of such groups with one generator parabolic and the other generator elliptic. There are exactly $14$ of these and exactly $5$ Bianchi groups in their commensurability class, namely $\mathrm{PSL}(2,{\mathcal O}_d)$ for $d=1,2,3,7$ and $15$. This complements our earlier identification of the $4$ arithmetic Kleinian groups generated by two parabolic elements.