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 PSL(2, θd) for d = 1, 2, 3, 7 and 15. This complements our earlier identification of the 4 arithmetic Kleinian groups generated by two parabolic elements.