Let d be a square-free positive integer and let O (= Od) be the ring of integers of the imaginary quadratic number field ℚ(-d). The groups PSL2(O) are called the Bianchi groups after Luigi Bianchi who made the first important contribution 1 to their study in 1892. Since then they have attracted considerable attention particularly during the last thirty years. Their importance stems primarily from their action as discrete groups of isometries on hyperbolic 3-space, H3. As a consequence they play an important role in hyperbolic geometry, low-dimensional topology together with the theory of discontinuous groups and automorphic forms. In addition they are of particular significance in the class of linear groups over Dedekind rings of arithmetic type. Serre9 has proved that in this class the Bianchi groups (along with, for example, the modular group, PSL2(z), where z is the ring of rational integers) have an exceptionally complicated (non-congruence) subgroup structure.
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