Bianchi Orbifolds of Small Discriminant

Let OD be the ring of integers in the imaginary quadratic field Q( √ D) of discriminant D < 0. Then PGL2(OD) is a discrete subgroup of the isometry group PSL2(C) (= PGL2(C)) of hyperbolic 3-space H3 . The quotient space H/PGL2(OD) = XD is topologically a noncompact 3-manifold whose cusps (ends) are of the form S × [0,∞). The number of cusps of XD is known to be hD , the class number of OD . So XD is a closed manifold X̂D with hD points removed. For small D , including the 31 discriminants in the range D > −100, R. Riley has done computer calculations of the Ford fundamental domain FD for the action of PGL2(OD) on H3 . (See [5] for an account of the techniques; for about half of these D ’s, Bianchi had calculated the fundamental domains — by hand, presumably — almost a century ago, in [2],[3].) Riley’s computer output includes how the faces of FD are identified by elements of PGL2(OD). So it becomes a pleasant exercise in geometric visualization to try to recognize the manifold X̂D . The results of carrying out this exercise for D > −100 are listed in Table I. (P 3 denotes real projective 3-space, ] is connected sum.)