LINK COMPLEMENTS ARISING FROM ARITHMETIC GROUP-ACTIONS

Let Gamma less than or equal to PSL(2)(2) be a torsion-free subgroup a cting discontinuously on 3-dimensional hyperbolic space H-3 = PSL(2)(C )/SU(2). Assume further that Gamma\H-3 has finite hyperbolic volume. T he quotient-space Gamma\H-3 is then a 3-manifold which can be compacti fied by the addition of finitely many 2-tori. This paper discusses a p rocedure which decides whether Gamma\H-3 is homeomorphic to the comple ment of a link in S-3. We apply our procedure to subgroups of low inde x in PSL(2)(O--7), where O--7 is the ring of integers in Q(root-7). As a result we find new link complements having a complete hyperbolic st ructure coming from an arithmetic group. Finally we prove that up to c onjugacy there are only finitely many commensurability classes of arit hmetic subgroups Gamma less than or equal to PSL(2)(C) so that Gamma\H -3 is homeomorphic to the complement of a link in S-3.