LINK COMPLEMENTS ARISING FROM ARITHMETIC GROUP ACTIONS

Abstract

Let [Formula: see text] be a torsion-free subgroup acting discontinuously on 3-dimensional hyperbolic space [Formula: see text]. Assume further that Γ\ℍ 3 has finite hyperbolic volume. The quotient-space Γ\ℍ 3 is then a 3-manifold which can be compactified by the addition of finitely many 2-tori. This paper discusses a procedure which decides whether Γ\ℍ 3 is homeomorphic to the complement of a link in S 3 . We apply our procedure to subgroups of low index in [Formula: see text], where [Formula: see text] is the ring of integers in [Formula: see text]. As a result we find new link complements having a complete hyperbolic structure coming from an arithmetic group. Finally we prove that up to conjugacy there are only finitely many commensurability classes of arithmetic subgroups [Formula: see text] so that Γ\ℍ 3 is homeomorphic to the complement of a link in S 3 .