Regular Tessellation Link Complements

By a regular tessellation, we mean a hyperbolic 3-manifold tessellated by ideal Platonic solids such that the symmetry group acts transitively on oriented flags. A regular tessellation has an invariant that we call the cusp modulus. For small cusp modulus, we classify all regular tessellations. For large cusp modulus, we prove that a regular tessellation has to be of infinite volume if its fundamental group is generated by peripheral curves only. This shows that there are at least 19 and at most 21 link complements that are regular tessellations (computer experiments suggest that at least one of the two remaining cases fails to be a link complement, but so far, we have no proof). In particular, we complete the classification of all principal congruence link complements given by Baker and Reid for the cases of discriminant D = -3 and D = -4. We describe only the manifolds arising as complements of links here, with a future publication Regular Tessellation Links giving explicit pictures of these links.