Commensurators of Cusped Hyperbolic Manifolds

This paper describes a general algorithm for finding the commensurator of a non-arithmetic hyperbolic manifold with cusps, and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding horosphere packings and canonical cell decompositions. For example, we use this to find the commensurators of all non-arithmetic hyperbolic once-punctured torus bundles over the circle. For hyperbolic 3-manifolds, the algorithm has been implemented using Goodman’s computer program Snap. We use this to determine the commensurability classes of all cusped hyperbolic 3-manifolds triangulated using at most 7 ideal tetrahedra, and for the complements of hyperbolic knots and links with up to 12 crossings.