Commensurators of Cusped Hyperbolic Manifolds

Abstract

This paper describes a general algorithm for finding the commensuratorof a nonarithmetic hyperbolic manifold with cuspsand for deciding when two such manifolds are commensurable.The method is based on some elementary observations regardinghorosphere packings and canonical cell decompositions. Forexample, we use this to find the commensurators of all nonarithmetichyperbolic once-punctured torus bundles over the circle. For hyperbolic 3-manifolds, the algorithm has been implementedusing Goodman's computer program Snap. We use thisto determine the commensurability classes of all cusped hyperbolic3-manifolds triangulated using at most seven ideal tetrahedra,and for the complements of hyperbolic knots and links withup to twelve crossings.