Journal article

Commensurators of Cusped Hyperbolic Manifolds

Oliver Goodman, Damian Heard, Craig Hodgson



This paper describes a general algorithm for finding the commensurator of a nonarithmetic 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 nonarithmetic 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 seven ideal tetrahedra, and for the complements of h..

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University of Melbourne Researchers


Funding Acknowledgements

We thank Ian Agol for pointing out a simplification to Our method of determining the commensurability of Euclidean tori, Gaven Martin for information on current Volume bounds, and Walter Neumann for several interesting discussions on this work. We also thank Alan Reid, Genevieve Walsh, and the referee for their helpful comments on the paper.This work was partially supported by grants from the Australian Research Council.