Commensurators of cusped hyperbolic manifolds. (English) Zbl 1338.57016

Summary: 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 hyperbolic knots and links with up to twelve crossings.

MathOverflow Questions:

Quadratic cusp shape


57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)


SnapPea; Orb; Snap
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