Commensurators of cusped hyperbolic manifolds

TL;DR

This paper introduces a new algorithm to compute the commensurator of non-arithmetic cusped hyperbolic manifolds and determine their commensurability, with practical implementations for various classes of 3-manifolds.

Contribution

It presents a general algorithm based on horosphere packings and cell decompositions, enabling the analysis of commensurability in hyperbolic 3-manifolds.

Findings

Computed commensurators of all non-arithmetic hyperbolic once-punctured torus bundles.

Determined commensurability classes of all cusped hyperbolic 3-manifolds with up to 7 tetrahedra.

Classified hyperbolic knot and link complements with up to 12 crossings.

Abstract

This paper describes a general algorithm for finding the commensurator of a non-arithmetic cusped hyperbolic manifold, 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.