Commensurators of Cusped Hyperbolic Manifolds

Abstract

Abstract. 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.

On commensurable hyperbolic Coxeter groups

“…In fact, F is a simply asymptotic orthoscheme [4,4,3] in H 3 whose edge lengths satisfy tanh q 1 q 2 = 1/2 and tanh q 1 q 3 = 1/ √ 2 (see [18]). In a similar way, the glue facet F = F(P 2 ) for P 2 is a simply asymptotic orthoscheme [6,3,3] in H 3 whose corresponding edge lengths satisfy tanh q 1 q 2 = 1/2 and tanh q 1 q 3 = 1/ √ 3. Now, suppose that the groups 1 and 2 are commensurable.…”

“…Geometrically, a fundamental polyhedron for 1 2 is the Coxeter polyhedron arising by glueing together P 1 and P 2 along their common facet F. In this context, the following general result of Karrass and Solitar [17, Theorem 10] will be useful. 6 , (3, ∞, 4)], respectively (see also Fig. 3 and the "Appendix").…”

“…is of finite index in [4,4,3], and the intersection K 2 := K ∩ [6,3,3] is of finite index in [6,3,3] . Hence, the groups K 1 resp.…”

“…It is known that for 1-cusped orbifolds Q = H n /G with discrete commensurator Comm(G), the cusp density is a commensurability invariant (see [6,Section 2]). However, since (transcendental) volume expressions are involved, this property is of limited value [see (2.9) and (2.10), for example].…”

“…3 (for k = 3, l = 4). It turns out to be the free product with amalgamation 3,4 = 3 4 of the two (noncommensurable) arithmetic Coxeter groups l = [3 2,1 , 3 6 , l], l = 3, 4, of infinite covolume (see Fig. 3; cf.…”