No abstract

Viewing Dehn's algorithm as a rewriting system, we generalize to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include finitely generated nilpotent groups, many relatively hyperbolic groups including geometrically finite groups and fundamental groups of certain geometrically decomposable 3-manifolds. The class has several nice closure properties. We also show that if a group has an infinite subgroup and one of exponential growth, and they commute, then it does not admit such an algorithm. We dub these Cannon's algorithms.

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.

The tight span of a finite metric space is essentially the ‘smallest’ path geodesic space into which the metric space embeds isometrically. In this situation, the tight span is also contractible and has a natural cell structure, so that it lends itself naturally to the study of the Cayley graph of a group. As a first step in this study, we consider the tight span of a metric space which arises from the graph metric of an antipodal graph. In particular, we develop techniques for the study of the tight span of a graph, which we then apply to antipodal graphs. In this way, we are able to find the polytopal structure of the tight span for special examples of antipodal graphs. Finally, we present computer generated examples of tight spans which were made possible by the techniques developed in this paper