Hamish Short

CMI, Université de Provence,
39 rue Joliot Curie,
13453 Marseille cedex, FRANCE.

Title: The homeomorphism problem for hyperbolic 3 manifolds

Abstract: We consider the homeomorphism problem for closed orientable 3-manifolds: given finite triangulations of two closed orientable 3-manifolds, give an algorithm to decide whether or not the two manifolds are homeomorphic. The problem solved for irreducible manifolds, by combining different results for the different geometries, and using the methods of normal surfaces developped by Jaco, Tollefson and others, and the Haken-Hemion solution for the case of Haken manifolds. To deal with the hyperbolic (irreducible) case, the usual course is to appeal to Sela's proof of the isomorphism problem for hyperbolic groups. In this joint work with Peter Scott, we give a geometric alternative to the use of the Sela's result. Given a triangulation of a hyperbolic 3-manifold, Jason Manning has given an algorithmic construction of a fundamental domain in hyperbolic space for the action of the fundamental group. Careful examination of the geometry of this construction gives bounds on lengths of the images of the generators to be examined when searching for an isomorphism between the groups of two such manifolds, thus providing an algorithmic solution to the problem.