The complexity of 3-manifolds and groups

A well-known aspect of group theory is that many natural questions that one might ask about a finitely presented group are algorithmically unsolvable. For example, it is known that there is no algorithm that can determine whether or not a group given by a finite presentation is trivial. This phenomenon extends into the theory of high dimensional manifolds. The homeomorphism problem for closed 4-manifolds is unsolvable, for example, and the proof of this fact uses solvability results from group theory. However, it is a recent development in the field that most reasonable questions about 3-manifolds have been shown to be solvable. What is often not clear, however, is the computational complexity of these problems. A notable case of this is the homeomorphism problem for compact 3-manifolds. There is an algorithm to solve this but the known upper bounds on its running time are huge, and the known lower bounds on the complexity of this problem are very weak indeed. To put this in a historical perspective, the Poincaré conjecture is a statement about the homeomorphism problem for closed, simply-connected 3-manifolds, a much smaller class, and it took over a century to prove (it was solved only 15 years ago).

The research project aims to investigate complexity questions about 3-manifolds and about groups. A specific problem about 3-manifolds that will be considered is whether one can find a good upper bound on the number of Pachner moves needed to pass from one triangulation of a hyperbolic 3-manifold to another. In group theory, particularly the study of groups that arise in low dimensional topology and geometry, there are still many challenges that ask if various natural questions are solvable at all. As mentioned above, general groups are completely intractable. But once one focuses on various geometrically defined groups, there is more cause for optimism. A specific problem that will be considered in this area is whether it is possible to determine whether a CAT(0) cube complex is virtually special. Another, more classical challenge, asks if there is an algorithm that can determine whether or not a group with a balanced presentation (the same number of generators as relations) is trivial; this is equivalent to asking for an algorithm that can determine whether or not a finite simplicial 2-complex is contractible.

There are more connections between the 3-manifold and group theoretic projects than is at first evident. The use of CAT(0) cube complexes has revolutionised the study of finite covers of 3-manifolds, and it is possible that they may have applications to the complexity of the homeomorphism problem. However, it is more likely that concrete geometric methods developed by Kuperberg will be most applicable.

This project falls within the EPSRC's 'Mathematical Sciences' research area. The EPSRC describes that one aim of its funding in this area is to 'support the health of the discipline'. The striking developments in both geometric group theory and the study of 3-manifolds over the past two decades have had a major influence on mathematics more widely. Thus this project forms part of a healthy and thriving area of research.