Grant: Discovery Grants Program - Individual

My research efforts for the next five years will focus on a series of related projects concerned with the topology of 3-dimensional manifolds. Many of the basic problems in this field can be analysed in terms of the Dehn filling operation. One of the main goals of my proposal is to contribute to our understanding of this operation and to apply this work to study open problems such as the exceptional filling conjecture and the generalised knot complement conjecture. Another one of my goals is to investigate the qualitative behaviour of infinite families of non-zero degree maps between 3-dimensional manifolds. In particular, I am interested in topologising the situation so as to determine precompactness conditions for such families and to understand the topological implications of divergence. A major line of contemporary research in 3-manifold theory concerns their virtual properties. These are properties which hold up to taking a finite degree cover. I will investigate various problems of this nature including the virtual Haken conjecture, the virtual fibring conjecture, and the virtual bi-orderability conjecture. Much of my work in 3-manifold topology depends on results regarding the geometric representation theory of their fundamental groups. I intend to continue working in this area, which will be useful not only for studying the problems discussed in this proposal, but also for opening up new areas of application as well. Finally, I will investigate various questions concerning 3-dimensional Poincaré duality groups (PD(3) groups) such as whether every finitely presented PD(3) group is the fundamental group of a 3-manifold. In particular I am interested in the linearity of such groups.