Geometry and Topology Seminar

Peter Scott's famous result states that the fundamental groups of hyperbolic surfaces are subgroup separable, which has many non-trivial consequences. For example, given any closed curve on such a surface, potentially with many self-intersections, there is always a finite cover to which the curve lifts to an embedding (i.e. lifts simply). It was shown recently that hyperbolic 3-manifold groups share this separability property, and this was a key tool in Ian Agol's resolution to the Virtual Haken and Virtual Fibering conjectures for hyperbolic 3-manifolds. I will begin this talk by giving some background on separability properties of groups, hyperbolic manifolds, and these two conjectures. There are also a number of interesting quantitative questions that naturally arise in the context of these topics. For example: given a closed curve on a surface, what is the minimal degree of a cover in which the curve lifts simply? This question fits into a recent trend in low-dimensional topology aimed at providing concrete topological and geometric information about hyperbolic manifolds that often cannot be gathered from existence results alone. After discussing various results partially answering this question for hyperbolic surfaces, I will focus on recent joint work with T. Aougab, J. Gaster, and J. Sapir in which we build a hyperbolic metric for a closed curve on a surface that yields an interesting relationship between its length and self-intersection number. I will then highlight how such a relationship can help answer the original question about lifting curves simply in finite covers.