Algebra/Topology Seminar

Speaker: Viveka Erlandsson (University of Bristol)

Title: Counting immersed graphs and curves bounding subsurfaces in hyperbolic surfaces

Abstract: Counting closed curves (geodesics) in a closed hyperbolic surfaces in a classical problem, and the first asymptotic result is due to Huber who showed that the number of closed curves of length at most L is asymptotic to e^L/L as L goes to infinity. This has since been generalized in many directions, both to other manifolds as well as to counting certain subsets of curves. Mirzakhani obtained the asymptotic growth of simple closed curves, giving a polynomial asymptotic growth (with exponent being a topological invariant). In fact, she obtained the growth rate for each topological type of simple curve, for example, those that bound a (embedded) subsurface of given genus. In this talk we will study the growth rate of those (non-simple) curves bounding immersed subsurfaces of given genus, or, closely related, those with given commutator length. This boils down to counting certain immersed graphs in the surface. Time allowing I will give some applications to the graph counting, such as a simple geometric proof of Huber’s theorem or properties of “random" surface subgroups and covers (due to Sophie Wright). This is joint work with Juan Souto.