Curves on Surfaces

Let S be a compact surface of negative Euler characteristic and Mod(S) its mapping class group. We say that two (closed) curves are of the same type if they lie in the same Mod(S)-orbit. If we equip S with a hyperbolic structure, we can study the number of geodesic curves of each type whose length is bounded by some number L. Due to a celebrated result by Mirzakhani, we know the precise asymptotics of this number, as L grows. This result has been generalised in various directions, including showing that the asymptotics also hold for other metrics on S. In this project we will study further directions, such as studying the orbits of curves under subgroups of Mod(S).