Towards new Mirzakhani-McShane identities

I will present a general recursive machinery which produces continuous functions on the moduli space of bordered Riemann surfaces, from a small amount of initial data. Under mild growth conditions on the initial data, the resulting functions are integrable with respect to the Weil-Petersson metric, and their integration produces functions of boundary lengths which satisfy the topological recursion. Conversely, any initial data for the topological recursion can be refined to initial data for this new machinery, which we call “geometric recursion” (GR). For suitable initial data, GR can produce the constant function 1 (via Mirzakhani-McShane identities), and linear statistics of the hyperbolic length spectra. The latter example can be thought as a new family of Mirzakhani-McShane identities. This is based on joint work with Andersen and Orantin