Abstract

Random covering spaces of a wedge sum of circles give us random regular graphs, a much-studied class of objects. For instance, a theorem of Friedman tells us that with high probability they are near-optimal expander graphs.

I’m going to talk about the extension of this fact to covering spaces of closed surfaces. The notion of `expander’ surface can be either combinatorial or Riemannian. In the Riemannian case we rely on a vast strengthening of near-optimal expander graphs known as `strong convergence’.

The launching off point is a study of the statistics of lifting curves from closed surfaces to uniformly randomly chosen degree-n covering spaces, and the fact that certain functions appearing here are `almost rational’ functions of the degree of the covering spaces.

This is combined with recent technology (the `polynomial method’) for establishing expansion properties of random graphs, and an interesting (new) fact about random walks on surface groups.

Based on joint works with Hide, Puder, de la Salle, and van Handel.