Random Construction of Riemann Surfaces

In this paper, we address the following question: What does a typical compact Riemann surface of large genus look like geometrically? We do so by constructing compact Riemann surfaces from oriented 3-regular graphs. The set for such Riemann surfaces is dense in the space of all compact Riemann surfaces, namely Belyi surfaces. And in this construction we can control the geometry of the compact Riemann surface by the geometry of the graph. We show that almost all such surfaces have large first eigenvalue and large Cheeger constant.