Hyperbolic surfaces with small diameter
What is the minimal possible diameter of a hyperbolic surface (i.e. with constant curvature equal to -1) of genus $g$? We will prove that it is asymptotic to $\log(g)$. While the lower bound follows from a simple volume growth argument, the upper bound is obtained by considering a model of random hyperbolic surfaces, which we analyse by adapting techniques from the study of random graphs. Based on joint work with Nicolas Curien and Bram Petri.