Bram Petri

Random regular graphs and the systole of a random surface: RANDOM REGULAR GRAPHS AND THE SYSTOLE OF A RANDOM SURFACE

We study the systole of a random surface, where by a random surface we mean a surface constructed by randomly gluing together an even number of triangles. We study two types of metrics on these surfaces, the first one coming from using ideal hyperbolic triangles and the second one using triangles that carry a given Riemannian metric. In the hyperbolic case we compute the limit of the expected value of the systole when the number of triangles goes to infinity (approximately 2.484). We also determine the asymptotic probability distribution of the number of curves of any finite length. This turns out to be a Poisson distribution. In the Riemannian case we give an upper bound to the limit supremum and a lower bound to the limit infimum of the expected value of the systole depending only on the metric on the triangle. We also show that this upper bound is sharp in the sense that there is a sequence of metrics for which the limit infimum comes arbitrarily close to the upper bound. The main tool we use is random regular graphs. One of the difficulties in the proof of the limits is controlling the probability that short closed curves are separating. To do this we first prove that the probability that a random cubic graph has a short separating circuit tends to 0 for the number of vertices going to infinity and show that this holds for circuits of a length up to

Mapping class group orbits of curves with self-intersections

\log_2

The Genus of Curve, Pants and Flip Graphs

of the number of vertices.

Finite length spectra of random surfaces and their dependence on genus

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity.We also consider the minimal genus of a subsurface that contains the curve. We determine the asymptotic number of orbits of curves with a fixed minimal genus and a bounded self-intersection number, as the complexity of the surface tends to infinity.As a corollary of our methods, we obtain that most curves that are homotopic are also isotopic. Furthermore, using a theorem by Basmajian, we get a bound on the number of mapping class group orbits on a given hyperbolic surface that can contain short curves. For a fixed length, this bound is polynomial in the signature of the surface.The arguments we use are based on counting embeddings of ribbon graphs.

Poisson approximation of the length spectrum of random surfaces

This article is about the graph genus of certain well studied graphs in surface theory: the curve, pants and flip graphs. We study both the genus of these graphs and the genus of their quotients by the mapping class group. The full graphs, except for in some low complexity cases, all have infinite genus. The curve graph once quotiented by the mapping class group has the genus of a complete graph so its genus is well known by a theorem of Ringel and Youngs. For the other two graphs we are able to identify the precise growth rate of the graph genus in terms of the genus of the underlying surface. The lower bounds are shown using probabilistic methods.

Subgroup growth of virtually cyclic right-angled Coxeter groups and their free products.

The main goal of this article is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric. Given suitable restrictions on the genus of the surface, we consider the number of appearances of fixed finite sets of combinatorial types of curves. Of any such set we determine the asymptotics of the probability distribution. It turns out that these distributions are independent of the genus in an appropriate sense. As an application of our results we study the probability distribution of the systole of random surfaces in a hyperbolic and a more general Riemannian setting. In the hyperbolic setting we are able to determine the limit of the probability distribution for the number of triangles tending to infinity and in the Riemannian setting we derive bounds.