saddle connections on a translation surface generalize both diagonals in a polygon and primitive vectors in a 2-dimensional lattice. their slopes thus contain geometric and algebraic information about the surface. slopes of saddle connections can be studied using the action of a horocycle subgroup of $\mathrm{sl}_2(\mathbb{r})$ on the moduli space of all translation surfaces. in particular, gaps between slopes are directly related to the return time function of a poincaré section for the horocycle flow.

in this talk, we will describe a poincaré section for horocycle flow in the smallest nontrivial stratum $\mathcal{h}(2)$ and see how to compute the return time function. then we will examine some consequences for gap distributions. this is joint work with anthony sanchez.