Counting Pairs of Saddle Connections (virtual)

We show that for almost every translation surface the number of pairs of saddle connections with bounded virtual area has asymptotic growth like cR^2 where the constant c depends only on the area and the connected component of the stratum. The proof techniques combine classical results for counting saddle connections with the crucial result that the Siegel-Veech transform is in L^2. In order to capture information about pairs of saddle connections, we consider pairs with bounded virtual area since the set of such pairs can be approximated by a fibered set which is equivariant under geodesic flow. In the case of lattice surfaces, small virtual area is equivalent to counting parallel pairs of saddle connections, which also have a quadratic growth of cR^2 where c depends in this case on the given lattice surface. This is joint work with Samantha Fairchild and Howard Masur.