Counting pairs of saddle connections

TL;DR

This paper establishes that for almost all translation surfaces, the number of saddle connection pairs with bounded cross product grows quadratically with the bound, revealing a universal asymptotic behavior linked to surface area and stratum component.

Contribution

It introduces a method to count pairs of saddle connections with bounded cross product, proving quadratic growth and connecting it to surface geometry and lattice structures.

Findings

Number of saddle connection pairs grows like c R^2 for almost every surface.

The constant c depends only on area and stratum component.

Similar quadratic growth for lattice surfaces with parallel saddle connections.

Abstract

We show that for almost every translation surface the number of pairs of saddle connections with bounded magnitude of the cross product has asymptotic growth like $c R^2$ 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 magnitude of the cross product 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 bounded magnitude of the cross product is equivalent to counting parallel pairs of saddle connections, which also have a quadratic growth of $c R^2$ where $c$ depends in this case on the given lattice surface.