Counting Saddle Connections on Hyperelliptic Translation Surfaces with a Slit

TL;DR

This paper studies the growth rate of certain saddle connections on hyperelliptic translation surfaces, establishing bounds that depend on the surface's stratum dimension, with implications for understanding geometric structures.

Contribution

It provides the first precise growth rate bounds for saddle connections avoiding a distinguished saddle, using horocycle renormalization techniques.

Findings

Upper bound holds for all surfaces in the hyperelliptic stratum.

Lower bound holds for almost every surface in the stratum.

Growth rate is proportional to L(log L)^{d-2} where d is the stratum's complex dimension.

Abstract

We consider saddle connections on a translation surface in a hyperelliptic connected component of a stratum that do not intersect the interior of a distinguished saddle connection. For this restricted set of saddle connections, we show that it satisfies an $L (\log L)^{d-2}$ growth rate, where $d$ is the complex dimension of the hyperelliptic stratum. The upper bound holds for all translation surfaces in the hyperelliptic stratum while the lower bound holds for almost every surface in the hyperelliptic stratum. The proof of the lower bound uses horocycle renormalization.