On the number of saddle connections in translation surfaces with poles

TL;DR

This paper studies translation surfaces with poles, characterizing when they have infinitely many saddle connections and providing bounds on their number, which differ from classical translation surfaces due to their infinite area.

Contribution

It offers a combinatorial characterization of strata with infinite saddle connections and establishes bounds on their quantities, advancing understanding of these structures.

Findings

Identifies conditions for infinite saddle connections in strata

Provides lower and upper bounds for saddle connections

Highlights differences from classical translation surfaces

Abstract

Translation surfaces with poles correspond to meromorphic differentials on compact Riemann surfaces. They appear in compactifications of strata of the moduli space of Abelian differentials and in the study of stability conditions. Such structures have different geometrical and dynamical properties than usual translation surfaces. In particular, they always have infinite area and can have a finite number of saddle connections. We provide a combinatorial characterization of the strata for which there can be an infinite number of saddle connections and give lower and upper bounds for the number of saddle connections and related quantities.