Counting saddle connections in flat surfaces with poles of higher order

TL;DR

This paper investigates flat surfaces with poles of higher order, establishing bounds on saddle connections and characterizing strata with potentially infinite saddle connections, extending understanding beyond finite-area surfaces.

Contribution

It introduces bounds for saddle connections on higher order pole surfaces and characterizes strata with infinite saddle connections for specific cases.

Findings

Established bounds for the number of saddle connections.

Provided a combinatorial characterization for infinite saddle connections.

Extended the theory to surfaces with poles of higher order.

Abstract

Flat surfaces that correspond to $k$-differentials on compact Riemann surfaces are of finite area provided there is no pole of order $k$ or higher. We denote by \textit{flat surfaces with poles of higher order} those surfaces with flat structures defined by a $k$-differential with at least one pole of order at least $k$. Flat surfaces with poles of higher order have different geometrical and dynamical properties than usual flat surfaces of finite area. In particular, they can have a finite number of saddle connections. We give lower and upper bounds for the number of saddle connections and related quantities. In the case $k=1\ or\ 2$, we provide a combinatorial characterization of the strata for which there can be an infinite number of saddle connections.