The Cantor-Bendixson Rank of Certain Bridgeland-Smith Stability Conditions

TL;DR

This paper investigates the set of directions with saddle connections on meromorphic quadratic differentials, proving its closure, finite Cantor-Bendixson rank, and constructing surfaces for all ranks, using translation surface techniques.

Contribution

It introduces a new proof of closure and finite rank of saddle connection directions, and constructs surfaces for all possible ranks, advancing understanding of meromorphic quadratic differentials.

Findings

Set of directions with saddle connections is closed.

Finite Cantor-Bendixson rank of this set, with a tight bound.

Construction of surfaces for all possible ranks.

Abstract

We provide a novel proof that the set of directions that admit a saddle connection on a meromorphic quadratic differential with at least one pole of order at least two is closed, which generalizes a result of Bridgeland and Smith, and Gaiotto, Moore, and Neitzke. Secondly, we show that this set has finite Cantor-Bendixson rank and give a tight bound. Finally, we present a family of surfaces realizing all possible Cantor-Bendixson ranks. The techniques in the proof of this result exclusively concern Abelian differentials on Riemann surfaces, also known as translation surfaces. The concept of a "slit translation surface" is introduced as the primary tool for studying meromorphic quadratic differentials with higher order poles.