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

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.