Realizing pairs of multicurves as cylinders on translation surfaces

TL;DR

This work analyzes when two multicurves on a closed surface can be realized simultaneously as cores of multicylinders on a translation surface. It combines the Thurston–Veech construction with horizontal grafting to translate combinatorial intersection data into concrete geometric realizations, yielding explicit criteria and constructive procedures. The authors prove a principal equivalence: a coherent pair of multicurves extends to a coherent filling pair if and only if there exists a translation surface realizing them as parallel multicylinders, with an obstruction-free condition and a method to remove singletons to reach fillability. They also show that pairwise coherence, while necessary, does not suffice in general, underscoring the subtlety of simultaneous cylindric realizations and providing concrete counterexamples. These results enable systematic construction of translation surfaces with prescribed cylinder decompositions and have potential implications for dynamics via associated pseudo-Anosov maps.

Abstract

Any pair of intersecting cylinders on a translation surface is "coherent," in that the geometric and algebraic intersection numbers of their core curves are equal (up to sign). In this paper, we investigate when a pair of multicurves can be simultaneously realized as the core curves of cylinders on some translation surface. Our main tools are surface topology and the "flat grafting" deformation introduced by Ser-Wei Fu.

Figures (9)

• Figure 1: An example of the Thurston--Veech construction applied to the pair of multi-curves $\alpha = a_1 \cup a_2$ and $\beta = b_1 \cup b_2 \cup b_3$.
• Figure 2: Replacing a non-coherent intersection with coherent ones
• Figure 3: Building a graph out of components of $S \setminus (\vec{\beta} \cup \mathbb{S}_{\vec{\beta}}(\vec{\alpha}))$.
• Figure 4: Adding curves to decrease the complexity of the complement of $\alpha \cup \beta$.
• Figure 5: A pair of coherent multicurves that cannot be jointly realized as parallel multicylinders, together with a realization with one of the multicylinders parallel. Observe that even if one repartitions and adds the singleton of $\beta$ to $\alpha$, then the pair is still not realizable as a pair of parallel multicylinders.

Theorems & Definitions (52)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• proof