Algebraic interaction strength for translation surfaces with multiple singularities

TL;DR

The paper computes the exact algebraic interaction strength $KVol$ for translation surfaces with multiple singularities by analyzing two families: regular $n$-gons with $n\equiv 2\pmod 4$ and Bouw–Möller surfaces $S_{m,n}$ with $1<\gcd(m,n)<n$. It develops a subdivision framework that decomposes saddle connections into sandwiched and non-sandwiched segments and derives sharp length and intersection estimates, enabling an exact bound $\dfrac{Int(\alpha,\beta)}{l(\alpha)l(\beta)}\le\dfrac{1}{2l_0^2}$ (with appropriate $l_0$) and identifying extremal configurations. For regular decagons ($n\ge10$, $n\equiv2\pmod4$), the maximal value is achieved by unions of two sides intersecting at both singularities with the same sign, giving $KVol(X_n)=\dfrac{n}{8\tan(\pi/n)}$, and similar two-side constructions yield equality for many Bouw–Möller surfaces with nontrivial gcd. These results provide the first exact computations of $KVol$ on translation surfaces with several singularities and extend prior single-singularity analyses to broader polygonal geometries, offering precise extremal curve configurations and a unified subdivision approach.

Abstract

We compute the maximal ratio of the algebraic intersection of two closed curves on two families of translation surfaces with multiple singularities. This ratio, called the interaction strength, is difficult to compute for translation surfaces with several singularities as geodesics can change direction at singularities. The main contribution of this paper is to deal with this type of surfaces. Namely, we study the interaction strength of the regular $n-$gons for $n \equiv 2 \pmod 4$ and the Bouw-Möller surfaces $S_{m,n}$ with $1 < \gcd(m,n) < n$. This answers a conjecture of the author from (Boulanger, Algebraic intersection, lengths and Veech surfaces, arXiv:2309.17165). and it completes the study of the algebraic interaction strength KVol on the regular polygon Veech surfaces. Our results on Bouw-Möller surfaces extends the results of (Boulanger-Pasquinelli, Algebraic intersections on Bouw-Möller surfaces, and more general convex polygons, arXiv:2409.01711). This is also the first exact computation of KVol on translation surfaces with several singularities, and the pairs of curves that achieve the best ratio are singular geodesics made of two saddle connections with different directions.

Figures (13)

• Figure 1: A triple cover of the double regular pentagon with three singularities (of angle $6\pi$) and for which the algebraic interaction strenght can be computed from BP24 - and is $\frac{3n}{4\tan(\pi /n)}$ -. The same picture can be extended to any number of singularities.
• Figure 2: The decomposition into three horizontal cylinders on the $14$-gon. On the right, the unfolding of the smallest cylinder $C_1$. The closed geodesic $\alpha$ is homologous to the union of two saddle connections.
• Figure 3: The sign of a transverse intersection.
• Figure 4: The cyclic order of the sides at the singularties. The segments $\gamma = 1 \cup 3$ and $\delta = 2 \cup 5$ intersect twice.
• Figure 5: The permutation $\sigma_0$ associated to the transitition diagram in sector $\Sigma_0$ for $n=10$.

Theorems & Definitions (51)

• Theorem 1.0.1
• Theorem 1.0.2
• Corollary 1.0.3
• Theorem 1.0.4
• Conjecture 1.0.5
• Remark 1.0.6
• Remark 2.2.1
• Definition 2.3.1
• Remark 2.3.3
• Definition 2.4.1