A model for horizontally restricted random square-tiled surfaces

TL;DR

This work introduces a horizontally restricted random model for square-tiled surfaces by fixing a conjugacy class 𝒦_{μ_n} with at most n^α cycles and sampling (σ,τ) uniformly from 𝒦_{μ_n}×S_n, enabling analysis of STS with at most n^α horizontal cylinders. It proves that the HR-induced distribution on A_n converges to uniform when α<1/2, derives moments and genus distributions, and establishes that STSs are asymptotically connected with a most likely stratum determined by parity, while also characterizing holonomy behavior, including a holonomy-torus probability tending to 1/e and a vanishingly small obstruction to visibility. The results extend the standard model (uniform on S_n×S_n) by providing HR-model counterparts for key topological and geometric statistics, supported by refined permutation- and representation-theoretic techniques. These findings illuminate how restricting horizontal cylinder configurations impacts the global geometry of STSs and saddle-connection data, with implications for volumes of strata and random-cover phenomena in translation-surface theory.

Abstract

A square-tiled surface (STS) is a (finite, possibly branched) cover of the standard square-torus with possible branching over exactly 1 point. Alternately, STSs can be viewed as finitely many axis-parallel squares with sides glued in parallel pairs. After a labelling of the squares by $\{1, \dots, n\}$, we can describe an STS with $n$ squares using two permutations $σ, τ\in S_n$, where $σ$ encodes how the squares are glued horizontally and $τ$ encodes how the squares are glued vertically. Hence, a previously considered natural model for STSs with $n$ squares is $S_n \times S_n$ with the uniform distribution. We modify this model to obtain a new one: We fix $α\in [0,1]$ and let $\mathcal{K}_{μ_n}$ be a conjugacy class of $S_n$ with at most $n^α$ cycles. Then $\mathcal{K}_{μ_n} \times S_n$ with the uniform distribution is a model for STSs with restricted horizontal gluings. Since horizontal cycles of the $σ$ permutation are related to the number of maximal horizontal cylinders, this new model serves as a random model for STSs with at most $n^α$ maximal horizontal cylinders. We deduce the asymptotic (as $n$ grows) number of components, genus distribution, most likely stratum and set of holonomy vectors of saddle connections for random STSs in this new model.

Figures (6)

• Figure 1: A square-tiled surface in $\hbox{\sf STS}_9$. Matching edge/vertex decorations are glued together. With the chosen labelling, the horizontal permutation is $\sigma = (1,2)(3,4,5)(6,7)(8,9)$ and the vertical permutation is $\tau = (1)(2,3)(4)(5,6,8)(7,9)$. The surface decomposes into three maximal horizontal cylinders indicated by the different gray shadings.
• Figure 2: Examples of horizontally one-cylinder surfaces. Note that in the surface (A) the maximal horizontal cylinder is comprised of a single band of squares (i.e. the height of the cylinder is 1). The maximal horizontal cylinders of surfaces (B), (C), and (D) are comprised of 2, 3 and 6 bands of squares respectively.
• Figure 3: Two translation surfaces. On the left is a surface formed by two hexagons on the right is one formed by identification of opposite sides of an octagon. Edge identifications are indicated by matching edge decorations. Cone points are represented by decorated vertices -- matching decorations mean that the vertices are identified under the edge gluings. The surfaces live in $\mathcal{H}(1,1)$ and $\mathcal{H}(2)$ respectively. The octagon surface also has two saddle connections on it (represented by dotted and dashed lines). Taking the Octagon sidelength to be 1, the holonomy vector corresponding to the dotted saddle connection is $\left(-\frac{1}{\sqrt{2}}, 1+\frac{1}{\sqrt{2}}\right)$ and the holonomy vector corresponding to the dashed saddle connection is $\left(1, 2+2\sqrt{2}\right)$.
• Figure 4: Three examples of square-tiled surfaces. The surfaces belong to $\hbox{\sf STS}_7 \cap \mathcal{H}(2,1,1)$, $\hbox{\sf STS}_5 \cap \mathcal{H}(2)$, $\hbox{\sf STS}_6 \cap \mathcal{H}(2)$ from left to right respectively. The surface on the left is a holonomy torus, the middle one is a visibility torus (but not a holonomy torus), and the one on the right is neither since it doesn't have a saddle connection corresponding to vector (1,0). The first two have 3 horizontal cyclinders each. The one on the right has two horizontal cylinders.
• Figure 5: Young diagram and tableau for the partition (5,3,3,2,1). From left to right: Young diagram; Tableau with hook lengths; Tableau with contents; Example of an SSYT that is not a BST (since, for instance, the squares filled with 2 is not connected); Example of a BST. The height is $h(T) = 0+0+1+0+0+0+0 = 1$ (since the only border-strip that touches two rows is the one corresponding to squares filled with 3).

Theorems & Definitions (96)

• Theorem 1.1: Asymptotic Probability of Single-Banded STSs
• Theorem 1.2: Moments of Number of Vertices
• Corollary 1.2: Expected Genus
• Theorem 1.3: Genus Distribution
• Theorem 1.4: Sufficient Criterion for Most Likely Stratum
• Corollary 1.4: Most Likely Stratum
• Theorem 1.5: Holonomy Theorem
• Proposition 2.1: Commutator to stratum data ShresRandom
• Remark 2.2
• Definition 2.3