二次微分层连通分支中具有给定实虚叶状结构的方铺曲面的有效计数

Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani's asymptotics for simple closed hyperbolic geodesics

We show that the number of square-tiled surfaces of genus $g$, with $n$ marked points, with one or both of its horizontal and vertical foliations belonging to fixed mapping class group orbits, and having at most $L$ squares, is asymptotic to $L^{6g-6+2n}$ times a product of constants appearing in Mirzakhani's count of simple closed hyperbolic geodesics. Many of the results in this paper reflect recent discoveries of Delecroix, Goujard, Zograf, and Zorich, but the approach considered here is very different from theirs. We follow conceptual and geometric methods inspired by Mirzakhani's work.

Square-tiled surfaces of fixed combinatorial type: equidistribution, counting, volumes of the ambient strata

We prove that square-tiled surfaces having fixed combinatorics of horizontal cylinder decomposition and tiled with smaller and smaller squares become asymptotically equidistributed in any ambient linear $GL(\mathbb R)$-invariant suborbifold defined over $\mathbb Q$ in the moduli space of Abelian differentials. Moreover, we prove that the combinatorics of the horizontal and of the vertical decompositions are asymptotically uncorrelated. As a consequence, we prove the existence of an asymptotic distribution for the combinatorics of a "random" interval exchange transformation with integer lengths…

Contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes

We compute explicitly the absolute contribution of square-tiled surfaces having a single horizontal cylinder to the Masur-Veech volume of any ambient stratum of Abelian differentials. The resulting count is particularly simple and efficient in the large genus asymptotics. Using the recent results of Aggarwal and of Chen-Moeller-Zagier on the long-standing conjecture about the large genus asymptotics of Masur-Veech volumes, we derive that the relative contribution is asymptotically of the order 1/d, where d is the dimension of the stratum. Similarly, we evaluate the contribution of one-cylinder…

Single-cylinder square-tiled surfaces and the ubiquity of ratio-optimising pseudo-Anosovs

In every connected component of every stratum of Abelian differentials, we construct square-tiled surfaces with one vertical and one horizontal cylinder. We show that for all but the hyperelliptic components this can be achieved in the minimum number of squares necessary for a square-tiled surface in that stratum. For the hyperelliptic components, we show that the number of squares required is strictly greater and construct surfaces realising these bounds. Using these surfaces, we demonstrate that pseudo-Anosov homeomorphisms optimising the ratio of Teichmüller to curve graph translation lengt…

等周期形式与连通分层的不变子簇

Isoperiodic forms and invariant subvarieties of connected strata

研究全纯1-形式的绝对周期叶状结构在连通分层中的叶闭包分类，排除高余维子簇。

方形平铺循环覆盖

Square-tiled cyclic covers

复射影线的循环复盖在四个适当的点分支具有一个正方形瓷砖表面的自然结构。我们描述了这样一个正方形瓷砖表面的组合，相应的Teichmüller曲线的几何形状，并计算测地流的Teichmüller曲线上的行列式丛的李雅普诺夫指数。本文包括一个新的例子（G. Forni和C. Matheus in \cite{Forni:Matheus}）的一个正方形平铺循环覆盖的Teichmüller曲线，该覆盖位于亏格4的阿贝尔微分层中，具有最大退化的Kontsevich-Zorich谱（Forni以前在亏格3中发现的唯一已知例子也对应于正方形平铺循环覆盖\cite{ForniSurvey}）。本文给出了具有最大退化Kontsevich-Zorich谱的全纯和亚纯二次微分层中Teichmüller曲线的几个新例子。这些例子大概涵盖了所有可能的具有最大简并谱的Teichmüller曲线。我们证明，这确实是类内的情况下，正方形平铺循环覆盖。