We define and study graphs associated to hexagon decompositions of surfaces by curves and arcs. One of the variants is shown to be quasi-isometric to the pants graph, whereas the other variant is quasi-isometric to (a Cayley graph of) the mapping class group.

We classify quasidiagonals of the $SL(2, R)$ action on products of strata or hyperelliptic loci. We use the technique of diamonds developed by Apisa and Wright in order to use induction on this problem.

We provide a complete description of realizable period representations for meromorphic differentials on Riemann surfaces with prescribed orders of zeros and poles, hyperelliptic structure and spin parity.

We prove an effective estimate with a power saving error term for the number of square-tiled surfaces in a connected component of a stratum of quadratic differentials whose vertical and horizontal foliations belong to prescribed mapping class group orbits and which have at most L squares. This result strengthens asymptotic counting formulas in the work of Delecroix, Goujard, Zograf, Zorich, and the author.

We study the vectorial length compactification of the space of conjugacy classes of maximal representations of the fundamental group $\Gamma$ of a closed hyperbolic surface $\Sigma$ in $\textrm{PSL}(2,{\mathbb{R}})^n$. We identify the boundary with the sphere ${\mathbb{P}}(({\mathcal{ML}})^n)$, where $\mathcal{ML}$ is the space of measured geodesic laminations on $\Sigma$. In the case $n=2$, we give a geometric interpretation of the boundary as the space of homothety classes of ${\mathbb{R}}^2$-mixed structures on $\Sigma$. We associate to such a structure a dual tree-graded space endowed with an ${\mathbb{R}}_+^2$-valued metric, which we show to be universal with respect to actions on products of two $\mathbb{R}$-trees with the given length spectrum.

We prove that the nonvarying strata of abelian and quadratic differentials in low genus have trivial tautological rings and are affine varieties. We also prove that strata of k-differentials of infinite area are affine varieties for all k. Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of the Hodge bundle for abelian and quadratic differentials of finite area is extremal in the sense that merging two zeros in each stratum leads to an extremal effective divisor in the boundary. A common feature throughout these results is a relation of divisor classes in strata of differentials as well as its incarnation in Teichmüller dynamics.

We consider the Weil–Petersson gradient vector field of renormalized volume on the deformation space of convex cocompact hyperbolic structures on (relatively) acylindrical manifolds. In this paper we prove the conjecture that the flow has a global attracting fixed point at the unique structure $M_{\rm geod}$ with minimum convex core volume.

This paper is about a type of quantitative density of closed geodesics and orthogeodesics on complete finite-area hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic and the shortest doubly truncated orthogeodesic that are $\varepsilon$-dense on a given compact set on the surface.

Following Bridgeman, we demonstrate several families of infinite dilogarithm identities associated with Fibonacci numbers, Lucas numbers, convergents of continued fractions of even periods, and terms arising from various recurrence relations.

We compute the gap distribution of directions of saddle connections for two classes of translation surfaces. One class will be the translation surfaces arising from gluing two identical tori along a slit. These yield the first explicit computations of gap distributions for non-lattice translation surfaces. We show that this distribution has support at zero and quadratic tail decay. We also construct examples of translation surfaces in any genus $d>1$ that have the same gap distribution as the gap distribution of two identical tori glued along a slit. The second class we consider are twice-marked tori and saddle connections between distinct marked points with a specific orientation. These results can be interpreted as the gap distribution of slopes of affine lattices. We obtain our results by translating the question of gap distributions to a dynamical question of return times to a transversal under the horocycle flow on an appropriate moduli space.

In this note we show that the expected value of the separating systole of a random surface of genus g with respect to Weil–Petersson volume behaves like $2\log g $ as the genus goes to infinity. This is in strong contrast to the behavior of the expected value of the systole which, by results of Mirzakhani and Petri, is independent of genus.

We derive a quadratic recursion relation for the linear Hodge integrals of the form $\langle \tau _{2}^{n}\lambda _{k}\rangle $ . These numbers are used in a formula for Masur-Veech volumes of moduli spaces of quadratic differentials discovered by Chen, Möller and Sauvaget. Therefore, our recursion provides an efficient way of computing these volumes.

A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with $2N$ crossings grows exponentially when $N$ grows, but the long-standing problem on the precise asymptotics is still out of reach.

We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as $N$ tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator.

The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinder. The proofs combine recent results on Masur–Veech volumes of moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics.

Given integers $g,n\geqslant 0$ satisfying $2-2g-n<0$, let ${\mathcal{M}}_{g,n}$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $g$ with $n$ cusps. We study the global behavior of the Mirzakhani function $B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$ which assigns to $X\in {\mathcal{M}}_{g,n}$ the Thurston measure of the set of measured geodesic laminations on $X$ of hyperbolic length ${\leqslant}1$. We improve bounds of Mirzakhani describing the behavior of this function near the cusp of ${\mathcal{M}}_{g,n}$ and deduce that $B$ is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of $B$ to statistics of counting problems for simple closed hyperbolic geodesics.

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer k, we are interested in the set of all closed geodesics with at least k (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in k (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like k for growing k.

J.-C. Yoccoz proposed a natural extension of Selberg’s eigenvalue conjecture to moduli spaces of abelian differentials. We prove an approximation to this conjecture. This gives a qualitative generalization of Selberg’s $\frac{3}{16}$ theorem to moduli spaces of abelian differentials on surfaces of genus ${\geqslant}2$.

Using Roelcke’s formula for the Green function, we explicitly construct a basis in the kernel of the adjoint Laplacian on a compact polyhedral surface $X$ and compute the $S$-matrix of $X$ at the zero value of the spectral parameter. We apply these results to study various self-adjoint extensions of a symmetric Laplacian on a compact polyhedral surface of genus two with a single conical point. It turns out that the behaviour of the $S$-matrix at the zero value of the spectral parameter is sensitive to the geometry of the polyhedron.

In this we exploit the arithmeticity criterion of Oh and Benoist–Miquel to exhibit an origami in the principal stratum of the moduli space of translation surfaces of genus three whose Kontsevich–Zorich monodromy is not thin in the sense of Sarnak.

We describe in this article the dynamics of a one-parameter family of affine interval exchange transformations. This amounts to studying the directional foliations of a particular dilatation surface introduced in Duryev et al [Affine surfaces and their Veech groups. Preprint, 2016, arXiv:1609.02130], the Disco surface. We show that this family displays various dynamical behaviours: it is generically dynamically trivial but for a Cantor set of parameters the leaves of the foliations accumulate to a (transversely) Cantor set. This study is achieved through analysis of the dynamics of the Veech group of this surface combined with a modified version of Rauzy induction in the context of affine interval exchange transformations.

Zhou et al. [‘On weakly non-decreasable quasiconformal mappings’, J. Math. Anal. Appl.386 (2012), 842–847] proved that, in a Teichmüller equivalence class, there exists an extremal quasiconformal mapping with a weakly nondecreasable dilatation. They asked whether a weakly nondecreasable dilatation is a nondecreasable dilatation. The aim of this paper is to give a negative answer to their problem. We also construct a Teichmüller class such that it contains an infinite number of weakly nondecreasable extremal representatives, only one of which is nondecreasable.