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 investigate the translation lengths of group elements that arise in random walks on the isometry groups of Gromov hyperbolic spaces. In particular, without any moment condition, we prove that non-elementary random walks exhibit at least linear growth of translation lengths. As a corollary, almost every random walk on mapping class groups eventually becomes pseudo-Anosov, and almost every random walk on $\mathrm {Out}(F_n)$ eventually becomes fully irreducible. If the underlying measure further has finite first moment, then the growth rate of translation lengths is equal to the drift, the escape rate of the random walk.

We then apply our technique to investigate the random walks induced by the action of mapping class groups on Teichmüller spaces. In particular, we prove the spectral theorem under finite first moment condition, generalizing a result of Dahmani and Horbez.

Singular directions in a Veech surface are shown to be exactly the directions of its saddle connections.

Recall that two geodesics in a negatively curved surface S are of the same type if their free homotopy classes differ by a homeomorphism of the surface. In this note we study the distribution in the unit tangent bundle of the geodesics of fixed type, proving that they are asymptotically equidistributed with respect to a certain measure ${\mathfrak {m}}^S$ on $T^1S$ . We study a few properties of this measure, showing for example that it distinguishes between hyperbolic surfaces.

Consider a component ${\cal Q}$ of a stratum in the moduli space of area-one abelian differentials on a surface of genus g. Call a property ${\cal P}$ for periodic orbits of the Teichmüller flow on ${\cal Q}$ typical if the growth rate of orbits with property ${\cal P}$ is maximal. We show that the following property is typical. Given a continuous integrable cocycle over the Teichmüller flow with values in a vector bundle $V\to {\cal Q}$ , the logarithms of the eigenvalues of the matrix defined by the cocycle and the orbit are arbitrarily close to the Lyapunov exponents of the cocycle for the Masur–Veech measure.

In this paper we investigate the range of validity of Ruelle’s property. First, we show that every finitely generated Fuchsian group has Ruelle’s property. We also prove the existence of an infinitely generated Fuchsian group satisfying Ruelle’s property. Concerning the negative results, we first generalize Astala and Zinsmeister’s results [Mostow rigidity and Fuchsian groups. C. R. Math. Acad. Sci. Paris311 (1990), 301–306; Teichmüller spaces and BMOA. Math. Ann.289 (1991), 613–625] by proving that all convergence-type Fuchsian groups of the first kind fail to have Ruelle’s property. Finally, we give some results about second-kind Fuchsian groups. [-3.2pc]

We prove several general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous $\operatorname {SL}(2, \mathbb {R})$ -action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $\operatorname {SL}(2, \mathbb {R})$ -action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the ‘weak convergence’ of push-forwards of horocycle measures under the geodesic flow and a short proof of weaker versions of theorems of Chaika and Eskin on Birkhoff genericity and Oseledets regularity in almost all directions for the Teichmüller geodesic flow.

We study the geometry of the Thurston metric on the Teichmüller space of hyperbolic structures on a surface $S$. Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite type; however, we focus particular attention on the case where the surface is a once-punctured torus. In that case, our results provide a detailed picture of the infinitesimal, local, and global behavior of the geodesics of the Thurston metric, as well as an analogue of Royden’s theorem.

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.

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 ${\geqslant}2$.

In this paper, we study an interesting curve, the so-called Manhattan curve, associated with a pair of boundary-preserving Fuchsian representations of a (non-compact) surface; in particular, representations corresponding to Riemann surfaces with cusps. Using thermodynamic formalism (for countable state Markov shifts), we prove the analyticity of the Manhattan curve. Moreover, we derive several dynamical and geometric rigidity results, which generalize results of Burger [Intersection, the Manhattan curve, and Patterson–Sullivan theory in rank 2. Int. Math. Res. Not.1993(7) (1993), 217–225] and Sharp [The Manhattan curve and the correlation of length spectra on hyperbolic surfaces. Math. Z.228(4) (1998), 745–750] for convex cocompact Fuchsian representations.

In this paper we show that a polygonal quasiconformal mapping always corresponds to a chord-arc curve. Furthermore, we find that the set of curves corresponding to polygonal quasiconformal mappings is path connected in the set of all bounded chord-arc curves.

We prove that for each Riemann surface of finite analytic type (p, n) with p ≥ 2, there exist uncountably many Teichmüller disks Δ in the Teichmüller space T(S), where S = - {a point a}, with these properties: (1) the natural projection j: T(S) → T() defined by forgetting a induces an isometric embedding of each Δ into T(); and (2) the stabilizer of each Teichmüller disk Δ in the a-pointed mapping class group of S is trivial.

Let S be a Riemann surface of finite type. Let ω be a pseudo-Anosov map of S that is obtained from Dehn twists along two families {A,B} of simple closed geodesics that fill S. Then ω can be realized as an extremal Teichmüller mapping on a surface of the same type (also denoted by S). Let ϕ be the corresponding holomorphic quadratic differential on S. We show that under certain conditions all possible nonpuncture zeros of ϕ stay away from all closures of once punctured disk components of S∖{A,B}, and the closure of each disk component of S∖{A,B} contains at most one zero of ϕ. As a consequence, we show that the number of distinct zeros and poles of ϕ is less than or equal to the number of components of S∖{A,B}.

In this paper, one model of the universal Teichmüller space is studied. By the method of construction, the lower bound of the inner radius of univalency by the Pre-Schwarzian derivative of quasidisks with infinity as an inner point (such as domains bounded by ellipses) is obtained.

This paper presents a rigidity theorem for infinite-dimensional Bergman spaces of hyperbolic Riemann surfaces, which states that the Bergman space $A^{1}(M)$, for such a Riemann surface $M$, is isomorphic to the Banach space of summable sequence, $l^{1}$. This implies that whenever $M$ and $N$ are Riemann surfaces that are not analytically finite, and in particular are not necessarily homeomorphic, then $A^{1}(M)$ is isomorphic to $A^{1}(N)$. It is known from V. Markovic that if there is a linear isometry between $A^{1}(M)$ and $A^{1}(N)$, for two Riemann surfaces $M$ and $N$ of non-exceptional type, then this isometry is induced by a conformal mapping between $M$ and $N$. As a corollary to this rigidity theorem presented here, taking the Banach duals of $A^{1}(M)$ and $l^{1}$ shows that the space of holomorphic quadratic differentials on $M,\ Q(M)$, is isomorphic to the Banach space of bounded sequences, $l^{\infty }$. As a consequence of this theorem and the Bers embedding, the Teichmüller spaces of such Riemann surfaces are locally bi-Lipschitz equivalent.

We make a detailed study of the relation of a euclidean convex region $\Omega \subset \mathbb C$ to $\mathrm{Dome} (\Omega)$. The dome is the relative boundary, in the upper halfspace model of hyperbolic space, of the hyperbolic convex hull of the complement of $\Omega$. The first result is to prove that the nearest point retraction $r: \Omega \to \mathrm{Dome} (\Omega)$ is 2-quasiconformal. The second is to establish precise estimates of the distortion of $r$ near $\partial \Omega$.

Let $S$ be a compact Riemann surface of genus ${\rm g}\,{>}\,1$. By using Teichmüller theory, a new approach is obtained to the existence of a Jenkins–Strebel differential with prescribed type and heights on $S$. Moreover, the geometric structures of horizontal trajectories of certain classes of quadratic differentials are discussed.

We consider an arbitrary Riemann surface $X$, possibly of infinite hyperbolic area. The Liouville measure of the hyperbolic metric defines a measure on the space $G(\tilde{X})$ of geodesics of the universal covering $\tilde{X}$ of $X$. As we vary the Riemann surface structure, this gives an embedding from the Teichmüller space of $X$ into the Fréchet space of Hölder distributions on $G(\tilde{X})$. We show that the embedding is continuously differentiable. In particular, we obtain an explicit integral representation of the tangent map.