According to the work of Kontsevich-Zorich, the invariant that classifies non-hyperelliptic connected components of the moduli spaces of Abelian differentials with prescribed singularities,is the parity of the spin structure.
We show that for the moduli space of quadratic differentials, the spin structure is constant on every stratum where it is defined. In particular this disproves the conjecture that it classifies the non-hyperelliptic connected components of the strata of quadratic differentials with prescribed singularities. An explicit formula for the parity of the spin structure is given.
Moduli spaces of quadratic differentials with prescribed singularities are not necessarily connected. We describe here all cases when they have a special hyperelliptic connected component. We announce the general classification theorem: up to the four exceptional cases in low genera the strata of meromorphic quadratic differentials are either connected, or have exactly two connected components, and one of the two components is hyperelliptic.
We present an explicit formula relating volumes of strata of meromorphicquadratic differentials with at most simple poles on Riemann surfacesand counting functions of the number of flat cylinders filled by closedgeodesics in associated flat metric with singularities. This generalizes the resultof Athreya, Eskin and Zorich in genus 0 to higher genera.
We construct an example of a uniquely ergodic measured foliation on a surface such that the associated translation flow on the orientation double cover is minimal but not uniquely ergodic. We then prove a geometric criterion for the horizontal foliation of a quadratic differential to be uniquely ergodic. The second theorem generalizes a result of Treviño for the horizontal flow on a translation surface, as well as Masur's criterion for unique ergodicity of the horizontal foliation.
In two fundamental classical papers, Masur and Veech have independently proved that the Teichmueller geodesic flow acts ergodically on each connected component of each stratum of the moduli space of quadratic differentials. It is therefore interesting to have a classification of the ergodic components. Veech has proved that these strata are not necessarily connected. In a recent work, Kontsevich and Zorich have completely classified the components in the particular case where the quadratic differentials are given by the global square of Abelian differentials. Here we are interested in the comp…
Configurations of rigid collections of saddle connections are connected component invariants for strata of the moduli space of quadratic differentials. They have been classified for strata of Abelian differentials by Eskin, Masur and Zorich. Similar work for strata of quadratic differentials has been done in Masur and Zorich, although in that case the connected components were not distinguished. We classify the configurations for quadratic differentials on the Riemann sphere and on hyperelliptic connected components of the moduli space of quadratic differentials. We show that, in genera greate…
The seminal work of Eskin-Masur-Zorich described the principal boundary of moduli spaces of abelian differentials that parameterizes flat surfaces with a prescribed generic configuration of short parallel saddle connections. In this paper we describe the principal boundary for each configuration in terms of twisted differentials over Deligne-Mumford pointed stable curves. We also describe similarly the principal boundary of moduli spaces of quadratic differentials originally studied by Masur-Zorich. Our main technique is the flat geometric degeneration and smoothing developed by Bainbridge-Che…
The volumes of strata of Abelian or quadratic differentials play an important role in the study of dynamics on flat surfaces, related to dynamics in polygonal billiards. This article reviews all known ways to compute volumes in the quadratic case and provides explicit values of volumes of the strata of meromorphic quadratic differentials with at most simple poles in all low dimensions.
Moduli spaces of Abelian and quadratic differentials are stratified by multiplicities of zeroes; connected components of the strata correspond to ergodic components of the Teichmuller geodesic flow. It is known that the strata are not necessarily connected; the connected components were recently classified by M. Kontsevich and the author and by E. Lanneau. The strata can be also viewed as families of flat metrics with conical singularities and with Z/2Z-holonomy. For every connected component of each stratum of Abelian and quadratic differentials we construct an explicit representative which i…
In this paper we introduce flat grafting as a deformation of quadratic differentials on a surface of finite type that is analogous to the grafting map on hyperbolic surfaces. Flat grafting maps are generic in the strata structure and preserve parallel measured foliations. We use flat grafting to construct paths connecting any pair of quadratic differentials. In other words, we characterize cone point splitting deformations. The slices of quadratic differentials closed under flat grafting maps with a fixed direction arise naturally and we prove rigidity properties with respect to the lengths of…
We consider commensurability of quadratic differentials on surfaces. Each commensurability class has a natural order by the covering relation. We show that each commensurability class contains a unique (orbifold) element. We also discuss the relationship between commensurability of quadratic differentials and fibered commensurability, a notion introduced by Calegari-Sun-Wang.
We study Veech surfaces of genus 2 arising from quadratic differentials that are not squares of abelian differentials. We prove that all such surfaces of type (2,2) and (2,1,1) are arithmetic. In (1,1,1,1) case, we reduce the question to abelian differentials of type (2,2) on hyperelliptic genus 3 surfaces with singularities at Weierstrass points, and we give an example of a non-arithmetic Veech surface.
In this paper, we study fundamental groups of strata of the moduli space of quadratic differentials. We use certain properties of the Abel-Jacobi map, combined with local surgeries on quadratic differentials, to construct quotient groups of the fundamental groups for a particular family of strata.
We give an algebraic way of distinguishing the components of the exceptional strata of quadratic differentials in genus three and four. The complete list of these strata is (9, -1), (6,3,-1), (3,3,3, -1) in genus three and (12), (9,3), (6,6), (6,3,3) and (3,3,3,3) in genus four. This result is part of a more general investigation of disjointness of Teichmueller curves with divisors of Brill-Noether type on the moduli space of curves. As a result we show that for many strata of quadratic differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for …
A meromorphic quadratic differential on a punctured Riemann surface induces horizontal and vertical measured foliations with pole-singularities. In a neighborhood of a pole such a foliation comprises foliated strips and half-planes, and its leaf-space determines a metric graph. We introduce the notion of an asymptotic direction at each pole, and show that for a punctured surface equipped with a choice of such asymptotic data, any compatible pair of measured foliations uniquely determines a complex structure and a meromorphic quadratic differential realizing that pair. This proves the analogue …
Consider the 1-dimensional Hurwitz space parameterizing covers of P^1 branched at four points. We study its intersection with divisor classes on the moduli space of curves. As an application, we calculate the slope of the Teichmuller curve parameterizing square-tiled cyclic covers and recover the sum of its Lyapunov exponents obtained by Forni, Matheus and Zorich. Motivated by the work of Eskin, Kontsevich and Zorich, we exhibit a relation among the slope of Hurwitz spaces, the sum of Lyapunov exponents and the Siegel-Veech constant for the moduli space of quadratic differentials.
We study relations between Rauzy classes coming from an interval exchange map and the corresponding connected components of strata of the moduli space of Abelian differentials. This gives a criterion to decide whether two permutations are in the same Rauzy class or not, without actually computing them. We prove a similar result for Rauzy classes corresponding to quadratic differentials.
This expository survey describes how holomorphic quadratic differentials arise in several aspects of Teichmüller theory, highlighting their relation with various geometric structures on surfaces. The final section summarizes results for non-compact surfaces of finite type, when the quadratic differential has poles of finite order at the punctures.
We compare the flat geometry associated to a quadratic differential with the hyperbolic geometry associated to the underlying Riemann surface. We show that if a curve is contained in a thick subsurface, then its hyperbolic length is comparable to its flat length times the flat size of the subsurface.
We describe the connected components of the complement of a natural "diagonal" of real codimension 1 in a stratum of quadratic differentials on CP1. We establish a natural bijection between the set of these connected components and the set of generic configurations that appear on such "flat spheres". We also prove that the stratum has only one topological end. Finally, we elaborate a necessary toolkit destined to evaluation of the Siegel-Veech constants.
We describe the space of measured foliations induced on a compact Riemann surface by meromorphic quadratic differentials. We prove that any such foliation is realized by a unique such differential $q$ if we prescribe, in addition, the principal parts of $\sqrt q$ at the poles. This generalizes a theorem of Hubbard and Masur for holomorphic quadratic differentials. The proof analyzes infinite-energy harmonic maps from the Riemann surface to $\mathbb{R}$-trees of infinite co-diameter, with prescribed behavior at the poles.
We state conjectures on the asymptotic behavior of the Masur-Veech volumes of strata in the moduli spaces of meromorphic quadratic differentials and on the asymptotics of their area Siegel-Veech constants as the genus tends to infinity.
We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmuller geodesic is in the principal stratum of quadratic differentials. We show that a Teichmuller geodesic typical with respect to the harmonic measure for such random walks, is recurrent to the thick part of the principal stratum. As a consequence, the vertical and horizontal foliations of such a random Teichmuller geodesic have no saddle connections.
Very few results are known about the topology of the strata of the moduli space of quadratic differentials. In this paper, we prove that any connected component of such strata has only one topological end. A typical flat surface in a neighborhood of the boundary is naturally split by a collection of parallel short saddle connections, but the discrete data associated to this splitting can be quite difficult to describe. In order to bypass these difficulties, we use the Veech zippered rectangles construction and the corresponding (extended) Rauzy classes.
A meromorphic quadratic differential with poles of order two, on a compact Riemann surface, induces a measured foliation on the surface, with a spiralling structure at any pole that is determined by the complex residue of the differential at the pole. We introduce the space of such measured foliations, and prove that for a fixed Riemann surface, any such foliation is realized by a quadratic differential with second order poles at marked points. Furthermore, such a differential is uniquely determined if one prescribes complex residues at the poles that are compatible with the transverse measure…
We construct geometrically a homeomorphism between the moduli space of polynomial quadratic differentials on the complex plane and light-like polygons in the 2-dimensional Einstein Universe. As an application, we find a class of minimal Lagrangian maps between ideal polygons in the hyperbolic plane.
We prove the existence of "half-plane differentials" with prescribed local data on any Riemann surface. These are meromorphic quadratic differentials with higher-order poles which have an associated singular flat metric isometric to a collection of euclidean half-planes glued by an interval-exchange map on their boundaries. The local data is associated with the poles and consists of the integer order, a non-negative real residue, and a positive real leading order term. This generalizes a result of Strebel for differentials with double-order poles, and associates metric spines with the Riemann …
In this paper, we study connected components of strata of the space of quadratic differentials lying over $\T_g$. We use certain general properties of sections of line bundles to put a upper bound on the number of connected components, and a generalized version of the Gauss map as an invariant to put a lower bound on the number of such components. For strata with sufficiently many zeroes of the same order we can state precisely the number of components.
