Anton Zorich

Connected components of the moduli spaces of Abelian differentials with prescribed singularities

Consider the moduli space of pairs (C,ω) where C is a smooth compact complex curve of a given genus and ω is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe connected components of this space. This classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons, and in the study of geometry of translation surfaces.

Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow

We compute the sum of the positive Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow. The computation is based on the analytic Riemann-Roch Theorem and uses a comparison of determinants of flat and hyperbolic Laplacians when the underlying Riemann surface degenerates.

DEVIATION FOR INTERVAL EXCHANGE TRANSFORMATIONS

Consider a long piece of trajectory x;T(x);T(T(x));:::;T ni1 (x) of an interval exchange transformation T. A generic interval exchange trans- formation is uniquely ergodic. Hence the ergodic theorem predicts that the numberi(x;n) of visits of our trajectory to the i-th subinterval would be approximately ‚in. Here ‚i is the length of corresponding subinterval of our unit interval X. In this paper we give an estimate for the deviation of the actual number of visits to the i-th subinterval Xi from one predicted by the ergodic theorem. We prove that for almost all interval exchange transformations the following bound is valid: max x2X 1•im limsup n!+1 logji(x;n) i ‚inj logn = µ2 µ1 < 1

SQUARE-TILED CYCLIC COVERS

A cyclic cover of the complex projective line branched at four appropriate points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding Teichmuller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichmuller curve with respect to the geodesic flow. This paper includes a new example (announced by G. Forni and C. Matheus in [17] of a Teichmuller curve of a square-tiled cyclic cover in a stratum of Abelian differentials in genus four with a maximally degenerate Kontsevich--Zorich spectrum (the only known example in genus three found previously by Forni also corresponds to a square-tiled cyclic cover [15]. We present several new examples of Teichmuller curves in strata of holomorphic and meromorphic quadratic differentials with a maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichmuller curves with maximally degenerate spectra. We prove that this is indeed the case within the class of square-tiled cyclic covers.

ZERO LYAPUNOV EXPONENTS OF THE HODGE BUNDLE

By the results of G. Forni and of R. Trevino, the Lyapunov spectrum of the Hodge bundle over the Teichmuller geodesic flow on the strata of Abelian and of qua- dratic differentials does not contain zeroes even though fo r certain invariant submanifolds zero exponents are present in the Lyapunov spectrum. In all previously known examples, the zero exponents correspond to those PSL(2,R)-invariant subbundles of the real Hodge bundle for which the monodromy of the Gauss—Manin connection acts by isometries of the Hodge metric. We present an example of an arithmetic Teichmuller curve, for which the real Hodge bundle does not contain any PSL(2,R)-invariant, continuous subbundles, and nevertheless its spectrum of Lyapunov exponents contains zeroes. We describe the mech- anism of this phenomenon; it covers the previously known situation as a particular case. Conjecturally, this is the only way zero exponents can appear in the Lyapunov spectrum of the Hodge bundle for any PSL(2,R)-invariant probability measure.

Lyapunov spectrum of invariant subbundles of the Hodge bundle

We study the Lyapunov spectrum of the Kontsevich–Zorich cocycle on SL(2,ℝ)-invariant subbundles of the Hodge bundle over the support of SL(2,ℝ)-invariant probability measures on the moduli space of Abelian differentials. In particular, we prove formulas for partial sums of Lyapunov exponents in terms of the second fundamental form (the Kodaira–Spencer map) of the Hodge bundle with respect to the Gauss–Manin connection and investigate the relations between the central Oseledets subbundle and the kernel of the second fundamental form. We illustrate our conclusions in two special cases.

Square Tiled Surfaces and Teichmüller Volumes of the Moduli Spaces of Abelian Differentials

We present an approach for counting the Teichmuller volumes of the moduli spaces of Abelian differentials on a Riemann surface of genus g. We show that the volumes can be counted by means of counting the “integer points” in the corresponding moduli space. The “integer points” are represented by square tiled surfaces — the flat surfaces tiled by unit squares. Such tilings have several conical singularities with 8, 12,... adjacent unit squares. Counting the leading term in the asymptotics of the number of tilings having at most N unit squares, we get the volumes of the corresponding strata of the moduli spaces.

Lower bounds for Lyapunov exponents of flat bundles on curves

Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top k Lyapunov exponents of the flat bundle is always greater or equal to the degree of any rank k holomorphic subbundle. We generalize the original context from Teichmueller curves to any local system over a curve with non-expanding cusp monodromies. As an application we obtain the large genus limits of individual Lyapunov exponents in hyperelliptic strata of Abelian differentials. Understanding the case of equality with the degrees of subbundle coming from the Hodge filtration seems challenging, e.g. for Calabi-Yau type families. We conjecture that equality of the sum of Lyapunov exponents and the degree is related to the monodromy group being a thin subgroup of its Zariski closure.

Volumes of Strata of Abelian Differentials and Siegel–Veech Constants in Large Genera

We state conjectures on the asymptotic behavior of the volumes of moduli spaces of Abelian differentials and their Siegel–Veech constants as genus tends to infinity. We provide certain numerical evidence, describe recent advances and the state of the art towards proving these conjectures.

Flat Surfaces and Dynamics on Moduli Space

Flat surfaces (also called translation surfaces) are pairs consisting of a Riemann surface together with a holomorphic one-form. The interest in flat surfaces stems from the dynamics on polygonal billiards. Instead of reflecting the trajectory at the boundary of the table one reflects the table and glues them along the reflection edges. The billiard paths become straight lines on the unfolded object and are thus much more easily accesible objects. If all the angles of the table are rational multiples of π, the resulting object is a flat surface. The