Alex Eskin

Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials

We compute the asymptotics of the number of connected branched coverings of a torus as their degree goes to infinity and the ramification type stays fixed. These numbers are equal to the volumes of the moduli spaces of pairs (curve, holomorphic differential) with fixed multiplicities of zeros of the differential and have several applications in ergodic theory.

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.

Invariant and stationary measures for the action on Moduli space

We prove some ergodic-theoretic rigidity properties of the action of on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of is supported on an invariant affine submanifold.The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work.

Asymptotic formulas on flat surfaces

We find asymptotics for the number of cylinders and saddle connections on flat surfaces. These results extend previous results of Veech.

Quasi-flats and rigidity in higher rank symmetric spaces

In this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces. Our results are powerful enough to apply to noncocompact lattices in higher rank symmetric spaces, such as SL(n, 2), n > 3: Theorem 8.1 is a major step towards the proof of quasiisometric rigidity of such lattices ([E]). We also give a different, and effective, proof of the theorem of Kleiner-Leeb on the quasi-isometric rigidity of higher rank symmetric spaces ([KL]).

Billiards in rectangles with barriers

We use Ratners theorem to compute the asymptotics of the number of (cylinders of) periodic trajectories in a rectangle with a barrier, assuming that the location p/q of the barrier is rational. We also show that as q tends to infinity, the constant in the asymptotic formula tends to the constant for the generic genus 2 flat surface.

Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces

Thus φ is bi-Lipschitz on large scales. If there exists a constant C ′ such that every point of Y is within a distance C′ of a point of φ(X), then there is a quasi-isometric embedding ψ : Y → X which is a “coarse inverse of φ”, i.e. supx∈X dX(x, ψ(φ(x))) <∞ and supy∈Y dY (y, φ(ψ(y))) < ∞. In this case φ is called a quasi-isometry and the spaces X and Y are said to be quasi-isometric. A basic example to keep in mind is that the fundamental group π1(M) (endowed with the word metric) of a compact Riemannian manifold M is quasi-isometric to the universal cover M̃ of M . In this paper we prove the following theorem:

Lattice point asymptotics and volume growth on Teichmüller space

We apply some of the ideas of the Ph.D. Thesis of G. A. Margulis (Mar70) to Teichmuller space. Let X be a point in Teichmuller space, and let BR(X) be the ball of radius R centered at X (with distances measured in the Teichmuller metric). We obtain asymptotic formulas as R tends to infinity for the volume of BR(X), and also for the cardinality of the intersection of BR(X) with an orbit of the mapping class group.

NON-DIVERGENCE OF TRANSLATES OF CERTAIN ALGEBRAIC MEASURES

Abstract. Let G and

Ergodic theoretic proof of equidistribution of Hecke points

H\subset G