ABSTRACTS OF PAPERS -- Alex Eskin

• A. Eskin, Z. Rudnick, and P. Sarnak. A proof of Siegel's weight formula.

  We compute the density of integer points on a hyperboloid in two ways. Comparing the results yields another proof of the Siegel wieght formula.

• A. Eskin and C. McMullen. Mixing, counting and equidistribution in Lie groups. Duke Math. J. 71(1993)

  We compute the asymptotics of the number of integer points on affine homogeneous varieties G/H, under the assumption that H is an affine symmetric subgroup of G. We use the Howe-Moore theorem and a certain geometric property of affine symmetric spaces.

• A. Eskin, S. Mozes, N. Shah. Unipotent flows and counting lattice points on homogeneous spaces. Annals of Math. 143(1996) 253-299.
  

  We study the density of integer points on affine homogeneos varieties G/H, such as the set of matrices with a given characteristic polynomial. Use of Ratner's theorem, the "linearization" technique of Dani and Margulis, and other techniques in unipotent flows allow us to relax the condition that H is an affine symmetric subgroup. Along the way we give a classifications of limits of translates of algebraic measures on locally symmetric spaces.

• A. Eskin, S. Mozes, N. Shah. Non-divergence of translates of certain algebraic measures. GAFA. 7(1997), pp. 48-80
  

  In this paper we investigate when the space of translates of an algebraic measure is relatively compact in the weak-star topology on a locally symmetric space. This is a technical result needed in the ``Unipotent flows....'' paper above. The result is closely related to the question of the the extent to which any finite dimensional representation irreducible over the rationals seperates subspaces.

• A. Eskin Y. Katznelson. Singular symmetric matrices. Duke Math. J. 79(1995)

  We compute the asymptotics as T tends to infinity, of the the number of integer symmetric matrices of determinant zero, and Hilbert-Schmidt norm less than T. We use elementary techniques in the geometry of numbers.

• A .Eskin, G. Margulis, S. Mozes. Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture. Annals of Math. 147 (1998), no. 1, 93--141
  

  The Oppenheim conjecture, proved by Margulis in 1986, states that the set of values at integral points of an indefinite quadratic form in three or more variables is dense, provided the form is not proportional to a rational form. In this paper we study the distribution of values of such a form. We show that if the signature of the form is not (2,1) or (2,2) then the values are uniformly distributed on the real line, provided the form is not proportional to a rational form. In the cases where the signature is (2,1) or (2,2) we show that no such universal formula exists, and give asymptotic upper bounds which are in general best possible.

• A. Eskin, B. Farb.   Quasi-flats and rigidity in symmetric spaces. Journal of the AMS 10(1997), pp. 653-692
  

  We use explicit geometric methods to show that the image of a (coarse) quasi-isometric embedding of n-dimesional Eucledian space into a symmetric space X of rank n is close to a finite union of flats. This fact generalizes Mostow's lemma in rank 1. We generalize this to maps of certain subsets of Eucledian space (flats with holes), and also give another proof of the theorem of Kleiner and Leeb on the quasi-isometric rigidity of higher-rank symmetric spaces.

• A. Eskin, B. Farb.   Quasi-flats in H2xH2. Preprint.

  The purpose of this paper is to exhibit most of the main ideas of the ``Quasi-flats...'' paper above in the familiar special case of a product of hyperbolic planes. We prove here that any (coarse) quasi-isometric embedding of 2 dimensional Eucledian space into a product of hyperbolic planes is close to a finite union of flats.

• A. Eskin.   Quasi-isometric rigidity of higher rank non-uniform lattices. Journal of the AMS. 11(1998), no. 2, pp. 321-361.

  We show that if L is a non-uniform lattice in a Lie group without rank 1 factors then any finitely generated group quasi-isometric to L is commensurable to L. The proof uses the result about ``Quasi-flats with holes'' proved in the ``Quasi-flats... '' paper as well as standard techniques in ergodic theory on semisimple groups.

• A. Eskin.   Counting problems and semisimple groups. Proc. of the ICM, Berlin, 1998.

  Some natural counting problems admit extra symmetries related to actions of Lie groups. For these problems, one can sometimes use ergodic and geometric methods, and in particular the theory of unipotent flows, to obtain asymptotic formulas. We will present counting problems related to diophantine equations, diophantine inequalities and quantum chaos, and also to the study of billiards on rational polygons.

• A. Eskin, H. Masur.   Pointwise asymptotic formulas on flat surfaces. Ergodic Theory Dynam. Systems 21 (2001), no. 2, 443--478.
  

  This paper concerns asymptotic formulas for the number of closed trajectories and saddle connections on a flat surface. It is inspired by some recent work of W. Veech, especially the recent paper "Siegel Measures". In particular, the analogy noted in "Siegel Measures" between the problem discussed in this paper and the quantitative version of the Oppenheim Conjecture allowed us to use some of the ideas developed by G.A. Margulis in the context of the Oppenheim conjecture to simplify and improve the quadratic upper bound of H. Masur for the number of saddle connections. This, together with an ergodic theorem proved for this purpose by A. Nevo, allowed us to find asymptotic quadratic growth rates for the number of closed trajectories on a generic surface. This sharpens results of W. Veech where these growth rates were found in the mean.

• A. Eskin, G.A. Margulis, S. Mozes.   Quadratic forms of signature (2,2) and eigenvalue spacings on flat 2-tori. To appear in Ann. Math.
  

  We study the problem mentioned in the paper "Upper bounds and asymptotics Quantatative version of the Oppenheim conjecture" in the exceptional case where the quadratic form has signature (2,2). We show that unless the quadratic form is extremely well approximated by a rational form, the values of the form are uniformly distributed, except for a point-mass at 0. This shows that the Berry-Tabor conjectures in the theory of quantum chaos hold for flat tori.

• A. Eskin, A. Okounkov.   Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. Invent. Math. 145 (2001), no. 1, 59--103.
  

  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.

• A. Eskin, S. Mozes, H. Oh. Uniform exponential growth for linear groups. Int. Math. Res. Not. 31(2002), 1675--1683.
  

  We announce the folowing result: Any finitely generated non virtually solvable linear group over a field of characteristic zero has uniform exponential growth. We show this by showing that for any such group $\Gamma$ there is an integer $N$ such that for any generating set for $\Gamma$ there exist two elements $A$ and $B$ which are words in the generators of length an most $N$, such that $A$ and $B$ generate a free semigroup.

• A. Eskin, S. Mozes, H. Oh. On Uniform exponential growth for linear groups. To appear in Invent. Math.
  

  This is the proof of the result announced above.
  

• A. Eskin, H. Oh. Integer points on a family of homogeneous varieties and unipotent flows. Preprint.

  We study a refined version of the Linnik problem on the asymptotic behavior of the number of representations of integer $m$ by an integral polynomial as $m$ tends to infinity. We assume that the polynomial arises from unvariant theory, and use methods from the theory of unipotent flows.

• A. Eskin, H. Oh. Ergodic theoretic proof of equidistribution of Hecke points. Preprint.
  

  We use the theory of unipotent flows to prove, in a general setting, the equidistribution of Hecke points. We follow the approach of Burger-Sarnak, and use a theorem of Mozes-Shah.

• A. Eskin, G. Margulis. Recurrence Properties of Random Walks on Finite Volume Homogeneous Manifolds. Preprint.

  We show that under some conditions on the defning measure, random walks on finite volume homogeneous manifolds have strong recurrence properties to compact sets. This behaviour is similar to the behaviour of unipotent flows.

• A. Eskin, H. Masur, M. Schmoll.   Billiards in rectangles with barriers. Duke Math. J. 118(2003), pp. 427--463
  

  We use Ratner's 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, and the height of the barrier is irrational. 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.

• A. Eskin, H. Masur, A. Zorich. Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems and the Siegel--Veech Constants. Publ. Math. Inst. Hautes Etudes Sci. 97 (2003), 61--179.
  
  A holomorphic 1-form on a compact Riemann surface $S$ naturally defines a flat metric on $S$ with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on $S$ having the same direction and the same length as the initial one. The similar phenomenon is valid for the families of parallel closed geodesics. We give a complete description of all possible configurations of parallel saddle connections (and of families of parallel closed geodesics) which might be found on a generic flat surface $S$. We count the number of saddle connections of length less than $L$ on a generic flat surface $S$; we also count the number of admissible configurations of pairs (triples,...) of saddle connections; we count the analogous numbers of configurations of families of closed geodesics. By the previous result of ~\cite{Eskin:Masur} these numbers have quadratic asymptotics $c \pi L^2$. Here we explicitly compute the constant $c$ for a configuration of every type. The constant $c$ is found from a Siegel--Veech formula. To perform this computation we elaborate the detailed description of the principal part of the boundary of the moduli space of holomorphic 1-forms and we find the numerical value of the normalized volume of the tubular neighborhood of the boundary. We use this for evaluation of integrals over the moduli space.
  
  
• A. Eskin, J. Markloff, D. Witte Morris. Unipotent flows on the space of branched covers of Veech surfaces. To appear in Ergodic Theory and Dynamical Systems.
  
  There is a natural action of $\SL(2,\real)$ on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup $U = \left\{ \begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix} \right\}$. We classify the $U$-invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the $\SL(2,\real)$-orbit of the set of branched covers of a fixed Veech surface.) For the $U$-action on these submanifolds, this is an analogue of Ratner's Theorem on unipotent flows. The result yields an asymptotic estimate of the number of periodic trajectories for billiards in a certain family of non-Veech rational triangles, namely, the isosceles triangles in which exactly one angle is $2 \pi/n$, with $n \ge 5$ and $n$ odd.
  
  
• A. Eskin, D. Fisher, K. Whyte. Quasi-isometries and rigidity of solvable groups. Preprint.
  
  In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to $R \ltimes R^n$ where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasi-isometries of the solvable Lie group. Our classification of self quasi-isometries for $R \ltimes R^n$ proves a conjecture made by Farb and Mosher in [FM4].
  
  Our techniques for studying quasi-isometries extend to some other classes of groups and spaces. In particular, we characterize groups quasi-isometric to any lamplighter group, answering a question of de la Harpe \cite{dlH}. Also, we prove that certain Diestel-Leader graphs are not quasi-isometric to any finitely generated group, verifying a conjecture of Diestel and Leader from [DL] and answering a question of Woess from [SW,Wo1]. We also prove that certain non-unimodular, non-hyperbolic solvable Lie groups are not quasi-isometric to finitely generated groups.
  
  The results in this paper are contributions to Gromov's program for classifying finitely generated groups up to quasi-isometry [Gr2]. We introduce a new technique for studying quasi-isometries, which we refer to as "coarse differentiation".