ABSTRACTS OF PAPERS -- Alex Eskin
We compute the density of integer points on a hyperboloid in two ways. Comparing the results yields another proof of the Siegel wieght formula.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
This is the proof of the result announced above.
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.
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.
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.
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.
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.
In this series of lectures, we describe some counting problems in moduli space and outline their connection to the dynamics of the SL(2,R) action on moduli space. Much of this is presented in analogy with the space of lattices SL(n,R)/SL(n,Z).
We prove that natural generating functions for enumeration of branched coverings of the pillowcase orbifold are level 2 quasimodular forms. This gives a way to compute the volumes of the strata of the moduli space of quadratic differentials.
We give a group-theoretic description of the parity of a pull-back of a theta characteristic under a branched covering. It involves lifting monodromy of the covering to the semidirect product of the symmetric and Clifford groups, known as the Sergeev group. As an application, we enumerate torus coverings with respect to their ramification and parity and, in particular, show that the corresponding all-degree generating functions are quasimodular forms.
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".
In this paper, we prove that certain spaces are not quasi-isometric to Cayley graphs of finitely generated groups. In particular, we answer a question of Woess and prove a conjecture of Diestel and Leader by showing that certain homogeneous graphs are not quasi-isometric to a Cayley graph of a finitely generated group. This paper is the first in a sequence of papers proving results announced in \cite{EFW0}. In particular, this paper contains many steps in the proofs of quasi-isometric rigidity of lattices in $\Solv$ and of the quasi-isometry classification of lamplighter groups. The proofs of those results are completed in \cite{EFW1}. The method used here is based on the idea of {\em coarse differentiation} introduced in \cite{EFW0}.
In this paper, which is the continuation of [EFW2], we complete the proof of the quasi-isometric rigidity of Sol and the lamplighter groups. The results were announced in [EFW1].
In this article we survey recent progress on quasi-isometric rigidity of polycyclic groups. These results are contributions to Gromov's program for classifying finitely generated groups up to quasi-isometry. The results discussed here rely on a new technique for studying quasi-isometries of finitely generated groups, which we refer to as coarse differentiation. We include a discussion of other applications of coarse differentiation to problems in geometric group theory and a comparison of coarse differentiation to other related techniques in nearby areas of mathematics.
This preliminary report contains a sketch of the proof of the following result: a slowly divergent Teichmuller geodesic satisfying a certain logarithmic law is determined by a uniquely ergodic measured foliation.
Masur showed that a Teichmuller geodesic that is recurrent in the moduli space of closed Riemann surfaces is necessarily determined by a quadratic differential with a uniquely ergodic vertical foliation. In this paper, we show that a divergent Teichmuller geodesic satisfying a certain slow rate of divergence is also necessarily determined by a quadratic differential with unique ergodic vertical foliation. As an application, we sketch a proof of a complete characterization of the set of nonergodic directions in any double cover of the flat torus branched over two points.
This is a series of lectures given at the Clay Institute Summer school in Pisa in 2007.
In this paper, we prove asymptotic formulas for the number of lattice points in a ball, and for the volume of a ball in Teichmuller space.
We compute the asymptotics, as R tends to infinity, of the number of closed geodesics in Moduli space of length at most R, or equivalently the number of pseudo-Anosov elements of the mapping class group of translation length at most R.
We compute the asymptotic growth rate of the number N(C, R) of closed geodesics of length less than R in a connected component C of a stratum of quadratic differentials. We prove that for any 0 < \theta < 1, the number of closed geodesics of length at most R that spend at least \theta-fraction of time outside of a compact subset of C is exponentially smaller than N(C, R). The theorem follows from a lattice counting statement. For points x, y in the moduli space M of Riemann surfaces, and for 0 < \theta < 1, we find an upper-bound for the number of geodesic paths of length less than R in C which connect a point near x to a point near y and spend a \theta-fraction of the time outside of a compact subset of C.
We provide partial results towards a conjectural generalization of a theorem of Lubotzky-Mozes-Raghunathan for arithmetic groups (over number fields or function fields) that implies, in low dimensions, both polynomial isoperimetric inequalities and finiteness properties. As a tool in our proof, we establish polynomial isoperimetric inequalities and finiteness properties for certain solvable groups that appear as subgroups of parabolic groups in semisimple groups, thus generalizing a theorem of Bux. We also develop a precise version of reduction theory for arithmetic groups whose proof is, for the most part, independent of whether the underlying global field is a number field or a function field.
A cyclic cover over the Riemann sphere branched at four points inherits a natural flat structure from the "pillow" flat structure on the basic sphere. We give an explicit formula for all individual Lyapunov exponents of the Hodge bundle over the corresponding arithmetic Teichmuller curve. The key technical element is evaluation of degrees of line subbundles of the Hodge bundle, corresponding to eigenspaces of the induced action of deck transformations.
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.
In this note we show that the results of H. Furstenberg on the Poisson boundary of lattices of semisimple Lie groups allow to deduce simplicity properties of the Lyapunov spectrum of the Kontsevich-Zorich cocycle of Teichmuller curves in moduli spaces of Abelian differentials without the usage of codings of the Teichmuller flow. As an application, we show the simplicity of some Lyapunov exponents in the setting of (some) Prym Teichmuller curves of genus $4$ where a coding-based approach seems hard to implement because of the poor knowledge of the Veech group of these Teichmuller curves.
Suppose N is an affine SL(2,R)-invariant submanfold of the moduli space of pairs (M,w) where M is a curve, and w is a holomorphic 1-form on M. We show that the Forni bundle of N (i.e. the maximal SL(2,R)-invariant isometric subbundle of the Hodge bundle of N) is always flat and is always orthogonal to the tangent space of N. As a corollary, it follows that the Hodge bundle of N is semisimple.
We use the relation between the volumes of the strata of meromorphic quadratic differentials with at most simple poles on $\mathbb{C}P^1$ and counting functions of the number of (bands of) closed geodesics in associated flat metrics with singularities to prove a very explicit formula for the volume of each such stratum conjectured by M. Kontsevich a decade ago. Applying ergodic techniques to the Teichmuller geodesic flow we obtain weak quadratic asymptotics for the number of (bands of) closed trajectories and for the number of generalized diagonals in almost all right-angled billiards.
We study the combinatorial geometry of "lattice" Jenkins--Strebel differentials with simple zeroes and simple poles on $\mathbb{C}P^1$ and of the corresponding counting functions. Developing the results of M. Kontsevich we evaluate the leading term of the symmetric polynomial counting the number of such "lattice" Jenkins-Strebel differentials having all zeroes on a single singular layer. This allows us to express the number of general "lattice" Jenkins-Strebel differentials as an appropriate weighted sum over decorated trees. The problem of counting Jenkins-Strebel differentials is equivalent to the problem of counting pillowcase covers, which serve as integer points in appropriate local coordinates on strata of moduli spaces of meromorphic quadratic differentials. This allows us to relate our counting problem to calculations of volumes of these strata . A very explicit expression for the volume of any stratum of meromorphic quadratic differentials recently obtained by the authors leads to an interesting combinatorial identity for our sums over trees.
We prove that the Birkhoff pointwise ergodic theorem and the Osceledets multiplicative ergodic theorem hold for every flat surface in almost every direction. The proofs rely on the strong law of large numbers, and on recent rigidity results for the action of the upper triangular subgroup of SL(2,R) on the moduli space of flat surfaces. Most of the results also use a theorem about continuity of splittings of the Kontsevich-Zorich cocycle recently proved by S. Filip.
We prove some ergodic-theoretic rigidity properties of the action of SL(2,R) on moduli space.
We prove results about orbit closures and equidistribution for the SL(2,R) action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows. The proofs of the main theorems rely on the measure classification theorem of [EMi2] and a certain isolation property of closed SL(2,\reals) invariant manifolds developed in this paper.
Let $X$ be quasi-isometric to either the mapping class group equipped with the word metric, or to Teichmuller space equipped with either the Teichmuller metric or the Weil-Petersson metric. We introduce a unified approach to study the coarse geometry of these spaces. We show that the quasi-Lipschitz image in $X$ of a box in $R^n$ is locally near a standard model of a flat in $X$. As a consequence, we show that, for all these spaces, the geometric rank and the topological rank are equal. The methods are axiomatic and apply to a larger class of metric spaces.
Let $G$ be a semisimple Lie group acting on a space $X$, let $\mu$ be a compactly supported measure on $G$, and let $A$ be a strongly irreducible linear cocycle over the action of $G$. We then have a random walk on $X$, and let $T$ be the associated shift map. We show that the cocycle $A$ over the action of $T$ is conjugate to a block conformal cocycle. This statement is used in the recent paper by Eskin-Mirzakhani on the classifications of invariant measures for the SL(2,R) action on moduli space. The ingredients of the proof are essentially contained in the papers of Guivarch and Raugi and also Goldsheid and Margulis.
We prove that the every quasi-isometry of Teichmuller space equipped with the Teichmuller metric is a bounded distance from an isometry of Teichmuller space. That is, Teichmuller space is quasi-isometrically rigid.
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.
We show that Sarnak's conjecture on Mobius disjointness holds for interval exchange transformations on three intervals (3-IETs) that satisfy a mild diophantine condition.
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 Teichmuller 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.
We compute the algebraic hull of the Kontsevich-Zorich cocycle over any GL(2,R) invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.