# Papers

## Rates of Mixing for the Weil-Petersson Geodesic Flow:Exponential Mixing in Exceptional Moduli spaces, to appear GAFA arxiv:1605.09037

(with Keith Burns, Carlos Matheus Santos, Amie Wilkinson)

Available as a pdf file

## Rate of Mixing for the Weil-Petersson Geodesic flow I:No rapid mixing in non-exceptional moduli space , to appear, Advances in Mathematics, arxiv 1312.6012s

(with Keith Burns, Carlos Matheus Santos, Amie Wilkinson)

Available as a pdf file

## Rigidity of Teichmuller space arxiv 1506.04774

### (with Alex Eskin, Kasra Rafi)

Available as a pdf file

## There exists an interval exchange with a non-ergodic generic measure arxiv 1410.1576

### (with Jon Chaika), Journal of Modern Dynamics, 2015

Available as a pdf file

## Limits in PMF of Teichmuller geodesics

### (with Jon Chaika, Mike Wolf, to appear Crelle Journal

Available as a pdf file

## Large Scale Rank of Teichmuller space arxiv: 1307.3733

### (with Alex Eskin, Kasra Rafi, to appear Duke Journal

Available as a pdf file

## Winning Games For Bounded geodesics in Teichmuller discs

### (with Jonathan Chaika, Yitwah Cheung Journal Modern Dynamics 2013 volume 7 395-428

Available as a pdf file

Abstract. We prove that for the surface defined by a holomorphic quadratic differential, the set of directions such that the corresponding Teichmuller geodesic lies in a compact set in the corresponding stratum is a winning set in Schmidt game. This generalizes a classical result in the case of the torus due to Schmidt and strengthens a result of Kleinbock and Weiss.

## Statistical hyperbolicity in Teichmuller space

### (with Spencer Dowdall, Moon Duchin) GAFA volume 24 (2014) 748-795 arXiv 1108.5416

Available as a pdf file

In this paper we explore the idea that Teichmuller space with the Teichmuller metric is hyperbolic on average." We consider several different measures on Teichm\"uller space and show that with respect to each one, the average distance between points in a ball of radius r is asymptotic to 2r, which is as large as possible.

## The Geometry of the Disc Complex

### (with Saul Schleimer) JAMS 26 (2013) 1-62 arXiv 1010.3174

Available as a pdf file

We give a distance estimate for the metric on the disk complex and show that it is Gromov hyperbolic. As another application of our techniques, we find an algorithm which computes the Hempel distance of a Heegaard splitting, up to an error depending only on the genus.

### (with Keith Burns, Amie Wilkinson), Annals of Math. 175 2012 835-908 arXiv 1004.5343

Available as a pdf file

In this paper we prove that the Weil-Petersson geodesic flow is ergodic on moduli space

## On train track splitting sequences

### (with Lee Mosher, Saul Schleimer), Duke Math. Journal 161 2012 1613-1656

Available as a pdf file

We show that the subsurface projection of a train track splitting sequence is an unparameterized quasi-geodesic in the curve complex of the subsurface. For the proof we introduce induced tracks, efficient position, and wide curves. This result is an important step in the proof that the disk complex is Gromov hyperbolic. As another application we show that train track sliding and splitting sequences give quasi-geodesics in the train track graph, generalizing a result of Hamenstadt

## Asymptotics of Weil-Petersson geodesics II:bounded geometry and unbounded entropy.

### (with Jeffrey Brock , Yair Minsky),to appear Geom Funct. Anal. arXiv 1004.4401

Available as a pdf file

We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil- Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equivalent condition for Weil-Petersson geodesics. As an application, we show the Weil-Petersson geodesic flow has compact invariant subsets with arbitrarily large topological entropy.

## Geometry of Teichmuller space with the Teichmuller metric

Available as a pdf file

This chapter is a survey of recent results in Teichmuller geometry

## Dichotomy for the Hausdorff dimension of the set of nonergodic directions.

### (with Yitwah Cheung, Pascal Hubert) Inventiones Math. (183) (2011) 337-383

Available as a pdf file

We consider billiards in a certain rectangle with a horizontal barrier This gives a one parameter family of flows in different directions. We study the Hausdorff dimension of the set of directions such that the flow in that direction is not ergodic. The dimension is computed explicitly in terms of the continued fraction expansion of the length of the barrier.

## Teichmuller geometry of moduli space, II: M(S) seen from far away

### (with Benson Farb) In the tradition of Ahlfors-Bers V 71-79 Contem Math. 510 American Math Soc. (2010)

Available as a pdf file

We construct a metric simplicial complex which is an almost isometric model of the moduli space M(S) of Riemann surfaces. We then use this model to compute the "tangent cone at infinity" of M(S): it is the topological cone on the quotient of the complex of curves C(S) by the mapping class group of S, endowed with an explicitly described metric. The main ingredient is Minsky's product regions theorem.

## Divergence of Teichmuller geodesics.

### (with Anna Lenzhen), Geom dedicata (2010) 114 191-210

Available as a pdf file

We study the asymptotic geometry of Teichm\"uller geodesic rays. The question of whether two rays through a given point stay bounded distance apart or not was settled except for one outstanding case. In this paper we settle that last case. We show that when the transverse measures to the vertical foliations of the quadratic differentials determining two different rays are topologically equivalent, but are not absolutely continuous with respect to each other, then the rays diverge in Teichm\"uller space.

## Teichmuller geometry of moduli space, I: Distance minimizing rays and the Deligne-Mumford compactification

### (with Benson Farb), Jour Diff Geom. (85) (2010) 187-227

Available as a pdf file

Let S be a closed, oriented surface with a finite (possibly empty) set of points removed. In this paper we relate two important but disparate topics in the study of the moduli space M(S) of Riemann surfaces: Teichm\"{u}ller geometry and the Deligne-Mumford compactification. We reconstruct the Deligne-Mumford compactification (as a metric stratified space) purely from the intrinsic metric geometry of M(S) endowed with the Teichm\"{u}ller metric. We do this by first classifying (globally) geodesic rays in M(S) and determining precisely how pairs of rays asymptote. We construct an iterated EDM ray space'' functor, which is defined on a quite general class of metric spaces. We then prove that this functor applied to M(S) produces the Deligne-Mumford compactification.

## Asymptotics of Weil-Petersson geodesics I: ending laminations, recurrence, and flows

### (with Jeffrey Brock, Yair Minsky), GAFA (19) (2010) 1229-1257

Available as a pdf file

We define an ending lamination for a Weil-Petersson geodesic ray. Despite the lack of a natural visual boundary for the Weil-Petersson metric, these ending laminations provide an effective boundary theory that encodes much of its asymptotic CAT(0) geometry. In particular, we prove an ending lamination theorem for the full-measure set of rays that recur to the thick part, and we show that the association of an ending lamination embeds asymptote classes of recurrent rays into the Gromov-boundary of the curve complex. As an application, we establish fundamentals of the topological dynamics of the Weil-Petersson geodesic flow, showing density of closed orbits and topological transitivity.

## Coarse and synthetic Weil-Petersson geometry: quasi-flats, geodesics, and relative hyperbolicity

### (with Jeffrey Brock), Geometry and Topology, (12) , 2008

This is available as a pdf file.

We analyze the coarse geometry of the Weil-Petersson metric on Teichm\"uller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil-Petersson metric via consideration of its coarse quasi-isometric model, the "pants graph." We show that in dimension~3 the pants graph is strongly relatively hyperbolic with respect to naturally defined product regions and show any quasi-flat lies a bounded distance from a single product. For all higher dimensions there is no non-trivial collection of subsets with respect to which it strongly relatively

## Topological dichotomy and strict ergodicity for translation surfaces

### (with Y.Cheung, P.Hubert ) Ergodic Theory Dynamical Systems {\bf 28} (2008) 1729-1748

This is available as a pdf file.

Hubert-Schmidt and McMullen have found examples of translation surfaces whose Veech group is infinitely generated. In this paper we show first that the Hubert-Schmidt examples satisfy the topological dichotomy property that for every direction either the flow in that direction is completely periodic or minimal. More significantly we show that they have minimal but non uniquely ergodic directions.

## Problems on flat surfaces and translation surfaces

### (with P.Hubert, T.Schmidt, A.Zorich)

This is a list of open problems in the subject and is available as a pdf file.

## Ergodic Theory of Translation surfaces

### to appear Handbook of Dynamical Systems, Elsevier

This survey is available as a pdf file.

## Minimal nonergodic directions on genus 2 translation surfaces

### (with Yitwah Cheung), to appear Ergodic Theory Dynamical Systems

The paper is available as a pdf file.

### Abstract

In this paper we show that every genus 2 translation surface which is not a Veech surface has a minimal direction which is not uniquely ergodic.

## A divergent Teichmuller geodesic with uniquely ergodic vertical foliation,

### Abstract

In this paper we construct an example of a quadratic differential whose vertical foliaiton is uniquely ergodic and yet the Teichmuller geodesci determined by the quadratic differential eventually leaves every compact set of moduli space.

## Multiple Saddle Connections on flat Surfaces and Principal Boundary of the Moduli Spaces of Quadratic Differentials

### Abstract

In this paper we consider the phenomenon of multiple homologous saddle connections on surfaces defined by quadratic differentials.

## The Pants Complex Has Only One End

### Abstract

In this paper we show that the pants complex of a closed surface of genus greater than $2$ has only one end.

### to appear, proceeding of Conference on Spaces of Kleinian groups London Math. Soc. Lec. Notes Cambridge University Press

The paper is available as a pdf file

## Quasiconvexity in the curve complex

### Abstract

In this paper we show that disc complex associated to a handlebody is a quasiconvex subset of the complex of curves.

Available as a postscript file

## Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems and the Siegel-Veech Constants .

### Abstract

In this paper we consider general counting problems for the number of saddle connections and cylinders of closed trajectories for Abelian differentials. Saddle connections and cylinders may occur with multiplicity. We discuss these issues and relate the constants to the Siegel-Veech formula. This is in turn is related to finding the principal boundary of the moduli space.

The paper is available as a Postscript file.

## Billiards in Rectangles with Barriers.

### Abstract

In this paper we consider a counting problem for closed orbits on a billiard table which is a rectangle with a barrier.

### Duke Mathematical Journal 118 427-463

The paper is available as a Postscript file.

### Abstract

In the paper we show that the isometry group of Teichmuller space with respect to the Weil-Petersson metric coincides with the mapping class group

### Geometriae Dedicata 93 177-190

Available as a as a dvi file.

## Rational billiards and flat structures

### to appear Handbook Dynamical Systems, Elsevier

This survey paper is available as a dvi file

## Asymptotic formulas on flat surfaces

### Erg. Th. Dyn. Sys. 21 443-478

The paper is available as a dvi file (137K).

## Unstable quasi-geodesics in Teichmuller space

### In the tradition of Ahlfors and Bers: Proceedings of the first Ahlfors-Bers Colloquium I.Kra, B.Maskit eds AMS Contemp Math. 256 (2000) 239-241

This paper is available as a dvi file

## Superrigidity and mapping class groups.

### Topology 37 1169-1176

The paper is available as a Postscript file (150K), or (without the figures) as a dvi file (35K).

## Geometry of the Complex of Curves I: Hyperbolicity

### Abstract

The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. It is not hard to see that the complex is finite-dimensional, but locally infinite. It was introduced by Harvey as an analogy, in the context of Teichmuller space, for Tits buildings for symmetric spaces, and has been studied by Harer and Ivanov as a tool for understanding mapping class groups of surfaces. In this paper we prove that, endowed with a natural metric, the complex is hyperbolic in the sense of Gromov.

In a certain sense this hyperbolicity is an explanation of why the Teichmuller space has some negative-curvature properties in spite of not being itself hyperbolic: Hyperbolicity in the Teichmuller space fails most obviously in the regions corresponding to surfaces where some curve is extremely short. The complex of curves exactly encodes the intersection patterns of this family of regions (it is the "nerve" of the family), and we show that its hyperbolicity means that the Teichmuller space is "relatively hyperbolic" with respect to this family. A similar relative hyperbolicity result is proved for the mapping class group of a surface.

The paper is available as a Postscript file (563K), or (without the figures) as a dvi file (180K).