Papers
Divergence of Teichmuller geodesics.
(with Anna Lenzhen)
Available as a pdf
file
Teichmuller geometry of moduli space, I:
Distance minimizing rays and the Deligne-Mumford compactification
(with Benson Farb)
Available as a pdf
file
Asymptotics of Weil-Petersson geodesics I:
ending laminations, recurrence, and flows
(with Jeffrey Brock, Yair Minsky)
Available as a pdf
file
Coarse and synthetic Weil-Petersson geometry: quasi-flats, geodesics, and relative hyperbolicity
(with Jeffrey Brock)
This is available as a pdf
file.
Topological dichotomy and strict ergodicity for translation surfaces
(with Y.Cheung, P.Hubert )
This is available as a pdf
file.
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,
(with Y.Cheung)
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.
to appear, Israel Journal of Mathematics
Available as a pdf
file
Multiple Saddle Connections on flat Surfaces and Principal
Boundary of the Moduli Spaces of Quadratic Differentials
(with A.Zorich)
Abstract
In this paper we consider the phenomenon of multiple homologous saddle connections on surfaces defined by quadratic differentials.
A current draft of the paper is available as a pdf
file
The Pants Complex Has Only One End
(with S.Schleimer)
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
(with Y.Minsky)
Contemporary Mathematics 355 309-320
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 .
(with Alex Eskin, Anton Zorich)
Publications IHES 97 61-179
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.
(with Alex Eskin, Martin Schmoll)
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.
Weil-Petersson isometry group.
(with Mike Wolf)
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
(with S. Tabachnikov)
to appear Handbook Dynamical Systems, Elsevier
This survey paper is available as a dvi file
Asymptotic formulas on flat surfaces
(with Alex Eskin)
Erg. Th. Dyn. Sys. 21 443-478
The paper is available as a dvi file (137K).
Unstable quasi-geodesics in Teichmuller space
(with Yair Minsky)
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.
(with Benson Farb)
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
(with Yair Minsky)
Invent.Math 138 (1999) 103-149
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).
Geometry of the Complex of Curves II: Heirarchical Structure
(with Yair Minsky)
to appear, GAFA
The paper (November 2000) is available as a
Postscript file (1180K).
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