Fields of interests

Teichmüller theory, mapping class groups, geometric group theory, hyperbolic 3-manifolds.

Preprints Papers In preparation Curriculum Vitæ Research Statement Collaborators

Preprints

• Counting geodesics in a stratum.
  (With Alex Eskin and Maryam Mirzakhani) [pdf]
  We compute the asymptotic growth rate of the number N(C,R) of closed geodesics of length ≤ R in a connected component C of a stratum of quadratic differentials. We prove that for any 0 ≤ θ ≤ 1, the number of closed geodesics γ of length at most R such that γ spends at least θ–fraction of its 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(S) of Riemann surfaces, and for 0 ≤ θ ≤ 1 we find an upper-bound for the number of geodesic paths of length ≤ R in C which connect a point near x to a point near y and spend a θ–fraction of the time outside of a compact subset of C .
• On hyperbolicity of free splitting and free factor complexes.
  (With Ilya Kapovich) [pdf]
  We show how to derive hyperbolicity of the free factor complex of F_N from the Handel-Mosher proof of hyperbolicity of the free splitting complex of F_N, thus obtaining an alternative proof of a theorem of Bestvina-Feighn. We also show that the natural map from the free splitting complex to free factor complex sends geodesics to quasi-geodesics.
• Hyperbolicity in Teichmüller space. [pdf]
  We review and organize some results describing the behavior of a Teichmüller geodesic and draw several applications: 1) We show that Teichmüller geodesics do not back track. 2) We show that a Teichmüller geodesic segment whose endpoints are in the thick part has the fellow travelling property. This fails when the endpoints are not necessarily in the thick part. 3) We show that if an edge of a Teichmüller geodesic triangle passes through the thick part, then it is close to one of the other edges.
• Diameter of the thick part of moduli space.
  
  (With Jing Tao) [pdf]
  We study the shape of the moduli space of a surface of finite type. In particular, we compute the asymptotic behavior of the Teichmüller diameter of the thick part of the moduli space. We show that the diameter grows like logarithm of the genus. A simillar result is true for the outer space.

Papers (published or accepted)

• Bounded combinatorics and the Lipschitz metric on Teichmüller space.
  (With Anna Lenzhen and Jing Tao)
  To appear: Geometriae Dedicata. [pdf]
  Considering the Teichmüller space of a surface equipped with Thurston’s Lipschitz metric, we study geodesic segments whose endpoints have bounded combinatorics. We show that these geodesics are cobounded, and that the closest point projection to these geodesics is strongly contracting. Consequently, these geodesics are stable. Our main tool is to show that one can get a good estimate for the Lipschitz distance by considering the length ratio of finitely many curves.
• Length of a curve is quasi-convex along a Teichmüller geodesic.
  (With Anna Lenzhen)
  J. Differential Geom. 88 (2011), no. 2, 267–295. [pdf]
  We show that for every simple closed curve α, the extremal length and the hyperbolic length of α are quasi-convex functions along any Teichmüller geodesic. As a corollary, we conclude that, in Teichmüller space equipped with the Teichmüller metric, balls are quasi-convex.
• Curve complexes with connected boundary are rigid.
  (With Saul Schleimer)
  Duke Math. J. Volume 158, Number 2 (2011), 225-246. [pdf]
  When the boundary of the curve complex is connected any quasi-isometry is bounded distance from a simplicial automorphism. As a consequence, when the boundary is connected the quasi-isometry type of the curve complex determines the homeomorphism type of the surface.
• Grafting rays fellow travel Teichmüller geodesics.
  (With Young-Eun Choi and David Dumas)
  International Mathematics Research Notices (2011); doi: 10.1093/imrn/rnr104 [pdf]
  Given a measured geodesic lamination L on a hyperbolic surface S, grafting the surface along sL (s > 0) defines a one-parameter family gr( sL, S) of conformal structures in the Teichmüller space, called the grafting ray. We show that every grafting ray, after reparametrization, is a Teichmüller quasi-geodesic and stays in a bounded neighborhood of a Teichmüller geodesic. As part of our approach, we show that grafting rays have controlled dependence on the base. That is, there exists a constant K such that for Riemann surfaces R and S, the distance between gr(sL, R) and gr(sL, S) is less than K time the distance between R and S.
• Length spectra and degeneration of flat metrics
  (With Moon Duchin and Christopher Leininger)
  Invent. Math. 182 (2010), no. 2, 231–277. [pdf]
  In this paper we consider flat metrics (semi-translation structures) on surfaces of finite type. There are two main results. The first is a complete description of when a set of simple closed curves is spectrally rigid, that is, when the length vector determines a metric among the class of flat metrics. Secondly, we give an embedding into the space of geodesic currents and use this to get a boundary for the space of flat metrics. The geometric interpretation is that flat metrics degenerate to mixed structures on the surface: part flat metric and part measured foliation.
• Divergence rate of geodesics in Teichmüller space and mapping class groups (with Moon Duchin)
  Geom. Funct. Anal. 19 (2009), no. 3, 722--742. [pdf]
  We say a function f(R) is a divergence function for two geodesic rays in a metric space that share a basepoint if points on these rays that are distance R from the basepoint can be connected along a path that remain distance at least R from the basepoint and has length less than f(R). We show that every two geodesic rays in the Teichmüller space that share a basepoint have a quadratic divergence function. Furthermore, we show that this esmiate is sharp by providing examples where every divergence function is at least quadratic. The same is also true for geodesic rays in the mapping class group.
• Covers and the curve complex.
  (With Saul Schleimer)
  Geom. Topol. 13 (2009), no. 4, 2141--2162. [pdf]
  We provide the first non-trivial examples of quasi-isometric embeddings between curve complexes. These are induced either by puncturing a closed surface or via finite-sheeted coverings. As a corollary, we give new quasi-isometric embeddings between mapping class groups.
• Lines of minima and Teichmüller geodesics.
  (With Young-Eun Choi and Caroline Series)
  Geom. Funct. Anal. 18 (2008), no. 3, 698--754. [pdf]
  We characterize the curves that are short along a line of minima in Teichmüller space and estimate their lengths. We find that the short curves coincide with the curves that are short along the corresponding Teichmüller geodesic. By deriving additional information about the twisting parameters around the short curves, we estimate the Teichmüller distance between the line of minima and the Teichmüller geodesic. We deduce that this distance can be arbitrarily large, but that if S is a once-punctured torus or four-times-punctured sphere, the distance is uniformly bounded.
• Lines of minima are uniformly quasigeodesic.
  (With Young-Eun Choi and Caroline Series)
  Pacific J. Math. 237 (2008), no. 1, 21-44. [pdf]
  We use the results in "Lines of minima and Teichmüller geodesics" and "A combinatorial model for the Teichmüller metric" to prove that every line of minima is a quasi-geodesics in the Teichmüller metric.
• Comparison between Teichmüller and Lipschitz metrics.
  (With Young-Eun Choi)
  J. Lond. Math. Soc. (2) 76 (2007), no. 3, 739-756. [pdf]
  We study the Lipschitz metric on Teichmüller space (defined by Thurston) and compare it with the Teichmüller metric. We show that in the thin part of Teichmüller space the Lipschitz metric is approximated up to bounded additive distortion by the sup metric on a product of lower-dimensional spaces (similar to the Teichmüller metric as shown by Minsky), and in the thick part, the two metrics are comparable within additive error. However, these metrics are not comparable in general; we construct a sequence of pairs of points in Teichmüller space whose distances approach zero in the Lipschitz metric while they approach infinity in the Teichmüller metric.
• A combinatorial model for the Teichmüller metric.
  Geom. Funct. Anal. 17 (2007), no. 3, 936-959. [pdf]
  We study how the length and the twisting parameter of a curve change along a Teichmüller geodesic. We then use our results to provide a formula for the Teichmüller distance between two hyperbolic metrics on a surface, in terms of the combinatorial complexity of curves of bounded lengths in these two metrics.
• Thick-thin decomposition for quadratic differentials.
  Math. Res. Lett. 14 (2007), no 2, 333-341. [pdf]
  We provide a concrete description of the geometry of a quadratic differential metric; the surface is decomposed into thick pieces that are glued along flat annuli. Every thick piece Y of S has an scaling factor such that, for every essential curve in Y , the quadratic differential length of this curve is equal to the scaling factor times its hyperbolic length (up to multiplicative constants depending only on the topology of S).
• A charaterization of short curves of a geodesic in Teichmüller space.
  Geom. Topol. 9 (2005) 179-202. [pdf] [ps]
  We provide a combinatorial condition characterizing curves that are short along a Teichmüller geodesic. This condition is closely related to the condition provided by Minsky for curves in a hyperbolic 3-manifold to be short. We show that short curves in a hyperbolic manifold homeomorphic to SxR are also short in the corresponding Teichmüller geodesic, and we provide examples demonstrating that the converse is not true.

In Preparation

• Coarse differentiation and the rank of Teichmüller space.
  (With Alex Eskin and Howard Masur)
  We give a local characterization flats in in Teichmüller space using the coarse differentiation techniques developed by Eskin, fisher and Whyte. As a corollary, we compute the rank of Teichmüller space.
• Essential tori and Dehn twists in Outer space.
  (With Matt Clay)
• Shadow of a Lipschitz geodesic in the curve comple.
  (With Anna Lenzhen and Jing Tao)
• Closed geodesics in the thin part of moduli space.
  For a surface S of genus g with p punctures, 3g + p > 4, we provide examples of closed geodesics in the moduli space that stay completely in the ε-thin part of moduli space. We provide a lower bound of ε Exp( (6g-7+2p) L ) for the number of such geodesics of length less than L and, in the case five-times-punctured torus, we provide an upper-bound of Exp( (3 + O(ε) ) L ).
• Teichmüller geodesics with non-uniquely ergodic vertical foliations.
  We construct examples of minimal non-uniquely ergodic foliations such that the limit set of the corresponding Teichmüller geodesic is equal to the entire simplex of measures supported on that foliation (examples for non-minimal foliations were originally produced by Lenzhen). We also examine the combinatorial properties of the vertical foliations of such geodesics and we show that, for all such examples, there is an infinite sequence of curves where the twisting coefficient of the vertical foliation around these curves goes to infinity.

Collaborators

Alex Eskin Anna Lenzhen Caroline Series Chris Leininger David Dumas Howard Masur Ilya Kapovich Jing Tao Maryam Mirzakhani Moon Duchin Saul Schleimer Young-Eun Choi