Teichmüller theory, mapping class groups, hyperbolic 3-manifolds,
geometric group theory.
Papers
Published/Accepted for publication
Lines of minima and Teichmüller geodesics.
( PDF )
(with Young-Eun Choi and Caroline Series)
To appear, Geometric and Functional Analysis
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.
Comparison between Teichmüller and Lipschitz metrics.
( PDF )
(with Young-Eun Choi) To appear, Journal of the London Mathematical Society
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.
( PDF )
To appear, Geometric and Functional Analysis
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.
( PDF )
Math. Res. Lett. 14 (2007), no 2, 333--341.
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,
( PDF , PS )
Geom. Topol. 9 (2005) 179--202.
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.
Preprints
Lines of minima are quasi-geodesics. ( PDF )
(with Young-Eun Choi and Caroline Series)
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.
On distinguishing curve complexes.
( PDF )
(with Saul Schleimer)
We show that if two surfaces have curve complexes that are one-ended and
quasi-isometric then these surfaces have equal complexity.
Covers and the curve complex.
( PDF )
(with Saul Schleimer)
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.
We show, for any geodesic G in the Teichmüller space of a surface S
and any subsurface of Y of S, that the projection of G to the complex of curves
of Y is an un-parametrize quasi-geodesic (that is, it always moves forward).
It is interesting to note that this is not true for the geodesics in the
Lipschitz metric on the Teichmüller space.
Divergence rate of geodesics in Teichmüller space and mapping class groups.
( PDF )
(with Moon Duchin)
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.
In Preparation
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 ).
Grafting lines fellow travel a Teichmüller geodesic.
(with Young-Eun Choi)
Given a measured geodesic lamination L on a hyperbolic surface S, grafting the surface along
tL (t > 0) defines a 1-parameter family F( t, L, S) of conformal structures in the
Teichmüller space of S. We show that there is a Teichmüller
geodesic ray which stays a bounded distance from F( t, L, S).
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.