arXiv subjects
Publications and source records attributed to Anna Lenzhen.
We study limit sets of Teichm\"uller disks in the Thurston boundary of Teichm\"uller space of a closed surface S of genus at least 2. It is well known that almost every Teichm\"uller geodesic ray converges to a point on the boundary. We show that unlike rays, Teichm\"uller disks with smallest possible limit sets are extremely rare.
We develop the notion of the active interval for a subsurface along a geodesic in the Thurston metric on Teichmuller space of a surface S. That is, for any geodesic in the Thurston metric and any subsurface R of S, we find an interval of times where the length of the boundary of R is uniformly bounded and the restriction of the geodesic to the subsurface R resembles a geodesic in the Teichmuller space of R. In particular, the set of short curves in R during the active interval represents a reparametrized quasi-geodesic in the curve graph of R (no backtracking) and the amount of movement in the curve graph of R outside of the active interval is uniformly bounded which justifies the name active interval. These intervals provide an analogue of the active intervals introduced by the third author in the setting of Teichmuller space equipped with the Teichmuller metric.
We study the geometry of the Thurston metric on the Teichmüller space $\mathcal{T}(S)$ of hyperbolic structures on a surface $S$. Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite type; however, we focus particular attention on the case where the surface is a once-punctured torus, $S_{1,1}$. In that case, our results provide a detailed picture of the infinitesimal, local, and global behavior of the geodesics of the Thurston metric, as well as an analogue of Royden's theorem.
We construct an example of a Teichmueller geodesic ray whose limit set in Thurston boundary of Teichmueller space is a d-dimensional simplex.
We prove that there are finite area flat surfaces whose Veech group is an infinite cyclic group consisting of hyperbolic elements
We study the geometry of the Thurston metric on Teichmuller space by examining its geodesics and comparing them to Teichmuller geodesics. We show that, similar to a Teichmuller geodesic, the shadow of a Thurston geodesic to the curve graph is a reparametrized quasi-geodesic. However, we show that the set of short curves along the two geodesics are not identical.
Suppose that $Σ$ is a hyperbolic surface and $f:\mathbb R_+\to\mathbb R_+$ a monotonic function. We study the closure in the projective tangent bundle $PTΣ$ of the set of all geodesics $γ$ satisfying $I(γ,γ)\leq f(\ell_Σ(γ))$. For instance we prove that if $f$ is unbounded and sublinear then this set has Hausdorff dimension strictly bounded between 1 and 3.
We describe a method for constructing Teichmüller geodesics where the vertical measured foliation $ν$ is minimal but is not uniquely ergodic and where we have a good understanding of the behavior of the Teichmüller geodesic. The construction depends on various parameters, and we show that one can adjust the parameters to ensure that the set of accumulation points of such a geodesic in the Thurston boundary is exactly the set of all possible measured foliations in the homotopy class of $ν$. With further adjustment of the parameters, one can even take $ν$ to be an ergodic measure on a non-uniquely ergodic foliation.
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.
The Hurwitz problem of composition of quadratic forms, or of "sum of squares identity" is tackled with the help of a particular class of $(\mathbb{Z}_2)^n$-graded non-associative algebras generalizing the octonions. This method provides an explicit formula for the classical Hurwitz-Radon identity and leads to new solutions in a neighborhood of the Hurwitz-Radon identity.
We study the asymptotic geometry of Teichmueller geodesic rays. 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 Teichmueller space.
We construct a Teichmuller geodesic which does not have a limit on the Thurston boundary of the Teichmuller space.
We show that for every simple closed curve α, the extremal length and the hyperbolic length of αare quasi-convex functions along any Teichmuller geodesic. As a corollary, we conclude that, in Teichmuller space equipped with the Teichmuller metric, balls are quasi- convex.