The quantum Teichmuller space as a noncommutative algebraic object

Visualizing the unit ball for the Teichmüller metric

We describe a method to compute the norm on the cotangent space to the moduli space of Riemann surfaces associated to the Finsler Teichmüller metric. Our method involves computing the periods of abelian double covers and is easy to implement for Riemann surfaces presented as algebraic curves using existing tools for approximating period matrices of plane algebraic curves. We illustrate our method by depicting the unit sphere in the cotangent space to moduli space at a particular surface of genus zero with five punctures and by corroborating the proof of a theorem of Royden's for our example.

Teichmüller空间、多项式轨迹与代数对应空间中的退化

Teichmüller spaces, polynomial loci, and degeneration in spaces of algebraic correspondences

该研究在代数对应语境下提出特征簇类似物，发现富克斯群与多项式的交配包含其中，通过固定多项式或富克斯群得到类Bers切片，证明其有界性并启动紧化研究，揭示新退化现象且对四次穿孔球面证明紧化自然同胚。

Amoebas, tropical varieties and compactification of Teichmuller spaces

In this paper we try to look at the compactification of Teichmuller spaces from a tropical viewpoint. We describe a general construction for the compactification of algebraic varieties, using their amoebas, and we describe the boundary via tropical varieties. When we apply this construction to the Teichmuller spaces we see that they can be mapped in a real algebraic hypersurface in such a way that the cone over the boundary is a subpolyhedron of a tropical hypersurface. We want to show how some properties of the boundary becomes straightforward if looked at from this point of view. For example…

On Grothendieck's construction of Teichmüller space

In his 1944 paper Veränderliche Riemannsche Flächen , Teichmüller defined a structure of complex manifold on the set of isomorphism classes of marked closed Riemann surfaces of genus g. The complex manifold he obtained is the space called today Teichmüller space. In the same paper, Teichmüller introduced the so-called universal Teichmüller curve -- a space over Teichmüller space where the fiber above each point is a Riemann surface representing that point. In fact, Teichmüller proved the existence of the Teichmüller curve as a space of Riemann surfaces parametrized by an analytic space, with a…

Euler characteristics of Teichmüller curves in genus two

We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmuller curve is proportional to the Euler characteristic of the Hilbert modular surface on which it lies. The idea is to use techniques from algebraic geometry to calculate the fundamental classes of these Teichmuller curves in certain compactifications of the Hilbert modular surfa…

同时均匀化与代数对应

Simultaneous Uniformization and Algebraic Correspondences

本文将Bers同时均匀化定理推广到代数对应框架，构造了同时均匀化一对非拓扑同胚的零亏格轨道形的代数对应，并给出穿孔球面Teichmüller空间的复解析实现。

The embedding of the space of negatively curved surfaces in geodesic currents

We prove by an algebraic method that the embedding of the Teichmuller space in the space of geodesic currents is totally linearly independent. We prove a similar result for all negatively curved surfaces using an ergodic argument.