Teichmuller theory of the punctured solenoid

A presentation for the baseleaf preserving mapping class group of the punctured solenoid

We give a presentation for the baseleaf preserving mapping class group $Mod(§)$ of the punctured solenoid $§$. The generators for our presentation were introduced previously, and several relations among them were derived. In addition, we show that $Mod(§)$ has no non-trivial central elements. Our main tool is a new complex of triangulations of the disk upon which $Mod(§)$ acts.

The outside of the Teichmuller space of punctured tori in Maskit's embedding

We consider the following question: Which parameters in the extension of a rational pleating ray across the boundary of $\Cal M$, the Maskit embedding of the Teichmüller space of once punctured tori correspond to a Kleinian group? Using methods of Keen and Series and Wright we prove a local result, stating that on each rational ray there is a sequence of parameters in $\overline\Bbb H\setminus\Cal M$ accumulating at the boundary point of $\Cal M$ on the ray. These are the unique parameters on the extended $p/q$ ray for which the special word $W_{p/q}$ is a primitive elliptic Möbius transformat…

Extremal maps of the universal hyperbolic solenoid

We show that the set of points in the Teichmuller space of the universal hyperbolic solenoid which do not have a Teichmuller extremal representative is generic (that is, its complement is the set of the first kind in the sense of Baire). This is in sharp contrast with the Teichmuller space of a Riemann surface where at least an open, dense subset has Teichmuller extremal representatives. In addition, we provide a sufficient criteria for the existence of Teichmuller extremal representatives in the given homotopy class. These results indicate that there is an interesting theory of extremal (and …

Pressure metrics and Manhattan curves for Teichmüller spaces of punctured surfaces

In this paper, we extend the construction of pressure metrics to Teichmüller spaces of surfaces with punctures. This construction recovers Thurston's Riemannian metric on Teichmüller spaces. Moreover, we prove the real analyticity and the convexity of Manhattan curves of the finite area type-preserving Fuchsian representations, and thus we obtain several related entropy rigidity results. Lastly, relating the two topics mentioned above, we show that one can derive the pressure metric by varying Manhattan curves.

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。

来自纤维化双曲3-流形群的有限商群的Taut多项式

Taut polynomials from finite quotients of fibered hyperbolic 3-manifold groups

本文证明，当单值映射为完全穿孔时，纤维化双曲3-流形群的有限商群能检测Thurston范数球的纤维化面的taut多项式，并发展了扭曲多变量Alexander多项式的profinite不变性的一般框架。