Shears for quasisymmetric maps

Quasisymmetric maps, shears, lambda lengths and flips

We study quasisymmetric maps, which act on the boundary of the hyperbolic plane, by looking at their action on the Farey triangulation. Our main results identify exactly which quasisymmetric maps correspond to pinched lambda lengths in terms of shearing coordinates and, separately, in terms of (simultaneous) flip distance of the Farey triangulation and its image by the map. These extend and clarify previous results of Penner and Sullivan.

Circle homeomorphisms and shears

We give parameterizations of homeomorphisms, quasisymmetric maps and symmetric maps of the unit circle in terms of shear coordinates for the Farey tesselation.

Bounded Earthquakes

We give a short proof of the fact that bounded earthquakes of the unit disk induce quasisymmetric maps of the unit circle. By a similar method, we show that symmetric maps are induced by bounded earthquakes with asymptotically trivial measures.

通过剪切刻画开单位圆盘的渐近Teichmuller空间

Characterization of the asymptotic Teichmuller space of the open unit disk through shears

该研究通过单位圆盘法里镶嵌上拟对称同胚诱导的剪切函数等价类，给出开单位圆盘渐近Teichmuller空间的参数化，定义三种新度量并证明其拓扑与Teichmuller度量拓扑等价，同时纠正了相关文献的一处证明错误。

相对双曲空间之间及其边界之间的映射

Maps between relatively hyperbolic spaces and between their boundaries

本文建立相对双曲群/空间间尊重外围结构的拟等距嵌入与边界拟对称嵌入的对应，并推广至多多项式失真情形，据此刻画相对虚拟幂零子群双曲的群。