Large‐scale rank and rigidity of the Teichmüller metric
Large‐scale rank and rigidity of the Teichmüller metric
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DOI：
10.1112/jtopol/jtw017
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发表时间：
2016-12
影响因子：
1.1
通讯作者：
B. Bowditch
中科院分区：
文献类型：
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作者：
B. Bowditch
我们研究紧致可定向曲面的泰希米勒空间在泰希米勒度量下的粗几何。我们描述了它何时允许欧几里得空间或欧几里得半空间的拟等距嵌入。我们证明了复杂度至少为2的曲面的泰希米勒空间的拟等距刚性：这是由埃斯金、马苏尔和拉菲独立证明的一个结果。我们推断出，除了一些众所周知的巧合情况外，泰希米勒空间是拟等距不同的。我们还表明泰希米勒空间满足二次等周不等式。证明这些结果的一个关键要素是泰希米勒空间允许一种三元运算，在有界距离内是自然的，这赋予该空间一个粗中位数空间的结构，其秩等于曲面的复杂度。由此，还可以推断出任何渐近锥都与一个CAT(0)空间是双利普希茨等价的，因此，特别地，是可缩的。
We study the coarse geometry of the Teichmüller space of a compact orientable surface in the Teichmüller metric. We describe when this admits a quasi‐isometric embedding of a euclidean space, or a euclidean half‐space. We prove quasi‐isometric rigidity for Teichmüller space of a surface of complexity at least 2: a result proved independently by Eskin, Masur and Rafi. We deduce that, apart from some well‐known coincidences, the Teichmüller spaces are quasi‐isometrically distinct. We also show that Teichmüller space satisfies a quadratic isoperimetric inequality. A key ingredient for proving these results is the fact that Teichmüller space admits a ternary operation, natural up to bounded distance, which endows the space with the structure of a coarse median space whose rank is equal to the complexity of the surface. From this, one can also deduce that any asymptotic cone is bilipschitz equivalent to a CAT(0) space, and so, in particular, is contractible.