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From Hierarchical to Relative Hyperbolicity
International Mathematics Research Notices ( IF 0.9 ) Pub Date : 2020-07-13 , DOI: 10.1093/imrn/rnaa141
Jacob Russell 1
Affiliation  

We provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, this criteria characterizes relative hyperbolicity. We apply our criteria to graphs associated to surfaces and prove that the separating curve graph of a surface is relatively hyperbolic when the surface has zero or two punctures. We also recover a celebrated theorem of Brock and Masur on the relative hyperbolicity of the Weil-Petersson metric on Teichmuller space for surfaces with complexity three.

中文翻译:

从分层到相对双曲线

我们通过证明层次结构方面的相对双曲性的新公式,为层次双曲空间提供了一个简单的组合标准,使其成为相对双曲的。在干净的分层双曲线组的情况下,该标准表征相对双曲线。我们将我们的标准应用于与曲面相关的图形,并证明当曲面具有零个或两个穿孔时,曲面的分离曲线图是相对双曲线的。我们还恢复了一个著名的 Brock 和 Masur 定理,该定理关于复杂度为 3 的表面的 Teichmuller 空间上的 Weil-Petersson 度量的相对双曲性。
更新日期:2020-07-13
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