Coarse and synthetic Weil–Petersson geometry: quasi-flats, geodesics and relative hyperbolicity
Coarse and synthetic Weil–Petersson geometry: quasi-flats, geodesics and relative hyperbolicity
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DOI：
10.2140/gt.2008.12.2453
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发表时间：
2007-07
影响因子：
2
通讯作者：
Jeffrey F. Brock;H. Masur
中科院分区：
文献类型：
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作者：
Jeffrey F. Brock;H. Masur
我们分析了Teichmümuller空间上Weil-Petersson度量的粗糙几何，着重于其在综合几何中的应用（特别是测地线的行为）。我们通过考虑Weil-Petersson度量的粗拟等距模型Pants图，解决了Weil-Petersson度量的强相对双曲性问题。我们表明，在3维的裤子图是强相对双曲自然定义的产品区域，并显示任何准平坦的谎言从一个单一的产品有界距离。对于所有更高的维度，没有非平凡的子集集合，关于它强烈相对双曲;这将[BDM]在6维和更高维度的定理扩展到中间范围（它是双曲的当且仅当维度是1或2 [BF]）。通过3维准测地线的稳定性和相对稳定性提供了对测地线行为的深刻理解，并通过曲线层次及其相关的边界层完整描述了Weil-Petersson度量的CAT 0边界。
We analyze the coarse geometry of the Weil-Petersson metric on Teichm¨ uller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil-Petersson metric via consideration of its coarse quasi-isometric model, the pants graph. We show that in dimension 3 the pants graph is strongly relatively hyperbolic with respect to naturally defined product regions and show any quasi-flat lies a bounded distance from a single product. For all higher dimensions there is no non-trivial collection of subsets with respect to which it strongly relatively hyperbolic; this extends a theorem of [BDM] in dimension 6 and higher into the intermediate range (it is hyperbolic if and only if the dimension is 1 or 2 [BF]). Stability and relative stability of quasi-geodesics in dimensions up through 3 provide for a strong understanding of the behavior of geodesics and a complete description of the CAT 0 -boundary of the Weil-Petersson metric via curve-hierarchies and their associated boundary laminations.