Divergence of Geodesics in Teichmüller Space and the Mapping Class Group
Divergence of Geodesics in Teichmüller Space and the Mapping Class Group
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Teichmüller空间中测地线的散度和映射类群
DOI：
10.1007/s00039-009-0017-3
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发表时间：
2006
影响因子：
2.2
通讯作者：
Kasra Rafi
中科院分区：
文献类型：
--
作者：
M. Duchin;Kasra Rafi
我们证明了Teichmüller空间（具有Teichmüller度量）和映射类群（具有字度量）都具有介于平坦空间的线性率和双曲空间的指数率之间的测地散度。对于Teichmüller空间中的每两条测地线，我们发现它们的发散至多是二次的。此外，这种估计是尖锐的，通过对射线的例子，正好二次发散。对于映射类组中的测地线射线，同样的陈述也是正确的。我们明确地描述了有效的路径“接近无穷大”在这两个空间。
We show that both Teichmüller space (with the Teichmüller metric) and the mapping class group (with a word metric) have geodesic divergence that is intermediate between the linear rate of flat spaces and the exponential rate of hyperbolic spaces. For every two geodesic rays in Teichmüller space, we find that their divergence is at most quadratic. Furthermore, this estimate is shown to be sharp via examples of pairs of rays with exactly quadratic divergence. The same statements are true for geodesic rays in the mapping class group. We explicitly describe efficient paths “near infinity” in both spaces.