Morse Boundaries of Proper Geodesic Metric Spaces
Morse Boundaries of Proper Geodesic Metric Spaces
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真测地线度量空间的莫尔斯边界
DOI：
10.4171/ggd/429
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Matthew Cordes
中科院分区：
文献类型：
--
作者：
Matthew Cordes
我们为恰当的测地空间引入一种新型边界，称为莫尔斯边界，它由识别该空间中“双曲方向”的射线构成。这个边界是一个拟等距不变量，因此为任何有限生成群产生一个定义良好的边界。在恰当的$\mathrm{CAT}(0)$空间的情形下，这个边界是查尼（Charney）和苏丹（Sultan）的收缩边界；在恰当的格罗莫夫（Gromov）双曲空间的情形下，这个边界是格罗莫夫边界。我们证明了关于泰希米勒（Teichmüller）空间的莫尔斯边界的三个结果。首先，我们表明一个曲面的映射类群的莫尔斯边界与该曲面的泰希米勒空间的莫尔斯边界是同胚的。其次，利用莱宁格（Leininger）和施莱默（Schleimer）的一个结果，我们表明泰希米勒空间的莫尔斯边界可以包含任意高维的球面。最后，我们表明存在一个从泰希米勒空间的莫尔斯边界到由射影测度叶状结构给出的泰希米勒空间的瑟斯顿（Thurston）紧化的单射连续映射。
We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the "hyperbolic directions" in that space. This boundary is a quasi-isometry invariant and thus produces a well-defined boundary for any finitely generated group. In the case of a proper $\mathrm{CAT}(0)$ space this boundary is the contracting boundary of Charney and Sultan and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. We prove three results about the Morse boundary of Teichm\"uller space. First, we show that the Morse boundary of the mapping class group of a surface is homeomorphic to the Morse boundary of the Teichm\"uller space of that surface. Second, using a result of Leininger and Schleimer, we show that Morse boundaries of Teichm\"uller space can contain spheres of arbitrarily high dimension. Finally, we show that there is an injective continuous map of the Morse boundary of Teichm\"uller space into the Thurston compactification of Teichm\"uller space by projective measured foliations.
Teichmüller 空间中的双曲空间
影响因子：
2.6
作者：
Leininger C
通讯作者：
Leininger C