李群的渐近锥面与超幂
Asymptotic cones and ultrapowers of Lie groups
摘要：
度量空间的渐近锥面最早是由格罗莫夫发明的。它们是度量空间，它捕获了底层度量空间的“大尺度结构”。后来，范登德里斯和威尔基给出了一个更一般的建设渐近锥面使用ultrapowers。关于渐近锥面的某些事实，如度量空间的完备性，现在可以很容易地从超幂的饱和性质中得出，在本综述中，我们想介绍范登德里斯-威尔基方法的两个应用。利用超幂，我们得到了半单李群的渐近锥面的一个明确的描述。从这一描述出发，利用半代数群和非标准方法，我们可以给出马古利斯猜想的一个简短证明。在第二个应用中，我们使用集合论来回答格罗莫夫的一个问题。
英文摘要：
Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the 'large-scale structure' of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the completeness of the metric space, now follow rather easily from saturation properties of ultrapowers, and in this survey, we want to present two applications of the van den Dries-Wilkie approach. Using ultrapowers we obtain an explicit description of the asymptotic cone of a semisimple Lie group. From this description, using semi-algebraic groups and non-standard methods, we can give a short proof of the Margulis Conjecture. In a second application, we use set theory to answer a question of Gromov.
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