标题： 强快捷空间 显示英文标题

标题： Strongly Shortcut Spaces

摘要： 我们为粗糙测地度量空间定义了强捷径性质，这是对强捷径图概念的推广。 我们证明了强捷径性质是一个粗糙相似不变量。 我们给出了强捷径性质的几个新特征，包括一个渐近锥特征。 我们利用这个特征证明了渐近CAT(0)空间是强捷径的。 我们证明，如果一个群在强捷径粗糙测地度量空间上进行度量适当且共紧的作用，那么它有一个强捷径Cayley图，因此是一个强捷径群。 因此我们证明了CAT(0)群是强捷径的。 为了证明这些结果，我们使用了一些中间结果，我们认为这些结果可能有独立的兴趣，包括我们所谓的圆收紧引理和精细Milnor-Schwarz引理。 圆收紧引理描述了如何通过对手一个粗糙Lipschitz映射进行手术来获得一个圆的拟等距嵌入，该映射将圆上的对径点映射得足够远。 精细Milnor-Schwarz引理是对Milnor-Schwarz引理的改进，它对群作用在空间上的拟等距的乘法常数提供了更精细的控制。 显示英文摘要

摘要： We define the strong shortcut property for rough geodesic metric spaces, generalizing the notion of strongly shortcut graphs. We show that the strong shortcut property is a rough similarity invariant. We give several new characterizations of the strong shortcut property, including an asymptotic cone characterization. We use this characterization to prove that asymptotically CAT(0) spaces are strongly shortcut. We prove that if a group acts metrically properly and coboundedly on a strongly shortcut rough geodesic metric space then it has a strongly shortcut Cayley graph and so is a strongly shortcut group. Thus we show that CAT(0) groups are strongly shortcut. To prove these results, we use several intermediate results which we believe may be of independent interest, including what we call the Circle Tightening Lemma and the Fine Milnor-Schwarz Lemma. The Circle Tightening Lemma describes how one may obtain a quasi-isometric embedding of a circle by performing surgery on a rough Lipschitz map from a circle that sends antipodal pairs of points far enough apart. The Fine Milnor-Schwarz Lemma is a refinement of the Milnor-Schwarz Lemma that gives finer control on the multiplicative constant of the quasi-isometry from a group to a space it acts on.