论文标题
强烈的快捷空间
Strongly Shortcut Spaces
论文作者
论文摘要
我们为粗糙的大地测量空间定义了强的快捷属性，从而推广了强捷径的概念。我们表明，强大的快捷属性是一个粗略的相似性。我们给出了强大捷径特性的几个新特征，包括渐近锥体表征。我们使用这种表征来证明渐近的CAT（0）空间是强烈的捷径。我们证明，如果一个组在强烈的捷径粗糙的测量公制空间上正确且均匀地进行指标，那么它具有强烈的快捷方式Cayley图，因此非常强烈的快捷键组也是如此。因此，我们表明CAT（0）组是强烈的捷径。 为了证明这些结果，我们使用了几个中间结果，我们认为这可能引起独立的兴趣，包括我们称之为圆圈拧紧引理和精细的Milnor-Schwarz引理。圆圈拧紧的引理描述了如何通过从一个圆圈上的粗糙Lipschitz地图上进行手术，从而从一个圆圈中进行手术，从而使圆圈的圆圈嵌入了圆圈，该圆圈将距离分开的距离对抗焦点对。精细的Milnor-Schwarz引理是Milnor-Schwarz引理的改进，可以更好地控制从一组到其作用的空间的准等级分析的乘法常数。
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.
