This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the complete classification of such groups up to quasi-isometries and we compare the quasi-isometric classification with the bi-Lipschitz classification. On the other hand, we study the problem whether two quasi-isometrically equivalent Lie groups may be made isometric if equipped with suitable left-invariant Riemannian metrics. We show that this is the case for three-dimensional simply connected groups, but it is not true in general for multiply connected groups. The counterexample also demonstrates that `may be made isometric' is not a transitive relation.

中文翻译：

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关于3D黎曼李群的拟等距和双Lipschitz分类

本笔记涉及维数为 3 或以下的连通李群的几何分类，赋予左不变黎曼度量。一方面，从文献中收集结果，我们对此类组的完整分类进行了回顾，直到准等距分类，并将准等距分类与双利普希茨分类进行了比较。另一方面，我们研究了如果配备合适的左不变黎曼度量，两个准等距等距的李群是否可以成为等距的问题。我们表明这是三维单连通群的情况，但对于多连通群通常不是这样。反例还表明“可以制成等距”不是传递关系。