论文标题
谎言组和粗几何二分法的庞加莱概况
Poincaré profiles of Lie groups and a coarse geometric dichotomy
论文作者
论文摘要
庞加莱概况是一个分析定义的粗不变式的家族，可以用作公制空间之间粗嵌入的障碍物。在本文中，我们计算了所有连接的非模样的谎言组，鲍姆斯拉格 - 统计组和瑟斯顿几何形状的庞加莱概况，展示了两种实质上不同类型的行为。就谎言组而言，我们获得了二分法，该二分法延伸了二分法，将等级的半岛半散谎言组和二分法分开，将二分法分隔为多项式生长和指数级生长的可溶解的可解决的单型谎言基团。我们提供了该二分法的等效代数，准等级和粗几何表述。 Our results have many consequences for coarse embeddings, for instance we deduce that for groups of the form $N\times S$, where $N$ is a connected nilpotent Lie group, and $S$ is a simple Lie group of real rank 1, both the growth exponent of $N$, and the Ahlfors-regular conformal dimension of $S$ are non-decreasing under coarse embeddings.即使在准静电设置中，这些结果也是新的，并给予准静电嵌入的障碍物，在许多情况下，这些嵌入比Buyalo-Schroeder先前获得的嵌入更强。
Poincaré profiles are a family of analytically defined coarse invariants, which can be used as obstructions to the existence of coarse embeddings between metric spaces. In this paper we calculate the Poincaré profiles of all connected unimodular Lie groups, Baumslag-Solitar groups and Thurston geometries, demonstrating two substantially different types of behaviour. In the case of Lie groups, we obtain a dichotomy which extends both the dichotomy separating rank one and higher rank semisimple Lie groups and the dichotomy separating connected solvable unimodular Lie groups of polynomial and exponential growth. We provide equivalent algebraic, quasi-isometric and coarse geometric formulations of this dichotomy. Our results have many consequences for coarse embeddings, for instance we deduce that for groups of the form $N\times S$, where $N$ is a connected nilpotent Lie group, and $S$ is a simple Lie group of real rank 1, both the growth exponent of $N$, and the Ahlfors-regular conformal dimension of $S$ are non-decreasing under coarse embeddings. These results are new even in the quasi-isometric setting and give obstructions to quasi-isometric embeddings which in many cases are stronger than those previously obtained by Buyalo-Schroeder.
