李群的 Poincaré 剖面与一种粗几何二分法
Poincaré profiles of Lie groups and a coarse geometric dichotomy
AI总结：
本文计算连通幺模李群、Baumslag–Solitar 群和 Thurston 几何的 Poincaré 剖面，建立代数、拟等距与粗几何等价的二分法，并推出粗嵌入及拟等距嵌入的新障碍。
摘要：
Poincaré 剖面是一族通过解析方式定义的粗不变量，可作为度量空间之间存在粗嵌入的障碍。本文计算了所有连通幺模李群、Baumslag–Solitar 群以及 Thurston 几何的 Poincaré 剖面，展示了两种本质不同的行为类型。在李群情形中，我们得到一种二分法，它既推广了区分秩一与高秩半单李群的二分法，也推广了区分多项式增长与指数增长的连通可解幺模李群的二分法。我们给出了这种二分法在代数、拟等距和粗几何方面的等价表述。我们的结果对粗嵌入有许多推论；例如，我们推断对于形如 $N\ imes S$ 的群，其中 $N$ 是连通幂零李群，而 $S$ 是实秩为 1 的单李群，$N$ 的增长指数与 $S$ 的 Ahlfors–regular 共形维数在粗嵌入下均不递减。即使在拟等距情形下，这些结果也是新的，并给出了拟等距嵌入的障碍；在许多情况下，这些障碍强于 Buyalo–Schroeder 此前得到的障碍。
英文摘要：
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.
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