Poincaré profiles of Lie groups and a coarse geometric dichotomy
Poincaré profiles of Lie groups and a coarse geometric dichotomy
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DOI：
10.1007/s00039-022-00617-4
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发表时间：
2020-11
影响因子：
2.2
通讯作者：
David Hume;J. M. Mackay;R. Tessera
中科院分区：
文献类型：
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作者：
David Hume;J. M. Mackay;R. Tessera
庞加莱轮廓是解析定义的不变量，它为度量空间之间的粗嵌入的存在提供了障碍。我们计算它们的所有连接么模李群，Baumslag-Solitar群和瑟斯顿几何，表现出两种基本不同类型的行为。对于李群，我们的二分法扩展了半单李群的秩一与高阶二分法和可解单模李群的多项式与指数增长二分法。我们提供了等价的代数，准等距和粗糙的几何配方的二分法。作为结果，我们推出的形式，其中N是一个连通的幂零李群，和S是一个秩一个简单的李群，无论是增长指数的N，和S的共形维数是不减下粗嵌入。这些结果是新的，即使是准等距嵌入，并给予障碍，在许多情况下，改善以前获得的Buyalo施罗德。
Poincaré profiles are analytically defined invariants, which provide obstructions to the existence of coarse embeddings between metric spaces. We calculate them for all connected unimodular Lie groups, Baumslag–Solitar groups and Thurston geometries, demonstrating two substantially different types of behaviour. For Lie groups, our dichotomy extends both the rank one versus higher rank dichotomy for semisimple Lie groups and the polynomial versus exponential growth dichotomy for solvable unimodular Lie groups. We provide equivalent algebraic, quasi-isometric and coarse geometric formulations of this dichotomy. As a consequence, we deduce that for groups of the form, whereNis a connected nilpotent Lie group, andSis a rank one simple Lie group, both the growth exponent ofN, and the conformal dimension ofSare non-decreasing under coarse embeddings. These results are new even for quasi-isometric embeddings and give obstructions which in many cases improve those previously obtained by Buyalo–Schroeder.