The Σ-invariants of Bieri–Neumann–Strebel and Bieri–Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Σ-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The “zeroth stage” of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the “nth stage” for any n.

中文翻译：

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CAT(0) 空间上 G 模块的更高水平极限集

Bieri-Neumann-Strebel 和 Bieri-Renz 的 Σ-不变量涉及离散群的作用G在几何上合适的空间米. 在早期版本中，米总是一个有限维欧几里得空间G由翻译执行。在这方面存在大量文献，将不变量与群论和热带几何（实际上是 Σ 理论所预期的）联系起来。最近，我们将这些不变量推广到以下情况米是适当的猫(0) 空间G通过等距作用。我们的论文 [BG16] 提出了这一“零阶段”。本论文提供了该理论对“n第阶段“对于任何n.