We introduce the notion of regular finite decomposition complexity of a metric family. This generalizes Gromov’s finite asymptotic dimension and is motivated by the concept of finite decomposition complexity (FDC) due to Guentner, Tessera and Yu. Regular finite decomposition complexity implies FDC and has all the permanence properties that are known for FDC, as well as a new one called Finite Quotient Permanence. We show that for a collection containing all metric families with finite asymptotic dimension, all other permanence properties follow from Fibering Permanence.

中文翻译：

---

正则有限分解复杂度

我们引入了度量族的规则有限分解复杂度的概念。这概括了 Gromov 的有限渐近维数，并受到 Guentner、Tessera 和 Yu 的有限分解复杂性 (FDC) 概念的启发。常规有限分解复杂性意味着 FDC，并具有 FDC 已知的所有持久性属性，以及一种称为有限商持久性的新属性。我们表明，对于包含所有具有有限渐近维度的度量族的集合，所有其他持久性属性都来自 Fibering Permanence。