Regular Finite Decomposition Complexity

TL;DR

This paper introduces regular finite decomposition complexity, a new concept that generalizes existing notions like finite asymptotic dimension and finite decomposition complexity, with strong permanence properties.

Contribution

It defines regular finite decomposition complexity, proves it implies FDC, and establishes new permanence properties including Finite Quotient Permanence.

Findings

Regular finite decomposition complexity generalizes Gromov's finite asymptotic dimension.

It implies finite decomposition complexity (FDC).

It possesses all known permanence properties of FDC plus Finite Quotient Permanence.

Abstract

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.