Decomposition complexity with respect to coarse properties

TL;DR

This paper introduces a formal framework for understanding decomposition complexity in metric spaces, generalizing existing concepts and providing conditions for spaces to satisfy Property A, which is important in coarse geometry.

Contribution

It formalizes the notion of uniform coarse properties and extends finite decomposition complexity, offering new insights into when metric spaces satisfy Property A.

Findings

Provides sufficient conditions for Property A

Generalizes finite decomposition complexity

Formalizes uniform coarse property families

Abstract

We formalize the concept of a family of metric spaces satisfying a coarse property uniformly and we generalize finite decomposition complexity of Erik Guentner, Romain Tessera, and Guoliang Yu. Of particular interest are results determining sufficient conditions for a metric space to satisfy Property A of Guoliang Yu.