Coarse property {C} and decomposition complexity

TL;DR

This paper introduces coarse analogs of property C and decomposition complexity in the large-scale geometry setting, establishing their relationships and properties similar to the classical metric case.

Contribution

It defines coarse versions of asymptotic property C and decomposition complexity, and proves their key properties and implications in the coarse category.

Findings

Coarse property C implies coarse property A.

Coarse versions are preserved under products and unions.

The paper extends large-scale geometric concepts to coarse spaces.

Abstract

The coarse category was established by Roe to distill the salient features of the large-scale approach to metric spaces and groups that was started by Gromov. In this paper, we use the language of coarse spaces to define coarse versions of asymptotic property C and decomposition complexity. We prove that coarse property C implies coarse property A; we also show that these coarse versions share many of the features of their metric analogs such as preservation by products or unions.