Decomposition theorems for asymptotic property C and property A

TL;DR

This paper introduces finite APC-decomposition complexity, a new concept combining finite decomposition complexity and asymptotic property C, which implies property A and is preserved under various group and space constructions.

Contribution

It defines finite APC-decomposition complexity, shows its relation to existing properties, and proves its preservation under key algebraic and geometric operations.

Findings

Finite APC-decomposition complexity implies property A.

Spaces with finite decomposition complexity or asymptotic property C have finite APC-decomposition complexity.

Finite APC-decomposition complexity is preserved under direct products, amalgamated products, and group extensions.

Abstract

We combine aspects of the notions of finite decomposition complexity and asymptotic property C into a notion that we call finite APC-decomposition complexity. Any space with finite decomposition complexity has finite APC-decomposition complexity and any space with asymptotic property C has finite APC-decomposition complexity. Moreover, finite APC-decomposition complexity implies property A for metric spaces. We also show that finite APC-decomposition complexity is preserved by direct products of groups and spaces, amalgamated products of groups, and group extensions, among other constructions.