On stability of asymptotic property C for products and some group extensions

TL;DR

This paper proves that Dranishnikov's asymptotic property C is stable under group operations like products and free products, and extends the property to certain group extensions, resolving several open questions.

Contribution

It demonstrates the stability of asymptotic property C under direct and free products, and characterizes its preservation in group extensions with specific conditions.

Findings

Asymptotic property C is preserved under direct products.

Asymptotic property C is preserved under free products.

Group extensions with finite asymptotic dimension kernel also preserve the property.

Abstract

We show that Dranishnikov's asymptotic property C is preserved by direct products and the free product of discrete metric spaces. In particular, if $G$ and $H$ are groups with asymptotic property C, then both $G \times H$ and $G * H$ have asymptotic property C. We also prove that a group~$G$ has asymptotic property C if $1\to K\to G\to H\to 1$ is exact, if $\operatorname{asdim} K<\infty$, and if $H$ has asymptotic property C. The groups are assumed to have left-invariant proper metrics and need not be finitely generated. These results settle questions of Dydak and Virk, of Bell and Moran, and an open problem in topology from the Lviv Topological Seminar.