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

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 $G\ast H$ 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, $asdimK<\infty$ and $H$ 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 (2016), of Bell and Moran (2015) and an open problem in topology.