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

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.