  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.
