  Suppose an amenable group $G$ is acting freely on a simply connected
simplicial complex $\tilde X$ with compact quotient $X$. Fix $n \geq 1$, assume
$H_n(\tilde X, \mathbb{Z})=0$ and let $(H_i)$ be a Farber chain in $G$. We
prove that the torsion of the integral homology in dimension $n$ of
$\tilde{X}/H_i$ grows subexponentially in $[G:H_i]$. By way of contrast, if $X$
is not compact, there are solvable groups of derived length 3 for which torsion
in homology can grow faster than any given function.
