On growth of homology torsion in amenable groups

Suppose an amenable group $G$ is acting freely on a simply connected simplicial complex $\tilde{X}$ with compact quotient $X$. Fix $n\ge 1$, 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 $\tilde{X}\mathrm{/}{H}_i$ 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.