  Let $(Z,d,\mu)$ be a compact, connected, Ahlfors $Q$-regular metric space
with $Q>1$. Using a hyperbolic filling of $Z$, we define the notions of the
$p$-capacity between certain subsets of $Z$ and of the weak covering
$p$-capacity of path families $\Gamma$ in $Z$. We show comparability results
and quasisymmetric invariance. As an application of our methods we deduce a
result due to Tyson on the geometric quasiconformality of quasisymmetric maps
between compact, connected Ahlfors $Q$-regular metric spaces.
