  Buyalo and Lebedeva have shown that the asymptotic dimension of a hyperbolic
group is equal to the dimension of the group boundary plus one. Among the work
presented here is a partial extension of that result to all groups admitting
$\mathcal{Z}$-structures; in particular, we show that $\hbox{asdim}G\geq
\hbox{dim}Z+1$ where $Z$ is the $\mathcal{Z}$-boundary.
