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Buyalo and Lebedeva have shown that the asymptotic dimension of a hyperbolicgroup is equal to the dimension of the group boundary plus one. Among the workpresented 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.

A Comparison of Large Scale Dimension of a Metric Space to the Dimension of its Boundary