  In \cite{BK02}, M. Bonk and B. Kleiner proved a rigidity theorem for
expanding quasi-M\"obius group actions on Ahlfors $n$-regular metric spaces
with topological dimension $n$. This led naturally to a rigidity result for
quasi-convex geometric actions on CAT$(-1)$-spaces that can be seen as a metric
analog to the "entropy rigidity" theorems of U. Hamenst\"adt and M. Bourdon.
Building on the ideas developed in \cite{BK02}, we establish a rigidity theorem
for certain expanding quasi-M\"obius group actions on spaces with different
metric and topological dimensions. This is motivated by a corresponding entropy
rigidity result in the coarse geometric setting.
