A coarse entropy-rigidity theorem and discrete length-volume inequalities

In [5], M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-Mobius 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. Hamenstadt and M. Bourdon. Building on the ideas developed in [5], we establish a rigidity theorem for certain expanding quasi-Mobius group actions on spaces with different metric and topological dimensions. This is motivated by a corresponding entropy-rigidity result in the coarse geometric setting.;Our analysis of these |``fractal" metric spaces depends heavily on a combinatorial inequality that relates volume to lengths of curves within the space. We extend such inequalities to a broader metric setting and obtain discrete analogs of some results due to W. Derrick. In the process, we shed light on a related question of Y. Burago and V. Zalgaller about pseudometrics on the n-dimensional unit cube.