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

Kinneberg, Kyle Edward (2022) A coarse entropy-rigidity theorem and discrete length-volume inequalities. Doctoral thesis, UIN SAIZU Purwokerto.

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Abstract

In [5], 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 [31] and M. Bourdon [10]. Building on the ideas developed in [5], we establish a rigidity theorem for certain expanding quasi-M¨obius group actions on spaces with different metric and topolog�ical 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 inequal�ity 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 [23, 24]. In the process, we shed light on a related question of Y. Burago and V. Zalgaller about pseudometrics on the n-dimensional unit cube.

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