The Combinatorial Loewner Property and Super-Multiplicativity Inequalities for Symmetric Self-Similar Metric Spaces

Abstract This paper introduces a general construction of self-similar metric spaces as a limit of discrete graphs. Our framework produces many classical examples, such as the Sierpiński carpet and the higher dimensional Menger sponges, but also a rich class of new examples. The main result of the work roughly speaking states: If the construction is sufficiently symmetric then the limiting object supports useful moduli estimates, namely the Combinatorial Loewner property of Bourdon–Kleiner and the super-multiplicativity inequality. We establish the super-multiplicativity on Menger sponges, for which it had not been previously known. The main new technique the work offers is a general framework of flows and resistance estimates on a rather general class of symmetric self-similar graphs.