The Combinatorial Loewner Property and super-multiplicativity inequalities for symmetric self-similar metric spaces

TL;DR

This paper develops a framework for constructing self-similar metric spaces from graphs, establishing the Combinatorial Loewner property and super-multiplicativity inequalities for these spaces, including new results for Menger sponges.

Contribution

It introduces a general method to produce self-similar spaces with symmetry, proving key properties like the Combinatorial Loewner property and super-multiplicativity inequalities.

Findings

Supports the Combinatorial Loewner property in symmetric self-similar spaces.

Establishes super-multiplicativity inequalities for Menger sponges.

Provides a new framework of flows and resistance estimates for analysis.

Abstract

This paper introduces a general construction of self-similar metric spaces as limits of discrete graphs. Our framework produces many classical examples, such as the Sierpi\'nski 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 inequalities. The latter are established 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.