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arXiv · 2408.15692

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

Riku Anttila·Sylvester Eriksson-Bique
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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.

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Riku Anttila, Sylvester Eriksson-Bique. 2024-08-28. The Combinatorial Loewner Property and super-multiplicativity inequalities for symmetric self-similar metric spaces. https://arxiv.org/abs/2408.15692

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