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Peer-reviewedCC BYPotential Analysis

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

  • Riku Anttila1
  • Sylvester David Eriksson-Bique1
  1. 1Department of Mathematics and Statistics, University of Jyväskylä, Seminaarinkatu 15, Jyväskylä, 40014, Finland
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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.

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Cite

BibTeXRIS
BibTeX
@article{anttila2026combinatorial,
  title = {{The Combinatorial Loewner Property and Super-Multiplicativity Inequalities for Symmetric Self-Similar Metric Spaces}},
  author = {Riku Anttila and Sylvester David Eriksson-Bique},
  journal = {Potential Analysis},
  year = {2026},
  month = may,
  doi = {10.1007/s11118-026-10310-y},
  url = {https://doi.org/10.1007/s11118-026-10310-y},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - The Combinatorial Loewner Property and Super-Multiplicativity Inequalities for Symmetric Self-Similar Metric Spaces
AU  - Riku Anttila
AU  - Sylvester David Eriksson-Bique
PY  - 2026
DA  - 2026/05/13
JO  - Potential Analysis
DO  - 10.1007/s11118-026-10310-y
UR  - https://doi.org/10.1007/s11118-026-10310-y
AB  - 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.
SN  - 0926-2601
SN  - 1572-929X
ER  -