PreprintCC BY 4.0arXiv
Polytopal substitutions and conformal dimension
Abstract
We introduce polytopal substitutions, a recursive procedure for constructing polyhedral approximations of self-similar topological spaces. For a class of substitutions called sub-hyperbolic, we show that the limit of these approximations is naturally homeomorphic to the limit space of the associated iterated monodromy group. We establish a relationship between the ℓp-contraction coefficients of sub-hyperbolic substitutions and the conformal dimension of their limit spaces. In particular, we characterize the conformal dimension in terms of asymptotic moduli and relate these moduli to p-energies of graphs.
Related
BibTeX
@misc{nekrashevych2026polytopal,
title = {{Polytopal substitutions and conformal dimension}},
author = {Volodymyr Nekrashevych},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31530},
eprint = {2610.31530},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31530},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Polytopal substitutions and conformal dimension
AU - Volodymyr Nekrashevych
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31530
UR - https://arxiv.org/abs/2610.31530
AB - We introduce polytopal substitutions, a recursive procedure for constructing polyhedral approximations of self-similar topological spaces. For a class of substitutions called sub-hyperbolic, we show that the limit of these approximations is naturally homeomorphic to the limit space of the associated iterated monodromy group. We establish a relationship between the ℓ^p-contraction coefficients of sub-hyperbolic substitutions and the conformal dimension of their limit spaces. In particular, we characterize the conformal dimension in terms of asymptotic moduli and relate these moduli to p-energies of graphs.
ER -