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Conformal dimension and boundaries of planar domains


Author: Kyle Kinneberg
Journal: Trans. Amer. Math. Soc. 369 (2017), 6511-6536
MSC (2010): Primary 30L10, 28A78; Secondary 30C20
DOI: https://doi.org/10.1090/tran/6944
Published electronically: May 11, 2017
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Abstract: Building off of techniques that were recently developed by M. Carrasco, S. Keith, and B. Kleiner to study the conformal dimension of boundaries of hyperbolic groups, we prove that uniformly perfect boundaries of John domains in $ \hat {\mathbb{C}}$ have conformal dimension equal to 0 or 1. Our proof uses a discretized version of Carrasco's ``uniformly well-spread cut point'' condition, which we call the discrete UWS property, that is well-suited to deal with metric spaces that are not linearly connected. More specifically, we prove that boundaries of John domains have the discrete UWS property and that any compact, doubling, uniformly perfect metric space with the discrete UWS property has conformal dimension equal to 0 or 1. In addition, we establish other geometric properties of metric spaces with the discrete UWS property, including connectivity properties of their weak tangents.


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Additional Information

Kyle Kinneberg
Affiliation: Department of Mathematics, Rice University, 6100 Main Street, Houston, Texas 77005
Email: kyle.kinneberg@rice.edu

DOI: https://doi.org/10.1090/tran/6944
Keywords: Conformal dimension, John domains, H\"older domains
Received by editor(s): August 17, 2015
Received by editor(s) in revised form: March 10, 2016
Published electronically: May 11, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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