Bi-Lipschitz geometry of quasiconformal trees
Abstract
A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. We study the geometry of these trees in two directions. First, we construct a catalog of metric trees in a purely combinatorial way, and show that every quasiconformal tree is bi-Lipschitz equivalent to one of the trees in our catalog. This is inspired by results of Herron-Meyer and Rohde for quasi-arcs. Second, we show that a quasiconformal tree bi-Lipschitz embeds in a Euclidean space if and only if its set of leaves admits such an embedding. In particular, all quasi-arcs bi-Lipschitz embed into some Euclidean space.
Keywords
Cite
@article{arxiv.2007.12297,
title = {Bi-Lipschitz geometry of quasiconformal trees},
author = {Guy C. David and Vyron Vellis},
journal= {arXiv preprint arXiv:2007.12297},
year = {2022}
}
Comments
40 pages. Added sections containing examples (Section 6) and a new result on spaces more general than trees (Section 7). Minor errors corrected and various clarifications added