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June 2022 Bi-Lipschitz geometry of quasiconformal trees
Guy C. David, Vyron Vellis
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Illinois J. Math. 66(2): 189-244 (June 2022). DOI: 10.1215/00192082-9936324

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 and Meyer and of 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.

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Guy C. David. Vyron Vellis. "Bi-Lipschitz geometry of quasiconformal trees." Illinois J. Math. 66 (2) 189 - 244, June 2022. https://doi.org/10.1215/00192082-9936324

Information

Received: 3 October 2021; Revised: 3 April 2022; Published: June 2022
First available in Project Euclid: 16 May 2022

Digital Object Identifier: 10.1215/00192082-9936324

Subjects:
Primary: 30L05
Secondary: 05C05 , 30L10 , 51F99

Rights: Copyright © 2022 by the University of Illinois at Urbana–Champaign

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Vol.66 • No. 2 • June 2022
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