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.
Citation
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
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