Bi-Lipschitz geometry of quasiconformal trees

TL;DR

This paper characterizes quasiconformal trees by constructing a combinatorial catalog of such trees and establishes conditions under which they bi-Lipschitz embed into Euclidean spaces, focusing on their leaf sets.

Contribution

It provides a combinatorial classification of quasiconformal trees and links their Euclidean embeddability to that of their leaves, extending previous results on quasi-arcs.

Findings

Every quasiconformal tree is bi-Lipschitz equivalent to a catalog tree.

A quasiconformal tree bi-Lipschitz embeds into Euclidean space iff its leaves do.

All quasi-arcs bi-Lipschitz embed into Euclidean space.

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.