BI-LIPSCHITZ EMBEDDINGS OF QUASICONFORMAL TREES

A quasiconformal tree is a doubling metric tree in which the di-ameter of each arc is bounded above by a fixed multiple of the distance be-tween its endpoints. In this paper we show that every quasiconformal tree bi-Lipschitz embeds in some Euclidean space, with the ambient dimension and the bi-Lipschitz constant depending only on the doubling and bounded turning constants of the tree. This answers Question 1.6 of David and Vellis [Illinois J. Math. 66 (2022), pp. 189-244].

Automated summary

This paper proves that a quasiconformal tree can be bi-Lipschitz embedded in some Euclidean space, with the ambient dimension and the bi-Lipschitz constant depending only on the doubling and bounded turning constants of the tree.