Universal quasiconformal trees

TL;DR

This paper proves the existence of a universal quasiconformal tree for each finite valence and shows all such trees can embed into the plane, answering key questions in the field.

Contribution

It establishes the existence of quasisymmetrically universal quasiconformal trees with bounded valence and demonstrates their embeddability into the plane, extending previous results.

Findings

Existence of universal quasiconformal trees for each finite valence

All such trees embed quasisymmetrically into 

Answers to open questions by Bonk and Meyer from 2020 and 2022

Abstract

A quasiconformal tree is a doubling (compact) metric tree in which the diameter of each arc is comparable to the distance of its endpoints. We show that for each integer $n\geq 2$, the class of all quasiconformal trees with uniform branch separation and valence at most $n$, contains a quasisymmetrically ''universal'' element, that is, an element of this class into which every other element can be embedded quasisymmetrically. We also show that every quasiconformal tree with uniform branch separation quasisymmetrically embeds into $\mathbb{R}^2$. Our results answer two questions of Bonk and Meyer from 2022, in higher generality, and partially answer one question of Bonk and Meyer from 2020.