# Quasiconformal and geodesic trees

## TL;DR

This paper proves that every quasiconformal tree can be transformed into a geodesic tree with Hausdorff dimension close to 1 using quasisymmetric mappings, linking these classes of metric trees.

## Contribution

It establishes a quasisymmetric equivalence between quasiconformal trees and geodesic trees with nearly one-dimensional Hausdorff measure.

## Key findings

- Quasiconformal trees are quasisymmetrically equivalent to geodesic trees.
- The Hausdorff dimension of these geodesic trees can be made arbitrarily close to 1.
- The result connects the geometric structure of quasiconformal trees to geodesic trees.

## Abstract

A quasiconformal tree is a metric tree that is doubling and of bounded turning. We prove that every quasiconformal tree is quasisymmetrically equivalent to a geodesic tree with Hausdorff dimension arbitrarily close to 1.