Transitive bi-Lipschitz group actions and bi-Lipschitz parameterizations

TL;DR

This paper characterizes when Ahlfors 2-regular quasisymmetric images of the plane are bi-Lipschitz equivalent to the plane, and when certain geodesic spaces are bi-Lipschitz equivalent to Carnot groups, based on group homogeneity properties.

Contribution

It establishes new criteria linking bi-Lipschitz parameterizations to group actions and homogeneity in metric spaces.

Findings

Ahlfors 2-regular quasisymmetric images are bi-Lipschitz to the plane if uniformly bi-Lipschitz homogeneous.

Certain geodesic spaces are bi-Lipschitz to Carnot groups under inversion-invariant homogeneity.

Provides a characterization of bi-Lipschitz equivalences via group actions.

Abstract

We prove that Ahlfors 2-regular quasisymmetric images of the Euclidean plane are bi-Lipschitz images of the plane if and only if they are uniformly bi-Lipschitz homogeneous with respect to a group. We also prove that certain geodesic spaces are bi-Lipschitz images of Carnot groups if they are inversion invariant bi-Lipschitz homogeneous with respect to a group.