Invertible Carnot Groups

TL;DR

This paper characterizes specific Carnot groups that admit a 1-quasiconformal metric inversion, linking their structure to Lie groups of Heisenberg type satisfying the $J^2$-condition, and extends the concept to certain non-fractal metric spaces.

Contribution

It provides a complete characterization of Carnot groups with 1-quasiconformal inversions and extends the framework to non-fractal metric spaces.

Findings

Carnot groups with 1-quasiconformal metric inversion are Lie groups of Heisenberg type satisfying the $J^2$-condition

Characterization of inversion invariant bi-Lipschitz homogeneity for certain non-fractal metric spaces

Link between algebraic conditions and geometric inversion properties

Abstract

We characterize Carnot groups admitting a 1-quasiconformal metric inversion as the Lie groups of Heisenberg type whose Lie algebras satisfy the $J^2$-condition, thus characterizing a special case of inversion invariant bi-Lipschitz homogeneity. A more general characterization of inversion invariant bi-Lipschitz homogeneity for certain non-fractal metric spaces is also provided.