Lipschitz extensions of maps between Heisenberg groups

TL;DR

This paper proves that for odd-dimensional Heisenberg groups, Lipschitz maps from a subset cannot always be extended to the whole group, highlighting a fundamental geometric limitation.

Contribution

It establishes the non-existence of Lipschitz extensions between Heisenberg groups of odd dimension, revealing a new geometric property of these groups.

Findings

Lipschitz extension property fails for odd-dimensional Heisenberg groups

Provides a counterexample to Lipschitz extension in this setting

Highlights differences between even and odd-dimensional cases

Abstract

Let $\H^n$ be the Heisenberg group of topological dimension $2n+1$. We prove that if $n$ is odd, the pair of metric spaces $(\H^n, \H^n)$ does not have the Lipschitz extension property.