Heisenberg quasiregular ellipticity

Following the Euclidean results of Varopoulos and Pankka–Rajala, we provide a necessary topological condition for a sub-Riemannian 3-manifold $M$ to admit a nonconstant quasiregular mapping from the sub-Riemannian Heisenberg group $\mathbb{H}$. As an application, we show that a link complement $S^3\backslash L$hasasub-Riemannianmetricadmittingsuchamappingonlyif$L$ is empty, the unknot or Hopf link. In the converse direction, if $L$ is empty, a specific unknot or Hopf link, we construct a quasiregular mapping from $\mathbb{H}$ to $S^3\mathrm{\backslash }L$. The main result is obtained by translating a growth condition on ${\pi }_1(M)$ into the existence of a supersolution to the $4$-harmonic equation, and relies on recent advances in the study of analysis and potential theory on metric spaces.