Heisenberg quasiregular ellipticity (Q2418895)

Summary: 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 \(\mathbb{S}^3\backslash L\) has a sub-Riemannian metric admitting such a mapping only if \(L\) is empty, an 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 \(\mathbb{S}^3\backslash L\). 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.