A Koebe distortion theorem for quasiconformal mappings in the Heisenberg group

We prove a Koebe distortion theorem for the average derivative of a quasiconformal mapping between domains in the sub-Riemannian Heisenberg group $\mathbb{H}_1$. Several auxiliary properties of quasiconformal mappings between subdomains of $\mathbb{H}_1$ are proven, including distortion of balls estimates and local BMO-estimates for the logarithm of the Jacobian of a quasiconformal mapping. Applications of the Koebe theorem include diameter bounds for images of curves, comparison of integrals of the average derivative and the operator norm of the horizontal differential, as well as the study of quasiconformal densities and metrics in domains in $\mathbb{H}_1$. The theorems are discussed for the sub-Riemannian and the Kor\'anyi distances. This extends results due to Astala--Gehring, Astala--Koskela, Koskela and Bonk--Koskela--Rohde.