Sharp geometric rigidity of isometries on Heisenberg groups

We prove sharp geometric rigidity estimates for isometries on Heisenberg groups. Our main result asserts that every $(1+\epsilon )$-quasi-isometry on a John domain of the Heisenberg group ${\mathbb{H}}^n$, $n>1$, is close to some isometry up to proximity order $\sqrt{\epsilon }+\epsilon$ in the uniform norm, and up to proximity order $\epsilon$ in the $L_p^1$-norm. We give examples showing the asymptotic sharpness of our results.