Structure of Almost-Isometries on the Heisenberg Group

We prove a structure theorem for almost-isometries on the three-dimensional Heisenberg group equipped with a Carnot-Caratheodory metric. The main theorem yields that almost-isometries of the Heisenberg group are of bounded distance from the composition of an isometry and a map which only alters the central coordinate. The proof of the main theorem relies on the technique of coarse differentiation, as well as the particular geometry of metric malls within the Heisenberg group.