Rickman rugs and intrinsic bilipschitz graphs

This paper studies the geometry of bilipschitz maps $f \colon $\mathbb{W}$ \to $\mathbb{H}$$,where$$\mathbb{H}$$isthefirstHeisenberggroup,and$$\mathbb{W}$ \subset $\mathbb{H}$$isaverticalsubgroupofco-dimension$1$\mathrm{.}Theimages$f($\mathbb{W}$)$ of such maps are called Rickman rugs in the Heisenberg group. The main theorem states that a Rickman rug in the Heisenberg group admits a corona decomposition by intrinsic bilipschitz graphs. As a corollary, Rickman rugs are countably rectifiable by intrinsic bilipschitz graphs. Here, an intrinsic bilipschitz graph is an intrinsic Lipschitz graph, which is simultaneously a Rickman rug. General intrinsic Lipschitz graphs need not be Rickman rugs, even locally, by an example of Bigolin and Vittone.