  This paper studies the geometry of bilipschitz maps $f \colon \mathbb{W} \to
\mathbb{H}$, where $\mathbb{H}$ is the first Heisenberg group, and $\mathbb{W}
\subset \mathbb{H}$ is a vertical subgroup of co-dimension $1$. The images
$f(\mathbb{W})$ 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.
