Rickman rugs and intrinsic bilipschitz graphs

TL;DR

This paper investigates the geometric structure of Rickman rugs in the Heisenberg group, showing they can be decomposed into intrinsic bilipschitz graphs and are countably rectifiable.

Contribution

It proves that Rickman rugs admit a corona decomposition by intrinsic bilipschitz graphs, establishing their countable rectifiability in the Heisenberg group.

Findings

Rickman rugs admit a corona decomposition by intrinsic bilipschitz graphs

Rickman rugs are countably rectifiable by intrinsic bilipschitz graphs

Intrinsic Lipschitz graphs are not necessarily Rickman rugs, even locally

Abstract

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.