[Paper Review] Characterizations of $k$-rectifiability in homogenous groups

Kennedy Obinna Idu, Valentino Magnani+1 more|arXiv (Cornell University)|Sep 8, 2020

Advanced Topology and Set Theory19 references4 citations

TL;DR

This paper establishes that in any homogeneous group, a Borel set is $k$-rectifiable if and only if it admits $(k,\mathbb{G})$-approximate tangent groups almost everywhere with respect to $\mathcal{H}^k$. The authors prove that the a.e. existence of such approximate tangent groups characterizes $k$-rectifiability, extending classical Euclidean results to the sub-Riemannian setting of homogeneous groups using tangent measures and density estimates on purely unrectifiable sets.

ABSTRACT

A well known notion of $k$-rectifiable set can be formulated in any metric space using Lipschitz images of subsets of $\mathbb{R}^k$. We prove some characterizations of $k$-rectifiability, when the metric space is an arbitrary homogeneous group. In particular, we show that the a.e. existence of the $(k,\mathbb{G})$-approximate tangent group implies $k$-rectifiability.

Motivation & Objective

• To characterize $k$-rectifiable sets in arbitrary homogeneous groups using geometric measure theory tools.
• To extend the classical characterization of rectifiability via approximate tangent planes to the sub-Riemannian setting of homogeneous groups.
• To establish that the a.e. existence of $(k,\mathbb{G})$-approximate tangent groups implies $k$-rectifiability, resolving a key question in geometric measure theory on stratified Lie groups.
• To unify and generalize prior results on rectifiability in Heisenberg groups and other homogeneous spaces by introducing a robust, intrinsic notion of tangent group.

Proposed method

• The authors use the notion of $(k,\mathbb{G})$-approximate tangent group, defined via weak limits of rescaled measures, to characterize rectifiability in homogeneous groups.
• They employ tangent measures and density estimates on purely unrectifiable sets to control the behavior of $\mathcal{H}^k$-measure near points of the set.
• The proof relies on compactness of the horizontal Grassmannian $\mathcal{H}(\mathbb{G},k)$ to cover the set with finitely many cones around model tangent groups.
• A key technical step involves comparing distances to different subgroups using the intrinsic metric and horizontal projections, leveraging the inequality $d(x,\mathbb{T}) \geq c_\mathbb{G} \| \pi_\mathbb{T}(x)^{-1} \pi_{\mathbb{T}^\perp}(x) \pi_\mathbb{T}(x) \|$.
• The authors apply Federer's rectifiability criterion by showing that the density of $\mathcal{H}^k \llcorner E$ in cones around tangent groups vanishes at scale zero, implying null $\mathcal{H}^k$-measure on unrectifiable parts.
• They use the fact that $\mathcal{H}^k \llcorner E$ is locally finite and apply a covering argument with cones of fixed aperture to control measure decay.

Experimental results

Research questions

• RQ1 Does the a.e. existence of $(k,\mathbb{G})$-approximate tangent groups imply $k$-rectifiability in homogeneous groups?
• RQ2 Can the classical characterization of rectifiability via approximate tangent planes be extended to non-Euclidean homogeneous groups?
• RQ3 How do density estimates and tangent measures interact in purely unrectifiable sets within sub-Riemannian spaces?
• RQ4 Is the existence of a unique $(k,\mathbb{G})$-approximate tangent group at $\mathcal{H}^k$-a.e. point sufficient to characterize $k$-rectifiability?
• RQ5 To what extent do intrinsic geometric structures like horizontal Grassmannians and intrinsic cones refine the notion of rectifiability in stratified Lie groups?

Key findings

• The existence of $(k,\mathbb{G})$-approximate tangent groups at $\mathcal{H}^k$-a.e. point $p \in E$ implies that $E$ is $k$-rectifiable.
• The equivalence between $k$-rectifiability and the a.e. existence of $(k,\mathbb{G})$-approximate tangent groups holds in any homogeneous group.
• The proof shows that the $\mathcal{H}^k$-measure of the purely unrectifiable part of $E$ is zero, by controlling cone decay via the intrinsic metric and horizontal projections.
• The authors establish that the density $\Theta^{*k}(E,p)$ is zero on the purely unrectifiable part, using a covering argument with cones of fixed aperture.
• The result generalizes Preiss-type theorems to homogeneous groups, showing that rectifiability is characterized by the existence of approximate tangent groups.
• The key inequality $\| \pi_{\mathbb{T}_p}(x)^{-1} \pi_{\mathbb{T}_p^\perp}(x) \pi_{\mathbb{T}_p}(x) \| \geq \left( \frac{2}{3} - \frac{\epsilon_0}{c_\mathbb{G}} \right) \|x\|$ ensures that points in $X(p,\mathbb{V}_i^\perp,\epsilon_0)$ lie outside $X(p,\mathbb{T}_p,\epsilon_0)$ for small $\epsilon_0$, enabling the cone comparison.