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Appendix C3 — The Tangent Cone: Carnot Groups and Growth Vectors

Zoom infinitely far into any sub-Riemannian geometry and it converges to a Carnot group — its metric tangent cone — labelled by a growth vector. This appendix defines that flag of brackets, the Ball–Box scaling that measures it, and the reach estimator the code actually runs.

By Igor Moiseev · 30 July 2026
From Caustics to Groups
  1. From Caustics to Groups: A Research Program
  2. The Forward Map: Caustics of the Model Groups
  3. The Inverse Map: Reading the Fingerprint
  4. In the Wild: DW-MRI, and Where the Method Stays Silent
Appendices — Theory Background
  1. C1. The Model Groups, by Example: Real-World Sub-Riemannian Systems
  2. C2. Caustics as Lagrangian Singularities (Arnol'd's ADE List)
  3. C3. The Tangent Cone: Carnot Groups and Growth Vectors ← you are here
  4. C4. The Conjugate Locus at the Pole: Astroids and Their Moduli
  5. C5. Abnormal Geodesics: The Yes/No Fingerprint
  6. C6. The Nilpotent-Deviation Statistic
What this appendix covers
The first leg of the fingerprint — the one that sorts the five groups into classes — is the growth vector of the tangent cone. This appendix says what the tangent cone is (a Carnot group), what the growth vector measures (bracket depth), how the Ball–Box theorem turns it into a scaling law you can fit, and how metric M1 in the code (src/growth.py) recovers it from geodesic-spreading data. It is the theory behind Part 2's figure.

Zoom in, and every geometry becomes a Carnot group

Take a sub-Riemannian manifold and a point $q_0$, and blow up the metric around $q_0$ by larger and larger factors. In the limit (Gromov–Hausdorff) the geometry converges to a model space called the metric tangent cone. Unlike the Riemannian case — where the tangent cone is always flat $\mathbb{R}^n$ — here it is a Carnot group: a nilpotent Lie group carrying a family of anisotropic dilations that stretch different coordinates by different powers. The tangent cone is the simplest geometry with the same infinitesimal bracket structure as the original, and it is the reference against which everything is measured (Appendix C6).

The growth vector: a flag of brackets

Let $\mathcal{D}$ be the distribution — the allowed directions, with $n_1 = \operatorname{rank} \mathcal{D}$. Add the directions you reach by one bracket, then two, and so on:

\[\mathcal{D} \;\subset\; \mathcal{D} + [\mathcal{D},\mathcal{D}] \;\subset\; \mathcal{D} + [\mathcal{D},\mathcal{D}] + [[\mathcal{D},\mathcal{D}],\mathcal{D}] \;\subset\; \cdots\]

The dimensions of this flag, $(n_1, n_2, n_3, \dots)$, are the growth vector. It reaches the full dimension at the step of the structure (the deepest bracket needed). For the series’ groups:

Group Growth vector Step Reading
Heisenberg, SE(2) $(2,3)$ 2 drive 2 ways; 1 coordinate one bracket deep
Engel $(2,3,4)$ 3 + 1 coordinate two brackets deep
Cartan $(2,3,5)$ 3 + 2 coordinates two brackets deep
SE(3) $(3,6)$ 2 drive 3 ways; 3 coordinates one bracket deep

This is exactly the “nonholonomic parking difficulty” of Appendix C1: a coordinate at flag level $k$ needs a bracket nested $k-1$ deep to reach.

Ball–Box: turning bracket depth into a scaling law

The growth vector is invisible to any single caustic germ (Appendix C2), but it is visible in how fast small balls grow. The Ball–Box theorem says: in dilation-adapted coordinates where coordinate $x_i$ has weight $w_i$ (its flag level), the sub-Riemannian ball of radius $r$ is comparable to the box $\prod_i {|x_i| \lesssim r^{w_i}}$. Two consequences the code uses:

\[Q \;=\; \sum_i i\,(n_i - n_{i-1}), \qquad n_0 := 0,\]

the homogeneous dimension (Heisenberg/SE(2): $Q=4$; Engel: $Q=7$; Cartan: $Q=10$; SE(3): $Q=9$). Note $Q$ exceeds the topological dimension — the hallmark of a genuinely sub-Riemannian space.

What the code measures (metric M1)

The estimator in src/growth.py reads the coordinate weights directly. It shoots unit-speed geodesics with vertical momenta spanning a wide band, measures each coordinate’s reach (a high quantile of $|x_i|$) at a grid of lengths $r$, and fits the exponent of $r$ — that exponent is the weight. Counting weights builds the growth vector; summing them gives $Q$. Two lessons the experiments forced (documented in docs/E0-growth-vector.md):

  1. the momenta must span a wide band, or a fixed distribution only samples the $wr \to 0$ regime at small $r$ and returns a spurious weight (Heisenberg’s weight-2 came out as 2.8 before the fix);
  2. the exponent fit must be noise-floor aware, $m(r) = \sqrt{(a\,r^w)^2 + b^2}$, or absolute position noise cliff-collapses the estimate.

With those, clean data recovers every growth vector exactly, and the noise/sample tradeoff is graded — with the higher-step groups (Cartan) the hardest, because their discriminating coordinate reaches only $\sim r^3$ and is the first thing noise erases.

Why groups can share a tangent cone

The tangent cone forgets everything except the bracket structure to leading order. So two genuinely different groups can share one: Heisenberg is the tangent cone of SE(2) — zoom into a parking car and the curvature of its steering washes out, leaving the flat Heisenberg model. That is why the growth vector alone cannot separate Heisenberg from SE(2) (both $(2,3)$), and why the series needs the finite-scale moduli of Appendix C6 to break that tie. The tangent cone fixes the species; the moduli fix the individual.

References

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