Carnot Groups: Structure & Applications

• Carnot groups are connected, simply connected nilpotent Lie groups with a stratified Lie algebra and automorphic dilations, fundamental in sub-Riemannian geometry.
• They feature the Carnot–Carathéodory metric defined via horizontal curves and are Ahlfors-regular, satisfying doubling and Poincaré inequalities.
• These groups provide concrete models, such as the Heisenberg and Engel groups, enabling insights into hypoelliptic PDEs, functional inequalities, and measure theory.

A Carnot group is a connected, simply connected, finite-dimensional nilpotent Lie group equipped with a left-invariant geodesic distance that is homogeneous under a 1-parameter family of automorphic dilations. These spaces are fundamental objects in sub-Riemannian geometry, metric measure theory, geometric analysis, and the theory of hypoelliptic PDEs, often serving as local models for general sub-Riemannian manifolds. The distinctive algebraic structure (stratification), geometric features (homogeneous and doubling), analytic apparatus (sub-Laplacian, Poincaré inequalities), and their rich metric geometry underlie a broad spectrum of applications and ongoing research problems.

1. Stratified Structure and Dilations

Let $G$ be a Carnot group with Lie algebra $\mathfrak{g}$. The essential structure is that of a stratified Lie algebra of step $s\geq1$: $$\mathfrak{g} = V_1 \oplus V_2 \oplus \cdots \oplus V_s,$$ with $[V_1,V_i] = V_{i+1}$ ($i=1,\ldots,s-1$), $V_{s+1}=\{0\}$, and $V_1$ generating $\mathfrak{g}$ as a Lie algebra. This stratification makes $\mathfrak{g}$ nilpotent and induces a canonical family of automorphic dilations: $$\delta_\lambda\left(\sum_{i=1}^s v_i\right) = \sum_{i=1}^s \lambda^i v_i, \quad v_i\in V_i,\, \lambda>0,$$ which exponentiate to group automorphisms $\delta_\lambda\colon G\to G$.

The exponential map $\exp\colon \mathfrak{g}\to G$ is a global diffeomorphism, allowing one to use exponential coordinates and express group multiplication through the truncated Baker–Campbell–Hausdorff formula. The subspace $V_1$ (the horizontal layer) can be equipped with an inner product or norm, extended by left-translation to a horizontal distribution $\Delta\subset TG$, which is bracket-generating.

The homogeneous (Hausdorff) dimension is $Q = \sum_{i=1}^s i\,\dim V_i$. This dimension governs scaling of the Haar measure under dilations: $\mu(\delta_\lambda(A)) = \lambda^Q \mu(A)$.

2. Carnot–Carathéodory Geometry and Metric Characterization

A fundamental geometric feature is the Carnot–Carathéodory (CC) distance, defined using admissible (horizontal) curves $\gamma\colon[0,1]\to G$ whose derivatives lie almost everywhere in the left-invariant distribution $\Delta$ derived from $V_1$. If the norm on $V_1$ is smooth, the length of such curves is

$L(\gamma) = \int_0^1 \|\dot\gamma(t)\|\,dt,$

and the CC distance is

$$d_{CC}(p,q) = \inf \{ L(\gamma) : \gamma(0)=p,\, \gamma(1)=q,\, \dot\gamma \in \Delta \}.$$

This is a left-invariant length distance, homogeneous under dilations: $$d_{CC}(\delta_\lambda(p),\delta_\lambda(q)) = \lambda d_{CC}(p,q)$$. With the induced measure, $(G,d_{CC},\mu)$ is a proper geodesic metric measure space, Ahlfors $Q$-regular, and doubling.

A core result is the metric characterization: among proper geodesic spaces, those which are isometrically homogeneous and admit at least one dilation are exactly the subFinsler Carnot groups. This was proven via Lie-theoretic and subFinsler geometry arguments (Donne, 2013), with further details relating to tangents and the Berestovskiĭ–Mitchell structure theorem (Donne, 2016).

3. Concrete Examples and Classification

The class of Carnot groups is broad, ranging from abelian groups (Euclidean spaces) to highly nonabelian nilpotent groups. Notable examples include:

• Heisenberg Group $H^n$: Step 2, $V_1=\mathbb{R}^{2n}$, $V_2=\mathbb{R}$, nontrivial bracket $[X_j,Y_j]=Z$. The classical sub-Riemannian model, with CC distance agreeing with the standard structure on $H^n$ (Donne, 2016, Donne et al., 2020).
• Engel Group: Step 3, with $V_1$ of dimension 2 and nonzero brackets $[X_1,X_2]=X_3$, $[X_1,X_3]=X_4$.
• Corank 1 Groups: Step-2 with $V_2$ one-dimensional, including all Heisenberg groups and other nontrivial 2-step nilpotents (Rizzi, 2015).
• Free-nilpotent Groups: All free nilpotent Lie algebras with a stratification are Carnot, and explicit lists exist in low dimensions (Donne et al., 2020).
• Filiform Groups: Maximally non-abelian step $n-1$ groups with a recursive bracket $[X_1,X_j]=X_{j+1}$.

A systematic classification exists up to dimension 7 (Donne et al., 2020), with further explicit lists and algebraic properties (center, growth vectors, etc.) provided.

4. Analytic and Geometric Properties

4.1 Doubling, Poincaré, and Isoperimetric Inequalities

Carnot groups with their CC structure are Ahlfors regular, satisfying $\mu(B(p,r))\sim r^Q$. By Jerison’s theorem, they admit $(1,1)$-Poincaré inequalities (Donne, 2016). The structure supports a rich calculus—left-invariant horizontal derivatives $X_i$ span $V_1$ and define the canonical sub-Laplacian $L = \sum X_i^2$. This operator is hypoelliptic (Baudoin et al., 2015), and the heat kernel satisfies two-sided Gaussian bounds.

Reverse Poincaré inequalities hold for the heat semigroup,

$$\Gamma(P_t f, P_t f) \leq \frac\Lambda t \left( P_t(f^2) - (P_t f)^2 \right),$$

with $\Lambda$ optimal and depending on the homogeneous dimension and the structure matrix of the heat kernel gradients. This underpins sharp isoperimetric inequalities and the $L^p$ boundedness of the (horizontal) Riesz transform (Baudoin et al., 2015).

4.2 Measure-Contraction, Geodesic Dimension, and Fatness

Carnot groups exhibit a nuanced relationship with measure contraction properties (MCP). For corank-1 Carnot groups, MCP$(K,N)$ holds iff $K\le0$ and $N\ge k+3$, where $k$ is the rank of $V_1$ (Rizzi, 2015). The geodesic dimension $\mathfrak{d}$, often strictly larger than the Hausdorff dimension, governs such properties: for corank-1, $\mathfrak{d}=k+3$.

Moreover, normal vs. abnormal geodesics and the concept of fatness (where $D_x+[X,D]_x=T_xM$ for any $X$) are tightly linked: a Carnot group is fat iff it is "ideal", i.e., has no nontrivial abnormal minimizers, and in this case the geodesic dimension matches the classical Carnot formula (Rizzi, 2015).

4.3 Poincaré and Log–Sobolev Inequalities

Recent advances have established that probability measures with suitably singular potentials (so-called "taming singularities" technique) on Carnot groups yield functional inequalities:

• Explicitly constructed measures secure Poincaré inequalities provided the coercivity function $$V_2(x) = \tfrac{1}{4}(|\nabla U|^2 - \Delta U) \to \infty$$ at infinity.
• Additional bounds on the potential $U$ (e.g., $\nabla U + U \le a(1+V_2)$) yield logarithmic Sobolev inequalities (Dagher et al., 2021).

5. Metric and Measure Geometry: Isodiametric Inequality, Rectifiability, and Boundaries

5.1 Isodiametric Inequality and Spherical Measures

Contrary to the Euclidean case, balls in general Carnot groups are not isodiametric for every homogeneous distance: for the $d_\infty$ and standard CC distances, balls can fail to maximize volume at fixed diameter, and thus the sharp isodiametric inequality fails generically except for specific cases (Rigot, 2010). The quotient of spherical to Hausdorff $Q$-measure, $C_d$, is strictly greater than 1 in these cases, with consequences for geometry and rectifiability.

5.2 Pure Unrectifiability and the $1/2$–Besicovitch Problem

Carnot groups are purely $Q$-unrectifiable: every Lipschitz image from $\mathbb{R}^Q$ has zero $\mathcal{H}_d^Q$-measure, due to the failure of the isodiametric inequality (Rigot, 2010). This provides counterexamples to the generalized $1/2$–Besicovitch density conjecture, as the density constant $\sigma_Q$ can be strictly greater than $1/2$.

5.3 Horofunction Boundary and Piecewise Linearity

The horofunction boundary, a metric compactification, can be described purely algebraically for Carnot groups with layered sup norms: every horofunction is a piecewise-linear function, obtained as “max–plus” combinations of Pansu derivatives on $V_1$ (Fisher, 12 Aug 2024). For higher Heisenberg groups, the boundary is full-dimensional (of codimension 1); however, for filiform groups with $n\geq 8$ the boundary exhibits a drop in dimension.

6. Regularity and Rigidity of Isometries

A cornerstone result is Pansu's affine rigidity theorem: every global isometry of a Carnot group with CC distance is a composition of a left translation and a stratification-preserving automorphism—hence smooth and affine in exponential coordinates (Donne, 2016). Pansu differentiability holds for every Lipschitz map between Carnot groups: the derivative is a group homomorphism that respects the stratification (Donne, 2013). Local isometries extend uniquely to affine maps (Donne, 2016).

Furthermore, the regularity theory has important consequences for sub-Laplacians and hypoelliptic PDEs, as isometries preserve the sub-Laplacian and hence are necessarily smooth by hypoelliptic regularity (Donne, 2016).

7. Recent Developments: Hypergenerated Groups and Flatness

A new algebraic concept is that of hypergenerated Carnot groups: stratified Lie algebras such that for any codimension-$k$ subspace $P\subset V_1$, certain inclusions of higher layers into the Lie algebra generated by $P$ are satisfied (Donne et al., 31 Mar 2025). Hypergenerated groups are exactly those for which boundaries with locally constant normal are locally flat hypersurfaces; equivalently, in these groups, the embedding of non-characteristic hypersurfaces is locally bi-Lipschitz. This algebraic–geometric rigidity extends to submanifolds of arbitrary codimension, with explicit structural criteria via the Kaplan operator for step-2 groups. Examples include all Heisenberg groups with $n\geq 2$ and certain higher-rank Carnot groups, while explicit non-hypergenerated counterexamples demonstrate the necessity of the condition (Donne et al., 31 Mar 2025).

• "A primer on Carnot groups: homogenous groups, CC spaces, and regularity of their isometries" (Donne, 2016)
• "Measure contraction properties of Carnot groups" (Rizzi, 2015)
• "Isodiametric inequality in Carnot groups" (Rigot, 2010)
• "A metric boundary theory for Carnot groups" (Fisher, 12 Aug 2024)
• "Reverse Poincare inequalities, Isoperimetry, and Riesz transforms in Carnot groups" (Baudoin et al., 2015)
• "Coercive Inequalities on Carnot Groups: Taming Singularities" (Dagher et al., 2021)
• "A metric characterization of Carnot groups" (Donne, 2013)
• "A Cornucopia of Carnot groups in Low Dimensions" (Donne et al., 2020)
• "Hypergenerated Carnot groups" (Donne et al., 31 Mar 2025)