# A note on pliability and the openness of the multiexponential map in Carnot groups

Frédéric Jean, Mario Sigalotti, Alessandro Socionovo

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Frédéric Jean, Mario Sigalotti, Alessandro Socionovo. A note on pliability and the openness of the multiexponential map in Carnot groups. 2025. hal-05157655v2

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Distributed under a Creative Commons Attribution 4.0 International License
# A NOTE ON PLIABILITY AND THE OPENNESS OF THE MULTIEXPONENTIAL MAP IN CARNOT GROUPS

FRÉDÉRIC JEAN, MARIO SIGALOTTI, AND ALESSANDRO SOCIONOVO

**ABSTRACT.** In recent years, several notions of non-rigidity of horizontal vectors in Carnot groups have been proposed, motivated, in particular, by the characterization of monotone sets and Whitney extension properties. In this note we compare some of these notions.

## 1. INTRODUCTION

Consider a Carnot group $\mathbb{G}$ with stratified Lie algebra $\mathfrak{g} = \mathfrak{g}_1 \oplus \cdots \oplus \mathfrak{g}_s$ (see Definition 2.1). The endpoint map $E : L^\infty([0, 1], \mathfrak{g}_1) \to \mathbb{G}$ (see Definition 2.2) plays a crucial role for studying metric and topological properties of $\mathbb{G}$. The goal of this paper is to explore and clarify the connections among several openness and submersivity properties of the endpoint map and of the multiexponential map

$$ \Gamma^{(p)} : (\mathfrak{g}_1)^p \to \mathbb{G}, \quad \Gamma^{(p)}(Y_1, \dots, Y_p) := \exp(Y_p) \cdots \exp(Y_1), $$

where $p \in \mathbb{N}$ and $\exp : \mathfrak{g}_1 \to \mathbb{G}$ is the exponential map, i.e., the restriction of $E$ to $\mathfrak{g}_1$, seen as the subspace of $L^\infty([0, 1], \mathfrak{g}_1)$ made of constant functions. In particular, given a vector $X \in \mathfrak{g}_1$, we are interested in the following properties of the multiexponential map and of $E_X(\cdot) := E(X + \cdot)$, the endpoint map based at $X$:

(H) there exists $p \in \mathbb{N}$ such that the map $\Gamma^{(p)}$ is open at $(X, \dots, X) \in (\mathfrak{g}_1)^p$;

(SbH) there exists $p \in \mathbb{N}$ such that the map $\Gamma^{(p)}$ is a submersion at $(X, \dots, X) \in (\mathfrak{g}_1)^p$;

(P) the map $E$ is open at $X$ (i.e., $E_X$ is open at 0).

One of the first appearances of these conditions in the literature is the remark by Cheeger and Kleiner in [3] that in the three-dimensional Heisenberg group every $X \in \mathfrak{g}_1$ satisfies condition (SbH) with $p = 2$. As a consequence, they deduced the following *cone property*: if a horizontally convex set $C$ contains a point $p$ in its closure and an open ball $B$, then it also contains the cone over $B$ with vertex at $p$.

In [2] Arena, Caruso, and Monti generalized the submersivity of the map $(Y_1, Y_2) \mapsto \exp(Y_2) \exp(Y_1)$ to the case where $\mathbb{G}$ is of step two of Métivier type and Morbidelli proved in [8] that the cone property holds for all Carnot groups of step two.

Then Montanari and Morbidelli realized in [7] that the cone property can be obtained assuming only that every $X \in \mathfrak{g}_1$ satisfies condition (H) (which is clearly
implied by (SbH)). In addition, they showed that if a vector $X \in \mathfrak{g}_1$ satisfies condition (SbH), then the Carnot-Carathéodory distance from the unit element of $\mathcal{G}$ is Pansu differentiable at $\exp(X)$. (They also showed that (SbH) implies the related notion of *deformable direction* introduced by Pinamonti and Speight in [9]).

Condition (H) appears again in the paper [10] by Rigot and it is used to characterise precisely monotone subsets of $\mathcal{G}$. We adapt the terminology introduced in [10], by distinguishing between the (H)-condition and the submersive (H)-condition (SbH) introduced above.

On the other hand, condition (P), called *pliability*, appeared for the first time in [4], where Juillet and the second author proved the $C^1$ Whitney extension property for horizontal curves in Carnot groups where every vector in $\mathfrak{g}_1$ satisfy the property (P) (see also the recent preprint [12] by Speight and Zimmerman for a generalization to horizontal curves tangent to pliable directions).

The first question addressed in this paper is the connection between pliability and (H)-conditions.

Later, Sacchelli and the second author introduced in [11] the notion of *strong pliability (SP)*, and they improved the results of [4] for horizontal curves in sub-Riemannian manifolds where every vector in $\mathfrak{g}_1$ of the tangent Carnot group satisfies (SP). Condition (SP) and the analogous notion of *strong (H)-condition* (SH) are defined as follows:

(SP) for all $\eta > 0$ there exists $Y \in \mathfrak{g}_1$ such that $\|Y\|_{\infty} < \eta$, $E_X(Y) = E_X(0)$, and the map $E_X$ is a submersion at $Y$;

(SH) for all $\eta > 0$ there exist an integer $p \ge 2$ and $W_1, \dots, W_p \in \mathfrak{g}_1$ such that $\|W_i - X\| < \eta$ for $i = 1, \dots, p$, $\Gamma^{(p)}(W_1, \dots, W_p) = \Gamma^{(p)}(X, \dots, X)$, and $\Gamma^{(p)}$ is a submersion at $(W_1, \dots, W_p)$.

Finally, note that (SP) holds true in particular for $X \in \mathfrak{g}_1$ that are regular points of the endpoint map (those $X$ are the ones such that the straight line $t \mapsto \exp(tX)$ is not abnormal). This property motivates a last condition:

(Reg) the map $E$ is a submersion at $X$ (i.e., $E_X$ is a submersion at $0$).

Our main result is the following.

**Theorem 1.1.** The following holds:

* *pliability, strong pliability, and strong (H)-condition are equivalent*;

* *regularity (Reg) and submersive (H)-condition are equivalent and imply (H)-condition, which, in turns, implies the preceding three conditions.*

We notice that the equivalence between (H) and submersive (H)-conditions does not hold, see Remark 3.12. Theorem 1.1 is proved in Section 3 in a slightly more general version, see Theorem 3.4.
**Acknowledgments.** We are thankful to Séverine Rigot for some useful discussions we had about the topics discussed in this paper, and, in particular, for pointing out a mistake that we made in a previous version of this work.

**Research funding.** This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101034255.

<figure><img src="image_2.png" /></figure>

This project has received funding from the Fondazione Ing. Aldo Gini of Padova, Italy.

## 2. PRELIMINARIES

Let us recall some basic facts about Carnot groups, endpoint, and exponential maps.

**Definition 2.1.** A Carnot group is a nilpotent, connected, and simply connected Lie group $\mathcal{G}$ whose Lie algebra $\mathfrak{g}$ admits a stratification, i.e., a direct sum decomposition

$$ \mathfrak{g} = \mathfrak{g}_1 \oplus \cdots \oplus \mathfrak{g}_s, $$

such that $[\mathfrak{g}_1, \mathfrak{g}_k] = \mathfrak{g}_{k+1}$ for $k = 1, \dots, s-1$ and $[\mathfrak{g}_1, \mathfrak{g}_s] = 0$.

The left translation by $g \in \mathcal{G}$ is the map $L_g : \mathcal{G} \to \mathcal{G}$ defined by $L_g(x) := gx$. Through this map, we have the identification

$$ (L_g)_* : \mathfrak{g} \to T_g\mathcal{G}, \quad \mathfrak{g} \ni X \mapsto (L_g)_* X \in T_g\mathcal{G}. $$

Every vector $X \in \mathfrak{g}$ defines the left-invariant vector field $X(g) := (L_g)_* X$. With abuse of notation, we identify the vector and the induced left-invariant vector field, in particular $X = X(1_G)$.

From now on, $I := [0, 1] \subset \mathbb{R}$ denotes the unit interval. A Lipschitz continuous curve $\gamma : I \to \mathcal{G}$ is said to be horizontal if $\dot{\gamma}(t) \in \mathfrak{g}_1$ for a.e. $t \in I$, that is,

$$ (L_{\gamma(t)})^* \dot{\gamma}(t) = Y(t) \quad \text{a.e.,} $$

for some function $Y \in L^\infty(I, \mathfrak{g}_1)$, called the control of $\gamma$.

**Definition 2.2.** The endpoint map is the mapping

$$ E : L^\infty(I, \mathfrak{g}_1) \to \mathcal{G}, \quad E(Y) := \gamma(1), $$

where $\gamma$ is the unique solution to the Cauchy problem

$$ \begin{cases} (L_{\gamma(t)})^* \dot{\gamma}(t) = Y(t), & t \in I, \\ \gamma(0) = 1_G. \end{cases} $$

Given $X \in \mathfrak{g}_1$, the endpoint map based at $X$ is the mapping $E_X : L^\infty([0, 1], \mathfrak{g}_1) \to \mathcal{G}$ defined by $E_X(\cdot) := E(X + \cdot)$.
In the following, we need the following well known result, which is proved, for instance, in [1, Chapter 8].

**Lemma 2.3.** For every $X \in \mathfrak{g}_1$, the endpoint map $E_X : L^\infty(I, \mathfrak{g}_1) \to \mathbb{G}$ is of class $C^1$ as soon as $L^\infty(I, \mathfrak{g}_1)$ is endowed with the $L^2$ topology.

If $X \in \mathfrak{g}$ and $\gamma : I \to \mathbb{G}$ is the integral curve of (the left-invariant vector field identified with) $X$ with $\gamma(0) = 1_{\mathbb{G}}$, then the exponential map is defined by

$$ \exp : \mathfrak{g} \to \mathbb{G}, \quad \exp(X) := \gamma(1). $$

Notice that $\gamma(t) = \exp(tX)$. Such a curve is called a *straight line*.

We denote the flow of a left-invariant vector field $X$ by $e^{tX} : \mathbb{G} \to \mathbb{G}$. We have that

$$ e^{tX}(g) = g \exp(tX). $$

For every $\lambda \in \mathbb{R}$, we define the dilation on the Lie algebra $\delta_\lambda : \mathfrak{g} \to \mathfrak{g}$ as the linear map such that

$$ \delta_\lambda(X) = \lambda^i X, \quad \text{for all } X \in \mathfrak{g}_i. $$

By abuse of notation, we use the same symbol to denote the dilation on the group $\delta_\lambda : \mathbb{G} \to \mathbb{G}$ defined by

$$ \delta_\lambda(\exp(X)) = \exp(\delta_\lambda(X)) \qquad (2.1) $$

for every $X \in \mathfrak{g}$, so that $(\delta_\lambda)_* = \delta_\lambda$. In particular, every $X \in \mathfrak{g}_1$ and $Y \in L^\infty(I, \mathfrak{g}_1) \to \mathbb{G}$ satisfy the identity

$$ \delta_\lambda(E_X(Y)) = E_{\lambda X}(\lambda Y). \qquad (2.2) $$

### 3. PROOF OF MAIN RESULT

Let us give a slightly more general form of the definitions of (P) and (SP).

**Definition 3.1.** Let $V$ be a vector subspace of $L^\infty(I, \mathfrak{g}_1)$. A vector $X \in \mathfrak{g}_1$ is $V$-pliable if the map $E_X$ restricted to $V$ is open at 0. $L^\infty(I, \mathfrak{g}_1)$-pliability coincides with pliability as defined by condition (P).

**Definition 3.2.** Let $V$ be a vector subspace of $L^\infty(I, \mathfrak{g}_1)$. The vector $X \in \mathfrak{g}_1$ is said to be *strongly $V$-pliable* if for all $\eta > 0$ there exists $Y \in V$ such that

i) $\|Y\|_\infty < \eta$;

ii) $E_X(Y) = E_X(0)$;

iii) $E_X|_V$ is a submersion at $Y$ (i.e., the differential of $E_X|_V$ at $Y$ is surjective).

$L^\infty(I, \mathfrak{g}_1)$-strong pliability coincides with strong pliability as defined by condition (SP).
Remark 3.3. In [4, Definition 3.4] and [11, Definition 4.1] the pliability and the strong pliability of a vector $X$ are defined in terms of the openness and the submersion property of the extended map $\mathcal{E}_X(Y) := (E_X(Y), Y(1))$ restricted to the space

$$ \mathcal{C}_0 := \{Y \in C^0([0, 1], \mathfrak{g}_1) \mid Y(0) = 0\}. $$

However, in [4, Proposition 3.7] the openness of $\mathcal{E}_X|_{\mathcal{C}_0}$ is shown to be equivalent to the $\mathcal{C}_0$-pliability (in the sense of Definition 3.1), while in [11, Lemma 4.2] the strong pliability as defined in [11, Definition 4.1] is shown to be equivalent to that introduced in Definition 3.2. As we will see in Theorem 3.4 (considering both $V = \mathcal{C}_0$ and $V = L^\infty([0, 1], \mathfrak{g}_1)$), it turns out that $X$ is $\mathcal{C}_0$-pliable if and only if it is pliable.

Theorem 1.1 is a corollary of the following slightly more general result.

**Theorem 3.4.** Let $\mathbb{G}$ be a Carnot group with Lie algebra $\mathfrak{g}$. Let $V$ be a dense vector subspace of $L^\infty(I, \mathfrak{g}_1)$ for the $L^2$ topology. For every $X \in \mathfrak{g}_1$, the following conditions are equivalent:

$V$-$\langle$-$P\rangle$ $X$ is $V$-pliable;

$V$-$\langle$-$SP\rangle$ $X$ is strongly $V$-pliable;

($\mathit{SH}$) $X$ satisfies the strong ($\mathit{H}$)-condition.

Moreover, $X$ satisfies the submersive ($\mathit{H}$)-condition ($\mathit{SbH}$) if and only if $X$ is a regular point of $E$ ($\mathit{Reg}$) condition), and both conditions imply

($\mathit{H}$) $X$ satisfies the ($\mathit{H}$)-condition;

which implies the three equivalent conditions $V$-$\langle$-$P\rangle$, $V$-$\langle$-$SP\rangle$, and ($\mathit{SH}$).

The idea of the proof of Theorem 3.4 is resumed in the following scheme.

<figure><img src="image_3.png" /></figure>

Here solid arrows denote implications that immediately follow by the definitions, while dashed arrows are proved via different lemmas.

The implication $V$-$\langle$-$P\rangle$ $\Rightarrow$ $V$-$\langle$-$SP\rangle$ is a consequence of Lemmas 3.5 and 3.7 (as explained in Remark 3.8). The implications ($\mathit{H}$) $\Rightarrow$ ($\mathit{P}$) and ($\mathit{SH}$) $\Rightarrow$ ($\mathit{SP}$) are proved in Lemma 3.9. The implication $V$-$\langle$-$SP\rangle$ $\Rightarrow$ ($\mathit{SH}$) is proved in Lemma 3.10. The equivalence ($\mathit{Reg}$) $\Leftrightarrow$ ($\mathit{SbH}$) is proved in Lemma 3.11. Finally, we notice that the implications ($\mathit{Reg}$) $\Rightarrow$ ($\mathit{SP}$) and ($\mathit{SbH}$) $\Rightarrow$ ($\mathit{SH}$) cannot be reversed, see Remark 3.12.

**Lemma 3.5.** A vector $X \in \mathfrak{g}_1$ is pliable if and only if it is strongly pliable.
*Proof.* The fact that strong pliability implies pliability is trivial. The converse can be proved in the following way.

Assume that $X \in \mathfrak{g}_1$ is pliable. It results from the homogeneity property (2.2) of $E_X$ that $\frac{X}{2}$ and $-\frac{X}{2}$ are also pliable. Let $g_1 = \exp(X)$ and $g_{\frac{1}{2}} = \exp(\frac{X}{2})$. Fix $\eta > 0$ and denote by $B_\eta$ be the ball of radius $\eta$ in $L^\infty(I, \mathfrak{g}_1)$ centered at 0. By pliability of the vector $-\frac{X}{2}$, the set $\mathcal{U}_- = g_1 E_{-\frac{X}{2}}(B_\eta)$ is a neighborhood of $g_{\frac{1}{2}}$. Now we can fix $0 < \eta' < \eta$ such that $\mathcal{U} := E_{\frac{X}{2}}(B_{\eta'}) \subset \mathcal{U}_-$. Since $\frac{X}{2}$ is pliable, $\mathcal{U}$ is a neighborhood of $g_{\frac{1}{2}}$.

Consider

$$ \dot{\gamma}(s) = \frac{X(\gamma(s))}{2} + Y_s(\gamma(s)), $$

which can be seen as a control system with control $s \mapsto Y_s$ taking values in the ball of center 0 and radius $\eta'$ in $\mathfrak{g}_1$. By a well-known result in geometric control theory (see, e.g., [5, Chapter 3, Theorem 1]), there exist $k \in \mathbb{N}$, $Y_1, \dots, Y_k \in \mathfrak{g}_1$ with $\|Y_i\| < \eta'$ for $i = 1, \dots, k$, and $t_1, \dots, t_k > 0$ such that $(s_1, \dots, s_k) \mapsto e^{s_k(\frac{X}{2}+Y_k)} \circ \dots \circ e^{s_1(\frac{X}{2}+Y_1)}(1_G)$ is a submersion at $(t_1, \dots, t_k)$, with $\sum_{i=1}^k t_i$ arbitrary small. Since $t_k$ can be extended without modifying the submersion property, we can assume without loss of generality that $\sum_{i=1}^k t_i = 1$.

Let us now notice that

$$ e^{(t_j+\epsilon_j)(\frac{X}{2}+Y_j)} = e^{t_j(\frac{X}{2}+\tilde{Y}_j(\epsilon_j))}, \quad \tilde{Y}_j(\epsilon) = Y_j + \frac{\epsilon}{t_j}(\frac{X}{2} + Y_j), $$

so that $e^{(t_k+\epsilon_k)(\frac{X}{2}+Y_k)} \circ \dots \circ e^{(t_1+\epsilon_1)(\frac{X}{2}+Y_1)} = e^{t_k(\frac{X}{2}+\tilde{Y}_k(\epsilon_k))} \circ \dots \circ e^{t_1(\frac{X}{2}+\tilde{Y}_1(\epsilon_1))}$. In particular, setting $Y' \in L^\infty(I, \mathfrak{g}_1)$ which is equal to $Y_1$ on $[0, t_1)$, then to $Y_2$ on $[t_1, t_1+t_2)$, and so on, we have that $Y' \in B_{\eta'}$ and $E_{\frac{X}{2}}$ is a submersion at $Y'$.

Letting $g'_{\frac{1}{2}} = E_{\frac{X}{2}}(Y') \in \mathcal{U}$, there exists $Y'' \in B_\eta$ such that $g_1 E_{-\frac{X}{2}}(Y'') = g'_{\frac{1}{2}}$. We set

$$ Y(t) = \begin{cases} 2Y'(2t), & t \in [0, \frac{1}{2}], \\ -2Y''(2(1-t)), & t \in [\frac{1}{2}, 1]. \end{cases} $$

Then, we have $E_X(Y) = g_1$ and, since $d_{Y'} E_{\frac{X}{2}}$ is surjective, also the differential at $d_Y E_X$ is surjective. $\square$

**Remark 3.6.** Let us stress that Lemma 3.5 does leverage on the properties of the endpoint map, in the sense that for the general case of a smooth open map $F$ at a point $X$, it is not true that there exist points arbitrarily close to $X$ whose image through $F$ coincide with $F(X)$ and at which $F$ is a submersion. Consider, for instance, $F : L^\infty(I, \mathbb{R}) \ni X \mapsto (\int_I X(t)dt)^3 \in \mathbb{R}$. The map $F$ is open at 0, but for every $X \in L^\infty(I, \mathbb{R})$ with $F(X) = 0$ the differential of $F$ at $X$ is the null map.

**Lemma 3.7.** Let $V \subset W \subset L^\infty(I, \mathfrak{g}_1)$. Assume that $V$ is dense in $W$ for the $L^2$-topology and that there exists a nondecreasing sequence $\{V_p\}_{p=1}^\infty$ of subsequences of $V$
such that $V = \bigcup_{p=1}^{\infty} V_p$. Then, a vector $X \in \mathfrak{g}_1$ is strongly W-pliable if and only if there exists $p \in \mathbb{N}$ such that $X$ is strongly $V_p$-pliable.

**Proof.** Fix $\eta > 0$. If $X$ is strongly W-pliable, then there exists $Y \in W$ with $\|Y\|_{\infty} < \eta$ such that $E_X(Y) = E_X(0)$ and $dE_X|_W(Y)$ is surjective. Pick $Y_1, \dots, Y_d \in W$ such that $dE_X|_W(Y)Y_1, \dots, dE_X|_W(Y)Y_d$ is a basis of $T_{E_X(0)}\mathbb{G}$, where $d$ is the dimension of $\mathbb{G}$. Since $V$ is dense in $W$ for the $L^2$-topology, $V = \bigcup_{i=1}^{\infty} V_p$ with $V_p \subset V_{p+1}$, and the map $Z \mapsto dE_X|_W(Y)Z$ is continuous in $L^2(I, \mathfrak{g}_1)$, we can assume that $Y_i \in V_p$, for all $i = 1, \dots, d$ and for some $p \ge 1$ large enough, by a standard approximation argument. Notice also that, since $V_p \subset V \subset W \subset L^{\infty}(I, \mathfrak{g}_1)$, we have $dE_X|_V(Y)Y_i = dE_X(Y)Y_i$, for all $i = 1, \dots, d$, and thus $dE_X(Y)Y_1, \dots, dE_X(Y)Y_d$ is a basis of $T_{E_X(0)}\mathbb{G}$.

By the inverse function theorem, there exists $r > 0$ such that $E_X$ is a diffeomorphism between $D = \{Y + \alpha_1 Y_1 + \dots + \alpha_d Y_d \mid |\alpha_1|, \dots, |\alpha_d| < r\}$ and $E_X(D)$. Without loss of generality,

$$r < \frac{\eta}{\|Y_1\|_{\infty} + \dots + \|Y_d\|_{\infty}}.$$

Let $Z \in V_p$, with $p$ large enough, be such that $\|Z - Y\|_2 < \epsilon$, for some $\epsilon > 0$ to be fixed later. Notice that, possibly replacing $Z$ by $t \mapsto Z(t) \min(1, \frac{\|Y\|_{\infty}}{\|Z(t)\|})$, we can also assume that $\|Z\|_{\infty} \le \|Y\|_{\infty}$. Using a classical degree argument, similarly to in [11, Lemma 5.5], and choosing $\epsilon > 0$ small enough, we have that $E_X(0) = E_X(w)$ for some $w \in \{Z + \alpha_1 Y_1 + \dots + \alpha_d Y_d \mid |\alpha_1|, \dots, |\alpha_d| < r\} \subset V_p$ and $dE_X(w)Y_1, \dots, dE_X(w)Y_d$ is a basis of $T_{E_X(0)}\mathbb{G}$.

Notice that

$$\|w\|_{\infty} \le \|Z\|_{\infty} + r(\|Y_1\|_{\infty} + \dots + \|Y_d\|_{\infty}) < 2\eta.$$

Therefore, Properties i), ii), and iii) in Definition 3.2 are satisfied by $w \in V_p$. $\square$

<figure><img src="image_4.png" /></figure>

**Remark 3.8.** Let $V \subset L^{\infty}$ be dense for the $L^2$ topology. Then, by the two previous results, it follows that

$$\text{str. V-pliable} \implies \text{V-pliable} \implies \text{pliable} \overset{\text{Lemma 3.5}}{\implies} \text{str. pliable} \overset{\text{Lemma 3.7}}{\implies} \text{V-pliable.}$$

Here we are using Lemma 3.7 with $V_p = V$ for every $p \in \mathbb{N}$ and $W = L^{\infty}$.

**Lemma 3.9.** If a vector $X \in \mathfrak{g}_1$ satisfies the (H)-condition (respectively, the (SH)-condition), then it is pliable (respectively, strongly pliable).

**Proof.** If $X \in \mathfrak{g}_1$ satisfies the (H)-condition, there exists $p \in \mathbb{N}$ such that the map

$$(Y_1, \dots, Y_p) \mapsto \exp(Y_p) \cdots \exp(Y_1)$$

is open at $(X, \dots, X)$. Let $\eta > 0$ be such that $\{e^{Y_1} \circ \dots \circ e^{Y_p}(1_{\mathbb{G}}) \mid \|Y_i - X\| < \eta \text{ for } i = 1, \dots, p\}$ is a neighborhood of $e^{pX}(1_{\mathbb{G}})$. Let $\delta_{\frac{1}{p}}$ be the dilation in $\mathbb{G}$ of parameter $1/p$. Then, thanks to (2.1), for every $(Y_1, \dots, Y_p) \in (\mathfrak{g}_1)^p$ we have

$$e^{\frac{1}{p}Y_1} \circ \dots \circ e^{\frac{1}{p}Y_p}(1_{\mathbb{G}}) = \delta_{\frac{1}{p}}(e^{Y_1} \circ \dots \circ e^{Y_p}(1_{\mathbb{G}})).$$
Hence, the image by $E_X$ of the set

$$ \{Y \in L^{\infty}(I, \mathfrak{g}_1) \mid \|Y\|_{\infty} < \eta, Y|_{(i/p, (i+1)/p)} \text{ constant } \forall i = 0, \dots, p-1\} $$

is a neighborhood of $e^X(1_G)$, which implies that $X$ is pliable.

The proof in the case of the (SH)-condition is similar. □

**Lemma 3.10.** If $X \in \mathfrak{g}_1$ is strongly pliable, then it satisfies the (SH) condition.

*Proof.* Let $X \in \mathfrak{g}_1$ be strongly pliable, and denote by $PC = PC(I, \mathfrak{g}_1)$ the subspace of $L^{\infty}(I, \mathfrak{g}_1)$ made of piecewise constant functions with rational discontinuity points. By Remark 3.8, $X$ is strongly PC-pliable.

Fix $\eta > 0$. By the strong PC-pliability, there exist $Z, Y_1, \dots, Y_d \in PC$, with $\|Z\|_{\infty} < \eta$, such that $E_X(Z) = E_X(0)$ and $dE_X(Z)Y_1, \dots, dE_X(Z)Y_d$ is a basis of $T_{E_X(0)}\mathbb{G}$. By definition of the space PC, there exists an integer $p \ge 2$ such that $Z|_{((i-1)/p, i/p)}, Y_1|_{((i-1)/p, i/p)}, \dots, Y_n|_{((i-1)/p, i/p)}$ are constant for every $i = 1, \dots, p$.

Denote by $Z_i$ and $Y_{j,i}$ the values of $Z$ and $Y_j$ on the interval $(\frac{i-1}{p}, \frac{i}{p})$. Then the map $\Gamma^{(p)}: (\mathfrak{g}_1)^p \to \mathbb{G}$ defined by

$$ \Gamma^{(p)}(W_1, \dots, W_p) := e^{W_1} \circ \dots \circ e^{W_p}(1_{\mathbb{G}}), $$

satisfies $\Gamma^{(p)}(X + Z_1, \dots, X + Z_p) = \Gamma^{(p)}(X, \dots, X)$. The proof is completed by showing that $\Gamma^{(p)}$ is a submersion at $(X + Z_1, \dots, X + Z_p)$.

We have that $Y_j = \sum_{i=1}^p Y_{j,i} \chi_{((i-1)/p, i/p)}$ and so

$$ dE_X(Z)Y_j = \sum_{i=1}^{p} dE_X(Z)Y_{j,i}\chi_{(\frac{i-1}{p}, \frac{i}{p})}. $$

Moreover,

$$ \begin{align} & dE_X(Z)Y_{j,i}\chi_{(\frac{i-1}{p}, \frac{i}{p})} \nonumber \\ &= \left. \frac{d}{d\varepsilon} \right|_{\varepsilon=0} \left( e^{\frac{X+Z_p}{p}} \circ \dots \circ e^{\frac{X+Z_{i+1}}{p}} \circ e^{\frac{X+Z_i+\varepsilon Y_{j,i}}{p}} \circ e^{\frac{X+Z_{i-1}}{p}} \circ \dots \circ e^{\frac{X+Z_1}{p}} \right) \nonumber \\ &\stackrel{(2.1)}{=} \left. \frac{d}{d\varepsilon} \right|_{\varepsilon=0} \delta_{\frac{1}{p}} \left( e^{X+Z_p} \circ \dots \circ e^{X+Z_{i+1}} \circ e^{X+Z_i+\varepsilon Y_{j,i}} \circ e^{X+Z_{i-1}} \circ \dots \circ e^{X+Z_1} \right) \nonumber \\ &= \left. (\delta_{\frac{1}{p}})_* \frac{d}{d\varepsilon} \right|_{\varepsilon=0} \left( e^{X+Z_p} \circ \dots \circ e^{X+Z_{i+1}} \circ e^{X+Z_i+\varepsilon Y_{j,i}} \circ e^{X+Z_{i-1}} \circ \dots \circ e^{X+Z_1} \right) \nonumber \\ &= \left. (\delta_{\frac{1}{p}})_* \frac{d}{d\varepsilon} \right|_{\varepsilon=0} \Gamma^{(p)}(X+Z_p, \dots, X+Z_{i+1}, X+Z_i+\varepsilon Y_{j,i}, X+Z_{i-1}, \dots, X+Z_1) \nonumber \\ &= (\delta_{\frac{1}{p}})_* d\Gamma^{(p)}(X+Z_1, \dots, X+Z_p) \cdot (0, \dots, 0, Y_{j,i}, 0, \dots, 0). \tag{3.1} \end{align} $$

So the image of the differential of $\Gamma^{(p)}$ at $(X+Z_1, \dots, X+Z_p)$ contains the image by $(\delta_p)_* = (\delta_{\frac{1}{p}})_*^{-1}$ of the span of $dE_X(Z)Y_1, \dots, dE_X(Z)Y_d$. Hence $\Gamma^{(p)}$ is a submersion at $(X+Z_1, \dots, X+Z_p)$. □
**Lemma 3.11.** A vector $X \in g_1$ satisfies the submersive (H)-condition (SbH) if and only if it is a regular point of the endpoint map E.

*Proof.* On the one hand, if $X \in g_1$ satisfies (SbH), then a computation analogous to (3.1) with $Z=0$ shows that $dE_X(0)$ is onto. On the other hand, note that the argument of Lemma 3.7 shows that, if $E_X$ is a submersion at 0, then $E_X$ restricted to PC is also a submersion at 0. Then the argument used in the proof of Lemma 3.10 shows that $X$ satisfies the submersive (H)-condition. $\square$

*Remark 3.12.* The (H)-condition is weaker than the submersive (H)-condition (SbH). In particular, also pliability, strong pliability, and the (SH)-condition are weaker than (SbH). Indeed, step 2 Carnot groups are known to satisfy the (H)-condition [8, Theorem 2.1], while non-Métivier step two Carnot groups admit abnormal curves of the form $t \mapsto \exp(tX)$ (see [6, Proposition 3.6]) and, hence, vectors $X$ that do not satisfy the submersive (H)-condition.

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UNITÉ DE MATHÉMATIQUES APPLIQUÉES, ENSTA, INSTITUT POLYTECHNIQUE DE PARIS,
91120 PALAISEAU, FRANCE

Email address: frederic.jean@ensta.fr

LABORATOIRE JACQUES-LOUIS LIONS, SORBONNE UNIVERSITÉ, UNIVERSITÉ DE PARIS, CNRS,
INRIA, PARIS, FRANCE

Email address: mario sigalotti@inria.fr, alessandro.socionovo@sorbonne-universite.fr