A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries

: Carnotgroupsaredistinguishedspacesthatarerichofstructure:theyarethoseLiegroupsequipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks. We consider them as special cases of graded groups and as homogeneous metric spaces. We discuss the regularity of isometries in the general case of Carnot-Carathéodory spaces and of nilpotent metric Lie groups.

metric measure theory, and other aspects of analysis or geometry, in Carnot groups. On the one hand, with so much structure many results in the Euclidean setting generalize to Carnot groups. The most celebrated example is Pansu's version of Rademacher's theorem for Lipschitz maps (see [85]). Other results that have been generalized are the Isoperimetric Inequality [82], the Poincaré Inequality [52], the Nash Embedding Theorem [60], the Myers-Steenrod Regularity Theorem [32], and, at least partially, the De-Giorgi Structure Theorem [37]. On the other hand, Carnot groups exhibit fractal behavior, the reason being that on such metric spaces the only curves of nite length are the horizontal ones. In fact, except for the Abelian groups, even if the distance is de ned by a smooth subbundle, the squared distance is not smooth and the Hausdor dimension of the space di ers from the topological dimension. Moreover, these spaces contain no subset of positive measure that is biLipschitz equivalent to a subset of a Euclidean space and do not admit biLipschitz embedding into re exive Banach spaces, nor into L , see [4,[29][30][31]93]. For this reason, nonAbelian Carnot groups, together with the classical fractals and boundaries of hyperbolic groups, are the main examples in an emerging eld called 'non-smooth analysis', in addition see [14,15,17,18,24,28,42,43,47,48,53,54,59,66,73,92]. A non-exhaustive, but long-enough list of other contribution in geometric measure theory on Carnot groups is [5-9, 12, 22, 33, 34, 47, 49, 56-58, 63, 67-69, 71, 76, 77, 79, 81, 86, 88, 89], and more can be found in [61,91].
In this essay, we shall distinguish between Carnot groups, homogeneous groups, and strati ed groups. In fact, the latter are Lie groups with a particular kind of grading, i.e., a strati cation. Hence, they are only an algebraic object. Instead, homogeneous groups¹ are Lie groups that are arbitrarily graded but are equipped with distances that are one-homogeneous with respect to the dilations induced by the grading. Finally, for the purpose of this paper, the term Carnot group will be reserved for those strati ed groups that are equipped with homogeneous distances making them Carnot-Carathéodory spaces. Namely, they are equipped with a subFinsler distance. It is obvious that up to biLipschitz equivalence every Carnot group has one unique geometric structure. This is the reason why this term sometimes replaces the term strati ed group.
From the metric viewpoint, Carnot groups are peculiar examples of isometrically homogeneous spaces. In this context we shall di erently use the term 'homogeneous': it means the presence of a transitive group action. In fact, on Carnot groups left-translations act transitively and by isometries. In some sense, these groups are quite prototypical examples of geodesic homogeneous spaces. An interesting result of Berestovskii gives that every isometrically homogeneous space whose distance is geodesic is indeed a Carnot-Carathéodory space, see Theorem 5.5. Hence, Carnot-Carathéodory spaces and Carnot groups are natural examples in metric geometry. This is the reason why they appear in di erent contexts.
A purpose of this essay is to present the notion of a Carnot group from di erent viewpoints. As we said, Carnot groups have very rich metric and geometric structures. However, they are easily characterizable as geodesic spaces with self-similarity. Here is an equivalent de nition, which is intermediate between the standard one (Section 3.3) and one of the simplest axiomatic ones (Theorem 6.2). A Carnot group is a Lie group G endowed with a left-invariant geodesic distance d admitting for each λ > a bijection δ λ : G → G such that d(δ λ (p), δ λ (q)) = λd(p, q), ∀p, q ∈ G. (0.1) In this paper we will pursue an Aristotelian approach: we will begin by discussing the examples, starting from the very basic ones. We rst consider Abelian Carnot groups, which are the nite-dimensional normed spaces, and then the basic non-Abelian group: the Heisenberg group. After presenting these examples, in Section 2 we will formally discuss the de nitions from the algebraic viewpoint, with some basic properties. In particular, we show the uniqueness of strati cations in Section 2.2. In Section 3 we introduce the homogeneous distances and the general de nition of Carnot-Carathéodory spaces. In Section 3.5 we show the continuity of homogeneous distances with respect to the manifold topology. Section 4 is devoted to present Carnot 1 Originally, the term homogeneous group has been used by Folland and Stein for those simply connected Lie groups whose Lie algebra is endowed with Lie algebra automorphisms of the form exp (log λ)A λ> , where A is diagonalizable and has positive eigenvalues, see [36, page 4]. The existence of one such a family is equivalent to the presence of a positive grading of the Lie algebra, the layers being the eigenspaces of A. Folland and Stein assume that, up to replacing A with αA, the smallest of the eigenvalues of A is 1. In fact, in this case such maps are dilations for some left-invariant distance, see [50]. groups as limits. In particular, we consider tangents of Carnot-Carathéodory spaces. In Section 5 we discuss isometrically homogeneous spaces and explain why Carnot-Carathéodory spaces are the only geodesic examples. In Section 6 we provide a metric characterization of Carnot groups as the only spaces that are locally compact geodesic homogeneous and admit a dilation. Finally, in Section 7 we overview several results that provide the regularity of distance-preserving homeomorphisms in Carnot groups and more generally in homogeneous groups and Carnot-Carathéodory spaces.

Prototypical examples
We start by reviewing basic examples of Carnot groups. First we shall point out that Carnot groups of step one are in fact nite-dimensional normed vector spaces. Afterwards, we shall recall the simplest noncommutative example: the Heisenberg group, equipped with various distances. Example 1.1. Let V be a nite-dimensional vector space, so V is isomorphic to R n , for some n ∈ N. In such a space we have • a group operation (sum) p, q → p + q, • dilations p → λp for each factor λ > .
We are interested in the distances d on V that are (i) translation invariant: (ii) one-homogeneous with respect to the dilations: These are the distances coming from norms on V. Indeed, setting v = d ( , v) for v ∈ V, the axioms of d being a distance together with properties (i) and (ii), give that · is a norm.
A geometric remark: for such distances, straight segments are geodesic. By de nition, a geodesic is an isometric embedding of an interval. Moreover, if the norm is strictly convex, then straight segments are the only geodesic.
An algebraic remark: each ( nite-dimensional) vector space can be seen as a Lie algebra (in fact, a commutative Lie algebra) with Lie product [p, q] = , for all p, q ∈ V. Via the obvious identi cation points/vectors, this Lie algebra is identi ed with its Lie group (V , +).
One of the theorem that we will generalize in Section 7 is the following.

Theorem 1.2. Every isometry of a normed space xing the origin is a linear map.
An analytic remark: the de nition of (directional) derivatives of a function f between vector spaces makes use of the group operation, the dilations, and the topology. We shall consider more general spaces on which these operations are de ned.
This operation gives a non-Abelian group structure on R . This group is called Heisenberg group.
We shall consider distances that are (i) left (translation) invariant: (ii) one-homogeneous with respect to the dilations: These distances, called 'homogeneous', are completely characterized by the distance from a point, in fact, just by the unit sphere at a point.
This function is δ λ -homogeneous and satis es the triangle inequality. To check that it satis es the triangle inequality we need to show that p · q ≤ p + q . First, and analogously for the y component. Second, The group structure induces left-invariant vector elds, a basis of which is where the in mum is over the piecewise smooth curves γ ∈ C ∞ pw ([ , ]; R ) with γ( ) = p, γ( ) = q, anḋ γ(t) = a(t)X γ(t) + b(t)Y γ(t) . Such a d is a homogenous distance and for all p, q ∈ R there exists a curve γ realizing the in mum. These distances are examples of CC distances also known as subFinsler distances. Example 1.3.3. (Korányi distance) Another important example is given by the Cygan-Korányi distance: The feature of such a distance is that it admits a conformal inversion, see [26, p.27]. The Korányi distance and the box distance are not CC distances.
The space R with the above group structure is an example of Lie group, i.e., the group multiplication and the group inversion are smooth maps. The space g of left-invariant vector elds (also known as the Lie algebra) has a peculiar structure: it admits a strati cation. Namely, setting V := span {X, Y} and V := span{Z}, As an exercise, verify that Z = [X, Y].

Strati cations
In this section we discuss Lie groups, Lie algebras, and their strati cations. We recall the de nitions and we point out a few remarks. In particular, we show that a group can be strati ed in a unique way, up to isomorphism.

. De nitions
Given a group G we denote by gh or g · h the product of two elements g, h ∈ G and by g − the inverse of g. A Lie group is a di erentiable manifold endowed with a group structure such that the map G×G → G, (g, h) → g − ·h is C ∞ . We shall denote by e the identity of the group and by Lg(h) := g · h the left translation. Any vector X in the tangent space at the identity extends uniquely to a left-invariant vector eldX, asXg = (dLg)e X, for g ∈ G.
The Lie algebra associated with a Lie group G is the vector space Te G equipped with the bilinear operation de ned by [X, Y] := [X,Ỹ]e, where the last bracket denotes the Lie bracket of vector elds, i.e., [X,Ỹ] := XỸ −ỸX.
The general notion of Lie algebra is the following: A Lie algebra g over R is a real vector space together with a bilinear operation [·, ·] : g × g → g, called the Lie bracket, such that, for all x, y, z ∈ g, one has All Lie algebras considered here are over R and nite-dimensional. In what follows, given two subspaces V , W of a Lie algebra, we set [V , W] := span{[X, Y]; X ∈ V , Y ∈ W}.
De nition 2.1 (Strati able Lie algebras). A strati cation of a Lie algebra g is a direct-sum decomposition g = V ⊕ V ⊕ · · · ⊕ Vs for some integer s ≥ , where Vs ≠ { } and [V , V j ] = V j+ for all integers j ∈ { , . . . , s} and where we set V s+ = { }. We say that a Lie algebra is strati able if there exists a strati cation of it. We say that a Lie algebra is strati ed when it is strati able and endowed with a xed strati cation called the associated strati cation.
A strati cation is a particular example of grading. Hence, for completeness, we proceed by recalling the latter. More considerations on this subject can be found in [68].
De nition 2.2 (Positively graduable Lie algebras). A positive grading of a Lie algebra g is a family (V t ) t∈( ,+∞) of linear subspaces of g, where all but nitely many of the V t 's are { }, such that g is their direct sum We say that a Lie algebra is positively graduable if there exists a positive grading of it. We say that a Lie algebra is graded (or positively graded, to be more precise) when it is positively graduable and endowed with a xed positive grading called the associated positive grading.
Given a positive grading g = ⊕ t> V t , the subspace V t is called the layer of degree t (or degree-t layer) of the positive grading and non-zero elements in V t are said to have degree t. The degree of the grading is the maximum of all positive real numbers t > such that V t ≠ { }.
Given two graded Lie algebras g and h with associated gradings g = ⊕ t> V t and h = ⊕ t> W t , a morphism of the graded Lie algebras is a Lie algebra homomorphism ϕ : g → h such that ϕ(V t ) ⊆ W t for all t > . Hence, two graded Lie algebras g and h are isomorphic as graded Lie algebras if there exists a bijection ϕ : g → h such that both ϕ and ϕ − are morphisms of the graded Lie algebras.
Recall that for a Lie algebra g the terms of the lower central series are de ned inductively by g ( ) = g, g (k+ ) = [g, g (k) ]. A Lie algebra g is called nilpotent if g (s+ ) = { } for some integer s ≥ and more precisely we say that g nilpotent of step s if g (s+ ) = { } but g (s) ≠ { }.

Remark 2.3.
A positively graduable Lie algebra is nilpotent (simple exercise similar to Lemma 2.16). On the other hand, not every nilpotent Lie algebra is positively graduable, see Example 2.8. A strati cation of a Lie algebra g is equivalent to a positive grading whose degree-one layer generates g as a Lie algebra. Therefore, given a strati cation, the degree-one layer uniquely determines the strati cation and satis es However, an arbitrary vector space V that is in direct sum with [g, g] (i.e., satisfying (2.4)) may not generate a strati cation, see Example 2.9. Any two strati cations of a Lie algebra are isomorphic, see Section 2.2.
A strati able Lie algebra with s non-trivial layers is nilpotent of step s. Every 2-step nilpotent Lie algebra is strati able. However, not all graduable Lie algebras are strati able, see Example 2.7. Strati able and graduable Lie algebras admit several di erent grading: given a grading ⊕V t , one can de ne the so-called s-power as the new grading ⊕W t by setting W t = V t/s , where s > . Moreover, for a strati able algebra it is not true that any grading is a power of a strati cation, see Example 2.6. Up to isomorphisms of graded Lie algebras and up to powers, these non-standard gradings give all the possible positive gradings of h that are not a strati cation.

Example 2.7.
Consider the -dimensional Lie algebra g generated by X , . . . , X with only non-trivial brack- This Lie algebra g admits a grading but it is not strati able.
Example 2.8. There exist nilpotent Lie algebras that admit no positive grading. For example, consider the Lie algebra of dimension 7 with basis X , . . . , X with only non-trivial relations given by [X , Example 2.9. We give here an example of a strati able Lie algebra g for which one can nd a subspace V in direct sum with [g, g] but that does not generate a strati cation. We consider g the strati able Lie algebra of step 3 generated by e , e and e and with the relation [e , e ] = . Then dim g = and a strati cation of g is generated by V := span{e , e , e }. De nition 2.10 (Positively graduable, graded, strati able, strati ed groups). We say that a Lie group G is a positively graduable (respectively graded, strati able, strati ed) group if G is a connected and simply connected Lie group whose Lie algebra is positively graduable (respectively graded, strati able, strati ed).
For the sake of completeness, in Theorem 2.14 below we present an equivalent de nition of positively graduable groups in terms of existence of a contractive group automorphism. The result is due to Siebert [95].
De nition 2.11 (Dilations on graded Lie algebras). Let g be a graded Lie algebra with associated positive grading g = ⊕ t> V t . For λ > , we de ne the dilation on g (relative to the associated positive grading) of factor λ as the unique linear map δ λ : g → g such that The family of all dilations (δ λ ) λ> is a one-parameter group of Lie algebra isomorphisms, i.e., δ λ • δη = δ λη for all λ, η > .
Exercise 2.12. Let g and h be graded Lie algebras with associated dilations δ g λ and δ h λ . Let ϕ : g → h be a Lie algebra homomorphism. Then ϕ is a morphism of graded Lie algebras if and only if ϕ Given a Lie group homomorphism ϕ : G → H, we denote by ϕ * : g → h the associated Lie algebra homomorphism. If G is simply connected, given a Lie algebra homomorphism ψ : g → h, there exists a unique Lie group homomorphism ϕ : G → H such that ϕ * = ψ (see [100,Theorem 3.27]). This allows us to de ne dilations on G as stated in the following de nition.
De nition 2.13 (Dilations on graded groups). Let G be a graded group with Lie algebra g. Let δ λ : g → g be the dilation on g (relative to the associated positive grading of g) of factor λ > . The dilation on G (relative to the associated positive grading) of factor λ is the unique Lie group automorphism, also denoted by δ λ : G → G, such that (δ λ ) * = δ λ .
For technical simplicity, one keeps the same notation for both dilations on the Lie algebra g and the group G. There will be no ambiguity here. Indeed, graded groups being nilpotent and simply connected, the exponential map exp : g → G is a di eomorphism from g to G (see [ For the sake of completeness, we give now an equivalent characterization of graded groups due to Siebert [95]. If G is a topological group and τ : G → G is a continuous group isomorphism, we say that τ is contractive if, for all g ∈ G, one has lim k→∞ τ k (g) = e. We say that G is contractible if G admits a contractive isomorphism.
For graded groups, dilations of factor λ < are contractive isomorphisms, hence positively graduable groups are contractible. Conversely, Siebert proved (see Theorem 2.14 below) that if G is a connected locally compact group and τ : G → G is a contractive isomorphism then G is a connected and simply connected Lie group and τ induces a positive grading on the Lie algebra g of G (note however that τ itself may not be a dilation relative to the induced grading).

Theorem 2.14. [95, Corollary 2.4] A topological group G is a positively graduable Lie group if and only if G is a connected locally compact contractible group.
Sketch of the proof. Regarding the non-trivial direction, by the general theory of locally compact groups one has that G is a Lie group. Hence, the contractive group isomorphism induces a contractive Lie algebra isomorphism ϕ. Passing to the complexi ed Lie algebra, one considers the Jordan form of ϕ with generalized eigenspaces Vα, α ∈ C. The t-layer V t of the grading is then de ned as the real part of the span of those Vα with − log |α| = t. Remark 2.15. A distinguished class of groups in Riemannian geometry are the so-called Heintze groups. They are those groups that admit a structure of negative curvature. It is possible to show that such groups are precisely the direct product of a graded group N times R where R acts on N via the grading, see [Hei74].

. Uniqueness of strati cations
We start by observing the following simple fact. Lemma 2.16. If g = V ⊕ · · · ⊕ Vs is a strati ed Lie algebra, then In particular, g is nilpotent of step s. Now we show that the strati cation of a strati able Lie algebra is unique up to isomorphism. Hence, also the structure of a strati ed group is essentially unique. Proposition 2.17. Let g be a strati able Lie algebra with two strati cations, Then s = t and there is a Lie algebra automorphism A : g → g such that A(V i ) = W i for all i.
Extend A to a linear map A : g → g. This is clearly a linear isomorphism and A(V i ) = W i for all i. We need to show that A is a Lie algebra morphism, i.e., [Aa, . On the other hand, we have Aa i ∈ W i and Ab j − b j ∈ g (j+ ) , so [Aa i , Ab j − b j ] ∈ g (i+j+ ) . Hence, we have

Metric groups
In this section we present homogeneous distances on groups. The term homogeneous should not be confused with its use in the theory of Lie groups. Indeed, clearly a Lie group is a homogeneous space since it acts on itself by left translations and in fact we will only consider distances that are left-invariant. We shall use the term homogeneous as it is done in harmonic analysis to mean that there exists a family of dilations. Around 1970 Stein introduced a de nition of homogeneous group as a simply connected Lie group whose Lie algebra is equipped with a family of maps of the form exp (log λ)A λ> , where A is a diagonalizable derivation² of the Lie algebra with³ eigenvalues ≥ . Equivalently, a homogeneous group is a graded group with trivial layers of degrees < . Even if in Stein's de nition there is no distance, it exactly leads to the class of Lie groups that admit⁴ a homogeneous distance, which is de ned as follows.
Some remarks are due: (1) Conditions (i) and (ii) imply that the distance is admissible, i.e., it induces the manifold topology, and in fact d is a continuous function and the group is connected⁵; (2) The maps {δ λ } λ> are of the form where A is a derivation of g with eigenvalues having real part ≥ , see [64]. However, there are homogeneous distances where the associated operator A is not diagonalizable (not even on the complex numbers). Also, on some Lie groups there are derivations with eigenvalues having real part ≥ which do not appear from a family of dilations of some homogeneous distance. One can nd examples in [64,102].
(3) Being the automorphisms δ λ , with λ ∈ ( , ), contractive, there exists a positive grading of the Lie algebra of the group, recall Theorem 2.14. Since the maps are of the form (3.2), a grading of the Lie algebra g can be de ned in terms of A as follows. Let g C be the complexi ed Lie algebra and Wα the generalized eigenspace of A : g C → g C with respect to α ∈ C. Then V t = g ∩ s∈R W t+is de nes the layers of a grading for g. Moreover, all layers of degree < are trivial.
(4) Every graded group with trivial layers of degree < admits a homogeneous distance with respect to the family of dilations induced by the grading, (recall De nition 2.11). This last result is due to Hebisch and Sikora, see [50], but also [68, Section 2.5].
(5) We point out that there are homogeneous distances that are not biLipschitz equivalent to any distance that is homogeneous with respect to the dilations relative to a positive grading. See [64,102].
2 Recall that the space of derivations naturally form the Lie algebra of the space of Lie algebra automorphisms, where the exponential map is the matrix exponential. 3 In the literature there is a bit of confusion whether in the de nition one assumes that the minimum eigenvalue is 1. However, this can be obtained up to rescaling A. 4 Namely, on the one hand, every homogeneous group admits a homogeneous distance with respect to the dilations relative to the grading. On the other hand, every Lie group admitting a homogeneous distance admits the structure of homogeneous group (however, the maps of the homogeneous-group structure may be di erent from the maps of the homogenous-distance structure, see remarks after De nition 3.1). 5 These claims are not trivial. However, in Section 3.5 we provide the proof in the case the group is positively graded and the maps {δ λ } are the dilations relative to the grading, see [64] for the general case.

. Abelian groups
The basic example of a homogeneous group is provided by Abelian groups, which admit the strati cation where the degree-one stratum V is the whole Lie algebra. Homogenous distances for this strati cation are the distances induced by norms, cf. Section 1.1.

. Heisenberg group
Heisenberg group equipped with CC-distance, or box distance, or Korányi distance is the next important example of homogeneous group, cf. Section 1.3.

. Carnot groups
Since on strati ed groups the degree-one stratum V of the strati cation of the Lie algebra generates the whole Lie algebra, on these groups there are homogeneous distances that have a length structure de ned as follows.
Let G be a strati ed group. So G is a simply connected Lie group and its Lie algebra g has a strati cation: Fix a norm · on V . Identify the space g as Te G so V ⊆ Te G. By left translation, extend V and · to a left-invariant subbundle ∆ and a left-invariant norm · : Using piecewise C ∞ curves tangent to ∆, we de ne the CC-distance associated with ∆ and · as Since ∆ and · are left-invariant, d is left-invariant. Since, δ λ v = λv for v ∈ V , and hence δ λ v = λ v for v ∈ ∆, the distance d is one-homogeneous with respect to δ λ .
The fact that V generates g, implies that for all p, q ∈ G there exists a curve γ ∈ C ∞ ([ , ]; G) withγ ∈ ∆ joining p to q. Consequently, d is nite valued.
We call the data (G, δ λ , ∆, · , d) a Carnot group, or, more explicitly, subFinsler Carnot group. Usually, the term Carnot group is reserved for subRiemannian Carnot group, i.e., when the norm comes from a scalar product.

. Carnot-Carathéodory spaces
Carnot groups are particular examples of a more general class of spaces. These spaces have been named after Carnot and Carathéodory by Gromov. However, in the work of Carnot and Carathéodory there is very little about these kind of geometries (one can nd some sprout of a notion of contact structure in [23] and in the adiabatic processes' formulation of Carnot). Therefore, we should say that the pioneer work has been done mostly by Gromov and Pansu, see [40,44,[81][82][83][84][85]97], see also [41,42].

De nition 3.4.
Let M be a smooth manifold. Let ∆ be a bracket-generating subbundle of the tangent bundle of M. Let · be a 'smoothly varying' norm on ∆. Analogously, using (3.3) with G replaced by M, one de nes the CC-distance associated with ∆ and · . Then (M, ∆, · , d) is called Carnot-Carathéodory space (or also CC-space or subFinsler manifold).

. Continuity of homogeneous distances
We want to motivate now the fact that a homogeneous distance on a group induces the correct topology. Such a fact is also true when one considers quasi distances and dilations that are not necessarily R-diagonalizable automorphisms. For the sake of simplicity, we present here the simpler case of distances that are homogeneous with respect to the dilations relative to the associated positive grading. The argument is taken from [68]. The more general proof will appear in [64].

Proposition 3.5. Every homogeneous distance induces the manifold topology.
Proof in the case of standard dilations. Let G be a Lie group with identity element e and Lie algebra graded by g = ⊕ s> Vs. Let d be a distance homogeneous with respect to the dilations relative to the associated positive grading. It is enough to show that the topology induced by d and the manifold topology give the same neighborhoods of the identity.
We show that if p converges to e then d(e, p) → . Since G is nilpotent, we can consider coordinates using a basis X , . . . , Xn of g adapted to the grading. Namely, rst for all i = , . . . , n there is d i > such that X i ∈ V d i , and consequently, δ λ (X) = λ d i X. There is a di eomorphism p → (P (p), . . . , Pn(p)) from G to R n such that for all p ∈ G p = exp(P (p)X ) · . . . · exp(Pn(p)Xn).
Notice that P i (e) = . Then, using the triangle inequality, the left invariance, and the homogeneity of the distance, we get We show that if d(e, pn) → then pn converges to e. By contradiction and up to passing to a subsequence, there exists ϵ > such that pn > ϵ, where · denotes any auxiliary Euclidean norm. Since the map λ → δ λ (q) is continuous, for all q ∈ G, then for all n there exists λn ∈ ( , ) such that δ λn (pn) = ϵ. Since the Euclidean ϵ-sphere is compact, up to subsequence, δ λn (pn) → q, with q = ϵ so q ≠ e and so d(e, q) > . However, < d(e, q) ≤ d(e, δ λn (pn)) + d(δ λn (pn), q) = λn d(e, pn) + d(δ λn (pn), q) → , where at the end we used the fact, proved in the rst part of this proof, that if qn converges to q then d(qn , q) → .

Remark 3.6.
To deduce that a homogeneous distance is continuous it is necessary to require that the automorphisms used in the de nition are continuous with respect to the manifold topology. Otherwise, consider the following example. Via a group isomorphism from R to R , pull back the standard dilations and the Euclidean metric from R to R. In this way we get a one-parameter family of dilations and a homogeneous distance on R . Clearly, this distance gives to R the topology of the standard R.

Limits of Riemannian manifolds . . A topology on the space of metric spaces
Let X and Y be metric spaces, L ≥ and C > .
If A, B ⊂ Y are subsets of a metric space Y and ϵ > , we say that A is an ϵ-net for B if De nition 4.1 (Hausdor approximating sequence). Let (X j , x j ), (Y j , y j ) be two sequences of pointed metric spaces. A sequence of maps ϕ j : X j → Y j with ϕ(x j ) = y j is said to be Hausdor approximating if for every R > and every δ > there exists (ϵ j ) j∈N such that 1. ϵ j → as j → ∞; 2. ϕ j | B(x j ,R) is a ( , ϵ j )-quasi isometric embedding; 3. ϕ j (B(x j , R)) is an ϵ j -net for B(y j , R − δ).

De nition 4.2.
We say that a sequence of pointed metric spaces (X j , x j ) converges to a pointed metric space (Y , y) if there exists an Hausdor approximating sequence ϕ j : (X j , x j ) → (Y , y).
This notion of convergence was introduced by Gromov [40] and it is also called Gromov-Hausdor convergence. It de nes a topology on the collection of (locally compact, pointed) metric spaces, which extends the notion of uniform convergence on compact sets for distances on the same topological space.

De nition 4.3.
If X = (X, d) is a metric space and λ > , we set λX = (X, λd). Let X, Y be metric spaces, x ∈ X and y ∈ Y. We say that (Y , y) is the asymptotic cone of X if for all λ j → , (λ j X, x) → (Y , y). We say that (Y , y) is the tangent space of X at x if for all λ j → ∞, (λ j X, x) → (Y , y).

Remark 4.4.
The notion of asymptotic cone is independent from x. In general, asymptotic cones and tangent spaces may not exist. In the space of boundedly compact metric spaces, limits are unique up to isometries.

. Tangents to CC-spaces: Mitchell Theorem
The following result states that Carnot groups are in nitesimal models of CC-spaces: the tangent metric space to an equiregular subFinsler manifold is a subFinsler Carnot group. We recall that a subbundle ∆ ⊆ TM is de nes a subbundle of TM. For example, on a Lie group G any G-invariant subbundle is equiregular. For the next theorem see [11,51,[72][73][74]. How to prove Theorem 4.6. In four steps: Step 1. Let (M, ∆, · , d) be the subFinsler structure. Consider a frame X , . . . , Xn of TM that is adapted to the subbundles (4.5). Consider exponential local coordinates. i.e., (x , . . . , xn) → exp(x X + · · · + xn Xn)(p).
These coordinates are examples of privileged coordinates, see [51].
Step 2. De ne approximate dilation maps δ λ with respect to privileged coordinates, and show that for all j one has λ − (δ λ ) * X j →X j , as λ → ∞, whereX j form a frame of vector elds.
Step 3. Show thatX , . . . ,Xn are a local frame at the origin. Moreover, they form a basis for the Lie algebra of a strati ed group.

. Asymptotic cones of nilmanifolds
We point out another theorem showing how Carnot groups naturally appear in Geometric Group Theory. In particular, they are important in the quasi-isometric classi cation of nilpotent groups, which is still an open problem, see [94]. Theorem 4.7 (Pansu). If G is a nilpotent simply connected Lie group equipped with a left-invariant subFinsler distance d, then the asymptotic cone of (G, d) exists and is a subFinsler Carnot group.
How to prove Theorem 4.7. In ve steps: Step 1. Fix direct summands V j such that g (k) = V k ⊕ g (k+ ) , so we get a direct sum decomposition of the Lie algebra g of G De ne the map δ λ : g → g to be the linear map such that δ λ (X) := λ j X, for every X ∈ V j and every j = , . . . , s. De ne a Lie bracket on g by Such a Lie bracket induces on G a product structure, denoted by *, that makes (G, *) a strati ed Lie group. Notice that if G is strati able and the V j 's are the strata of a strati cation, then the group structure does not change.
Step 2. Consider the norm on V by the push forward of the subFinsler structure on g via the projection π : g → V with kernel V ⊕· · ·⊕Vs. Namely, if the subFinsler structure on G is determined by a Lie bracket generating subspace V ⊆ g and a norm · on V, then the pushed-forward norm · * of v ∈ V is v * := inf{ w : w ∈ V , π(w) = v}.
Extend the norm left-invariantly with respect to *. Thus we get an associated subFinsler distance d∞. Hence, (G, *, d∞) is a subFinsler Carnot group, and it will be the asymptotic cone of (G, d). What one wants to prove is that ϵ(δϵ) * d → d∞ uniformly on compact sets, as ϵ → .
Step 3. For the sake of simplicity, for the rest of this sketched argument we assume that the group structure does not change. For p ∈ G, let γ : [ , ] → G be a geodesic for d with γ( ) = id and γ( ) = p. If γ ∈ g denotes the control of γ, consider the curve σ : [ , ] → G from the identity with control π(γ ). So L d (γ) ≥ L d∞ (σ). Show that d(γ( ), σ( )) = o(d(id, p)), as p → ∞, and use this to deduce that Step 4. Let σ : [ , ] → G be a geodesic for d∞ from the identity to a point p. By de nition of the norm · * , there exists a curveσ from the identity for which π • σ = π •σ and L d (σ) ≤ L d∞ (σ). Since the two curves σ and σ have same projection under π one proves that the nal point ofσ di ers from p by o (d(id, p)).
Step 5. From step 2 and 3, one deduces that This implies the conclusion.
Theorem 4.7 is due to Pansu, see [83], where the more general cases of nilmanifolds and of nitely generated nilpotent groups are considered. Nowadays, there are other proofs of such result that are more quanti ed in the sense that a bound of rate of convergence is given, see [20,21].

Isometrically homogeneous geodesic manifolds . Homogeneous metric spaces
Let X = (X, d) be a metric space. Let Iso(X) be the group of self-isometries of X, i.e., distance-preserving homeomorphisms of X.
The main examples are the following. Let G be Lie group and H < G compact subgroup. So the collection of the cosets G/H := {gH : g ∈ G} is an analytic manifold on which G acts (on the left) by analytic maps. Since H is compact, there always exists a G-invariant Riemannian distance d on G/H.
We present some regularity results for isometries of such spaces. Later, in Section 7.1 we shall give more details. Ottazzi). The isometries of G/H as above are analytic maps.
The above result is due to the author and Ottazzi, see [65]. The argument is based on the structure theory of locally compact groups, see next two sections. Later, in Theorem 7.1, we will discuss a stronger statement.
We present now an immediate consequence saying that in these examples the isometries are also Riemannian isometries for some Riemannian structure. In general, the Riemannian isometry group may be larger, e.g., in the case of R with -norm. Examples of intrinsic distances are given by length structures, subRiemannian structures, Carnot-Carathéodory spaces, and geodesic spaces. A metric space whose distance is intrinsic is called geodesic if the in mum in De nition 5.4 is always attained. The signi cance of the next result is that CC-spaces are natural objects in the theory of homogeneous metric spaces.

Corollary 5.3. For each admissible G-invariant distance d on G/H there is a Riemannian G-invariant distance d R on G/H such that
Theorem 5.5 (Berestovskii, [13]). If a homogeneous Lie space G/H is equipped with an admissible G-invariant intrinsic distance d, then d is subFinsler, i.e., there exist a G-invariant bracket-generating subbundle ∆ and a G-invariant norm · such that d is the CC distance associated to ∆ and · .
How to prove Berestovskii's result. In three steps: Step 1. Show that locally d is ≥ to some Riemannian distance d R .
Step 2. Deduce that curves that have nite length with respect to d are recti able (in any coordinate system) and de ne the horizontal bundle ∆ using velocities of such curves. Similarly, de ne · using tangents of nite-length curves.
Step 3. Conclude that d is the CC distance for ∆ and · . A posteriori we know that ∆ is bracket-generating, since d is nite-valued.
The core of the argument is in Step 1. Hence we describe its proof in an exemplary case.
Simpli ed proof of Step 1. We only consider the following simpli cation: As a consequence, the isometrically homogeneous metric spaces considered in Theorem 6.1 are all of the form discussed in Section 5 after De nition 5.1.
How to prove Theorem 6.1. In many steps: Step 1. Iso(X) is locally compact, by Ascoli-Arzelá Theorem.
There exists an open (and closed) subgroup G of Iso(X) that can be approximated by Lie groups:

(inverse limit of continuous epimorphisms with compact kernel).
This step is called Main Approximation Theorem in the structure of locally compact groups. It is due mainly to Gleason and Yamabe and it is based on Peter-Weyl Theorem, see [103].
Step 3. G X transitively, by Baire Category Theorem; so X = G/H = lim Step 4. Since the topological dimension of X is nite and X is locally connected, for i large G i /H i → G/H is a homeomorphism.
Step 5. G = G i for i large, so G is a Lie group so G is NSS, i.e., G has no small subgroups.
Step 6. the fact that G is NSS implies that Iso(X) is NSS.
Step 7. Locally compact groups with NSS are Lie groups, by Gleason Theorem [39].

. A metric characterization of Carnot groups
In what follows, we say that a metric space (X, d) is self-similar if there exists λ > such that the metric space (X, d) is isometric to the metric space (X, λd). In other words, there exists a homeomorphism f : X → X such that d(f (p), f (q)) = λd(p, q), for all p, q ∈ X.
When this happens for all λ > (and all maps f = f λ x a common point) X is said to be a cone. Homogeneous groups are examples of cones.
The following result is a corollary of the work of Gleason-Montgomery-Zippin, Berestovskii, and Mitchell, see [62]. It gives a metric characterization of Carnot groups. Sketch of the proof. Each such metric space X is connected and locally connected. Using the conditions of local compactness, self-similarity, and homogeneity, one can show that X is a doubling metric space. In particular, X is nite dimensional. By the result of Gleason-Montgomery-Zippin (Theorem 6.1) the space X has the structure of a homogeneous Lie space G/H and, by Berestovskii's result (Theorem 5.5), as a metric space X is an equiregular subFinsler manifold. By Mitchell's result (Theorem 4.6), the tangents of X are subFinsler Carnot groups. Since X is self-similar, X is isometric to its tangents.

Isometries of metric groups . Regularity of isometries for homogeneous spaces
Let X be a metric space that is connected, locally connected, locally compact, with nite topological dimension, and isometrically homogeneous. By Montgomery-Zippin, G := Iso(X) has the structure of (analytic) Lie group, which is unique. Fixing x ∈ X, the stabilizing subgroup H := Stab G (x ) is compact. Therefore, G/H has an induced structure of analytic manifolds. We have that X is homeomorphic to G/H and G acts on G/H analytically. One may wonder if X could have had a di erent di erentiable structure. The next result says that this cannot happen. .

Isometries of Carnot groups
We now explain in further details what are the isometries of Carnot groups and some of their generalizations. We summarize our knowledge with the following results.
• [78] implies that the global isometries are smooth, since Carnot groups are homogeneous spaces.
• [65] implies that isometries between open subsets of subFinsler Carnot groups are a ne, i.e., composition of translations and group homomorphisms. • [55] implies that isometries of nilpotent connected metric Lie groups are a ne.
In the rest of this exposition we give more explanation on the last three points.
. Local isometries of Carnot groups Theorem 7.3 (LD, Ottazzi, [65]). Let G , G be subFinsler Carnot groups and for i = , consider Ω i ⊂ G i open sets. If F : Ω → Ω is an isometry, then there exists a left translation τ on G and a group isomorphism ϕ between G and G , such that F is the restriction to Ω of τ • ϕ, which is a global isometry.
Sketch of the proof. In four steps: Step 1. We may assume that the distance is subRiemannian, regularizing the subFinsler norm.
Step 2. F is smooth; this is a PDE argument using the regularity of the subLaplacian, see the next section.
Step 3. F is completely determined by its horizontal di erential (dF) He G , see the following Corollary 7.6.
Step 4. The Pansu di erential (PF)e exists and is an isometry with the same horizontal di erential as F at e, see [85].
We remark that in Theorem 7.3 the assumption that Ω i are open is necessary, unlike in the Euclidean case. However, these open sets are not required to be connected.

. Isometries of subRiemannian manifolds
The regularity of subRiemannian isometries should be thought of as a two-step argument, where as an intermediate result one obtains the preservation of a good measure: the Popp measure. A good introduction to the notion of Popp measure can be found in [19]. Both of the next results are due to the author in collaboration with Capogna, see [32]. . SubRiemannian isometries are determined by the horizontal di erential Corollary 7.6. Let M and N be two connected equiregular subRiemannian manifolds. Let p ∈ M and let ∆ be the horizontal bundle of M. Let f , g : M → N be two isometries. If f (p) = g(p) and df | ∆p = dg| ∆p , then f = g.
The proof can be read in [65,Proposition 2.8]. Once we know that the isometries xing a point are a compact Lie group of smooth transformations, the argument is an easy exercise in di erential geometry.

. Isometries of nilpotent groups
The fact that Carnot isometries are a ne (Theorem 7.3) is a general feature of the fact that we are dealing with a nilpotent group. In fact, isometries are a ne whenever they are globally de ned on a nilpotent connected group. Here we obviously require that the distances are left-invariant and induce the manifold topology. For example, this is the case for arbitrary homogeneous groups. Theorem 7.7 (LD, Kivioja). Let N and N be two nilpotent connected metric Lie groups. Any isometry F : N → N is a ne.
This result is proved in [55] with algebraic techniques by studying the nilradical, i.e., the biggest nilpotent ideal, of the group of self-isometries of a nilpotent connected metric Lie group. The proof leads back to a Riemannian result of Wolf, see [101].

. Two isometric non-isomorphic groups
In general isometries of a subFinsler Lie group G may not be a ne, not even in the Riemannian setting. As counterexample, we take the universal covering groupG of the group G = E( ) of Euclidean motions of the plane. This group is also called roto-translation group. One can see that there exists a Riemannian distance onG that makes it isometric to the Euclidean space R . In particular, they have the same isometry group. However, a straightforward calculation of the automorphisms shows that not all isometries xing the identity are group isomorphisms ofG. As a side note, we remark that the groupG admits a left-invariant subRiemannian structure and a map into the subRiemannian Heisenberg group that is locally biLipschitz. However, these two spaces are not quasi-conformal, see [35].
Examples of isometric Lie groups that are not isomorphic can be found also in the strict subRiemannian context. There are three-dimensional examples, see [1]. Also the analogue of the roto-translation construction can be developed.