Direct limits of infinite-dimensional Carnot groups
aa r X i v : . [ m a t h . M G ] J a n DIRECT LIMITS OF INFINITE-DIMENSIONAL CARNOT GROUPS
TERHI MOISALA AND ENRICO PASQUALETTO
Abstract.
We give a construction of direct limits in the category of complete metric scalablegroups and provide sufficient conditions for the limit to be an infinite-dimensional Carnot group.We also prove a Rademacher-type theorem for such limits.
Contents
1. Introduction 11.1. Overview 11.2. Construction of direct limits 21.3. A Rademacher-type theorem 31.4. Structure of the paper 42. Preliminaries 42.1. Basic notions in metric geometry 42.2. Reminder on direct and inverse limits 62.3. Scalable groups 73. Direct limits of (complete) metric scalable groups 93.1. Scalable groups with distances 93.2. Metric scalable groups 113.3. Complete metric scalable groups 144. Infinite-dimensional Carnot groups 174.1. Direct limits of Carnot groups 194.2. Example of a degenerate direct system of Carnot groups 215. A Rademacher-type theorem for direct limits of CMSGs 22Appendix A. Inverse limits of complete metric scalable groups 24References 261.
Introduction
Overview.
The main purpose of this paper is to study direct limits of Carnot groups. Carnotgroups are characterized as the locally compact, geodesic metric spaces, that are isometricallyhomogeneous and admit dilations (see [LD13]). During the last decades, they have provided afruitful framework for Geometric Analysis and Metric Geometry (see [LD17] for a comprehensiveintroduction to Carnot groups). Recently, there has been growing interest in generalizing Carnotgroups into infinite dimensions. A notion of infinite-dimensional Heisenberg group based on anabstract Wiener space was introduced in [DG08, BGM13] and Lie groups generalizing those in[Mel09]. A form of infinite-dimensional sub-Riemannian geometry from control theoretic viewpoint
Date : January 11, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Carnot group, Gˆateaux differential, Rademacher theorem, direct limit, scalable group. was suggested in [GMV15]. In [MR13, MPS19], a Rademacher-type theorem has been proven whenthe target is a so called Banach homogeneous group, which is a Banach space equipped with asuitable non-abelian group structure. In a recent paper [DZ19], the authors study inverse limitsof free nilpotent Lie groups.By studying direct limits of Carnot groups we continue, from a constructive viewpoint, thedevelopment of infinite-dimensional generalization of Carnot groups. Our starting point is thenotion of infinite-dimensional Carnot group introduced by E. Le Donne, S. Li, and the first namedauthor in [LDLM19]. The direct limits are considered – in the categorical sense – in the context of (complete) metric scalable groups (briefly, a (C)MSG). Metric scalable groups are metric groupsendowed with a family { δ λ } λ ∈ R ⊆ Aut( G ) of dilations of G (see Definition 3.6). By an infinite-dimensional Carnot group G we mean a CMSG admitting a filtration by Carnot subgroups , meaningthat there exists an increasing sequence N ⊆ N ⊆ . . . ⊆ G of Carnot groups whose union isdense in G ; the precise definition is recalled in Definition 4.2.It is rather straightforward to show that every (infinite-dimensional) Carnot group is a directlimit of finite-dimensional Carnot groups (see Proposition 4.6) in the category of CMSGs. A morechallenging task is to understand when a direct system of infinite-dimensional Carnot groups hasa direct limit in the category of CMSGs, and when the limit is an infinite-dimensional Carnotgroup itself. We give now an overview of our results.1.2. Construction of direct limits.
We start our consideration from scalable groups, that inthis paper are groups equipped with dilation automorphisms. It turns out (not surprisingly) thatdirect limits of scalable groups always exist, since they are purely algebraic objects and no topologyis involved. Their existence will be proved (via an explicit construction) in Theorem 2.7.In the setting of MSGs the situation is much more delicate, as we are now going to describe. Inanalogy with the case of Banach spaces, one might heuristically expect that – calling (cid:0) G ′ , { ϕ ′ i } i ∈ I (cid:1) the direct limit of our direct system in the category of scalable groups – the direct limit as a MSGcan be built as follows: first, we define the pseudodistance d ′ (i.e., distinct points may have zerodistance) on G ′ as d ′ ( x, y ) := inf n d i ( x i , y i ) (cid:12)(cid:12)(cid:12) i ∈ I, x i , y i ∈ G i , ϕ ′ i ( x i ) = x, ϕ ′ i ( y i ) = y o for every x, y ∈ G ′ ;second, we consider the quotient G ′ / ∼ , where x ∼ y if and only if d ′ ( x, y ) = 0, which inherits anatural structure of metric space. This sort of construction will be investigated in Section 3.2 andis called the metric limit (see Definition 3.7). In the case of CMSGs, the candidate direct limit G is given by the metric completion ( G, d ) of the metric limit G ′ / ∼ .In Banach spaces all the vector space operations are automatically Lipschitz-continuous. Infact, direct limits of Banach spaces always exist and are exactly the completions of metric limits.However, in the context of (C)MSGs, the metric limit does not necessarily provide us with a(C)MSG. The obstruction comes from the fact that there is no reason why right translations,inversion, and dilation on G ′ should be continuous with respect to d ′ ; only left translations mustbe continuous due to the left-invariance of d ′ . Consequently, one cannot expect the scalable groupstructure to carry over to G ′ / ∼ and to its completion G , and even if it does, the resulting space G does not need to be a topological group. We give an example of this troublesome phenomenonin Section 4.2.In Definitions 3.8 and 3.13 we introduce some conditions on the continuity of the operationson G ′ that guarantee the well-posedness of the metric limit and its metric completion. Whenthe direct system in the category of MSGs or CMSGs satisfies these conditions, we say that it is non-degenerate in the corresponding category. Our main result is the following: IRECT LIMITS OF INFINITE-DIMENSIONAL CARNOT GROUPS 3
Theorem 1.1 (Direct limits of (C)MSGs) . Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) be a direct system of (complete)metric scalable groups. Then the direct limit exists and equals the (completion of the) metric limitif and only if the direct system is non-degenerate. The non-degeneracy assumptions can be formulated as a quantitative condition on the directsystem (see Propositions 3.9 and 3.14). In the general case the formulation is quite involved,but in some specific circumstances of interest many simplifications occur; for instance, when theelements of the direct system have uniformly bounded nilpotency step (Corollary 3.18) or have aCarnot group structure (Theorem 4.8). We remark that these kinds of questions on continuity ofgroup operations have also been of independent interest, see e.g. [Mon36] for a classical result and[Tka14] for a survey on semitopological groups.We stress that the non-degeneracy condition is necessary and sufficient for the fact that themetric completion of the metric limit is a metric scalable group, but it is not clear if this fact isequivalent to the existence of the direct limit. A problem that still remains open is the following:P1) Is there a direct system of CMSGs that is degenerate, whose direct limit exists?Regarding the question if infinite-dimensional Carnot groups are stable under the operation oftaking direct limits, we provide a sufficient criterion. In this paper we say that a direct system iscountable if the indexing set I is countable. Theorem 1.2 (Direct limits of infinite-dimensional Carnot groups) . Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) bea countable non-degenerate direct system of infinite-dimensional Carnot groups in the categoryof CMSGs. If each group G i is nilpotent, then the direct limit is an infinite-dimensional Carnotgroup. The complete picture of the relation between nilpotency of the groups G i and non-degeneracyof the direct system remains unclear; we record the following open problems.P2) Can we remove the nilpotency assumption in the statement of Theorem 1.2?P3) If (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) is a direct system of infinite-dimensional Carnot groups such thatthe nilpotency steps of the groups G i are uniformly bounded, is it non-degenerate?Let us also mention that inverse limits of CMSGs can be treated in a similar way. Indeed, inverselimits of scalable groups always exist (see Theorem 2.8), but at the level of CMSGs, the expectedconstruction of an inverse limit might fail to work. We characterize when inverse limits exist andare of the desired form (Theorem A.4) in terms of a suitable non-degeneracy condition (DefinitionA.1). Appendix A will be dedicated to inverse limits of CMSGs, but we do not insist further onthem, as in this paper we will not provide any of their possible applications. For the same reason,we leave the study of general limits and colimits in this category for future research.1.3. A Rademacher-type theorem.
We study the stability of CMSGs satisfying a Rademachertheorem under taking direct limits. One of the main results that have been achieved in [LDLM19],concerning infinite-dimensional Carnot groups, is a variant of the classical Rademacher’s theoremon the almost everywhere differentiability of Lipschitz functions. Roughly speaking, the result saysthat each Lipschitz function on a given infinite-dimensional Carnot group is Gˆateaux differentiableat almost every point, where the notion of ‘negligible set’ is expressed in terms of the filtrationby Carnot subgroups. A key ingredient in the proof is the celebrated Pansu–Rademacher theoremfor Lipschitz functions on Carnot groups [Pan89].In this paper, we say that a couple ( G, N ) – where G is a CMSG and N is a σ -ideal of nullsets , in the sense of Definition 5.1 – has the Rademacher property if for any Lipschitz function f : G → R the set of points where f is not Gˆateaux differentiable (in the sense of Definition 5.3) TERHI MOISALA AND ENRICO PASQUALETTO belongs to N ; see Definition 5.5. The main result regarding the Rademacher property reads asfollows: Theorem 1.3 (Rademacher theorem for direct limits) . Let (cid:8) ( G i , N i ) (cid:9) i ∈ I be a countable directsystem of complete metric scalable groups having the Rademacher property. If the system admitsa direct limit G , then ( G, N ) has the Rademacher property for a natural σ -ideal N of null sets(depending on the direct system). Theorem 1.3 generalizes the Rademacher’s theorem for infinite-dimensional Carnot groupsproven in [LDLM19]. Indeed, by Proposition 4.6, if G is an infinite-dimensional Carnot groupwith a filtration ( G i ) i ∈ N by Carnot groups, then G is obtained as the direct limit of ( G i ) i ∈ N . Thetheorem in [LDLM19] is recovered by choosing ( N i ) i ∈ N to be the σ -ideals of null sets of some Haarmeasures on the Carnot groups G i , where the Rademacher property of the pairs ( G i , N i ) is givenby [Pan89].Our definition of the σ -ideal N on the direct limit G closely follows along the constructionpresented in [LDLM19], which is in turn inspired by what done in [Aro76]. However, while in[Aro76, LDLM19] it is shown that N is non-trivial (in the sense that its elements have emptyinterior, so in particular every Lipschitz function has at least one differentiability point), it seemsthat – at our level of generality – nothing about N can be said. It would be interesting to findother sufficient conditions for the σ -ideal N to be non-trivial.1.4. Structure of the paper.
In Sections 2.1 and 2.2 we recall some results on Cauchy-continuityand the notions of direct and inverse limits in the categorical sense, respectively. The existenceof limits in the category of scalable groups is proven in Section 2.3. In Section 3.1 we provesome auxiliary continuity results on scalable groups with compatible distances. Sections 3.2 and3.3 are devoted to the direct limits of MSGs and CMSGs, respectively. Direct limits of infinite-dimensional Carnot groups are discussed in Section 4. The problem of stability of CMSGs satis-fying a Rademacher theorem under taking direct limits will be addressed in Section 5. Finally, inAppendix A we briefly investigate the existence of inverse limits of (C)MSGs.2.
Preliminaries
Basic notions in metric geometry.
Let ( I, ≤ ) be a directed set , meaning that ≤ is apreorder (i.e., a reflexive and transitive binary relation) on I with the property that any twoelements have an upper bound (i.e., for any i, j ∈ I there exists k ∈ I such that i ≤ k and j ≤ k ).A net indexed over I in a given set X is any map x : I → X. We shall often denote x by { x i } i ∈ I .If X is a topological space, then the net { x i } i ∈ I is said to converge to a point ¯ x ∈ X provided forevery neighborhood U of ¯ x there exists i ∈ I such that x i ∈ U for every i ∈ I such that i ≥ i .In such case, we write ¯ x = lim i ∈ I x i . Definition 2.1 (Cauchy-continuity) . Let (X , d X ) , (Y , d Y ) be pseudometric spaces. Let ϕ : X → Y be continuous. Then we say that ϕ is Cauchy-continuous provided it satisfies the following property: ( x n ) n ⊆ X is Cauchy = ⇒ (cid:0) ϕ ( x n ) (cid:1) n ⊆ Y is Cauchy. By a metric completion of a given metric space (X , d ) we mean any couple ( ¯X , ι ), where ( ¯X , ¯ d )is a complete metric space, while ι : X ֒ → ¯X is an isometric embedding having dense image. Itholds that ( ¯X , ι ) is uniquely determined up to unique isomorphism, meaning that for any othercouple ( ¯X ′ , ι ′ ) satisfying the same property there exists a unique isometric bijection Φ : ¯X → ¯X ′ such that Φ ◦ ι = ι ′ . Occasionally, we will implicitly identify X with the subspace ι (X) of ¯X, andwe will just say that ¯X is the metric completion of X (without mentioning the embedding ι ). IRECT LIMITS OF INFINITE-DIMENSIONAL CARNOT GROUPS 5
Let us recall the following well-known fact. For the reader’s convenience, we sketch its proof.
Theorem 2.2 (Extension theorem) . Let (X , d X ) , (Y , d Y ) be metric spaces. Suppose Y is complete.Let ϕ : X → Y be a continuous map. Denote by ¯X the metric completion of X . Then ϕ admits a(unique) continuous extension ¯ ϕ : ¯X → Y if and only if it is Cauchy-continuous.Proof. Let us denote by ι : X ֒ → ¯X the isometric embedding that comes with the metric completion. Necessity.
Suppose ϕ admits a continuous extension ¯ ϕ : ¯X → Y. Let ( x n ) n be a Cauchy sequencein X. Then there exists ¯ x ∈ ¯X such that ι ( x n ) → ¯ x . By using the continuity of the map ¯ ϕ , wededuce that ϕ ( x n ) = ¯ ϕ (cid:0) ι ( x n ) (cid:1) → ¯ ϕ (¯ x ). In particular, (cid:0) ϕ ( x n ) (cid:1) n is a Cauchy sequence in Y. Sufficiency.
Suppose ϕ is Cauchy-continuous. We define the map ¯ ϕ : ¯X → Y as follows: givenany ¯ x ∈ ¯X, we define ¯ ϕ (¯ x ) ∈ Y as the limit of ϕ ( x n ) as n → ∞ , where ( x n ) n ⊆ X is any sequencesatisfying ι ( x n ) → ¯ x . Notice that such a sequence ( x n ) n exists (by density of ι (X) in ¯X) and isCauchy, the sequence (cid:0) ϕ ( x n ) (cid:1) n ⊆ Y is Cauchy (thanks to the Cauchy-continuity of ϕ ), and thelimit lim n ϕ ( x n ) ∈ Y exists because of the completeness of Y. The well-posedness of ¯ ϕ can beeasily checked, while its continuity follows from a standard diagonalization argument. To showthat the map ¯ ϕ extends ϕ , just consider for any point ¯ x = ι ( x ) ∈ ι (X) the sequence ( x n ) n that isconstantly equal to x . Finally, the uniqueness of ¯ ϕ is granted by the density of ι (X) in ¯X. (cid:3) Definition 2.3 (Modulus of continuity) . Let (X , d X ) and (Y , d Y ) be two pseudometric spaces. Fixany map ϕ : X → Y . Then for any x ∈ X and ε > we define the quantity ω ϕ ( x ; ε ) ∈ [0 , + ∞ ] as ω ϕ ( x ; ε ) := sup n η > (cid:12)(cid:12)(cid:12) d Y (cid:0) ϕ ( x ) , ϕ ( x ′ ) (cid:1) < ε for every x ′ ∈ X with d X ( x, x ′ ) < η o . The function ω ϕ ( x ; · ) : (0 , + ∞ ) → [0 , + ∞ ] is said to be the modulus of continuity of ϕ at x . Notice that ϕ is continuous at x if and only if ω ϕ ( x ; ε ) > ε >
0. Similarly, ϕ isuniformly continuous on a set B ⊆ X if and only if inf x ∈ B ω ϕ ( x ; ε ) > ε > T of a pseudometric space (X , d ) is said to be totally bounded provided forany radius r > x , . . . , x n ∈ T such that T ⊆ S ni =1 B r ( x i ). Lemma 2.4 (Characterizations of Cauchy-continuity) . Let (X , d X ) , (Y , d Y ) be pseudometricspaces and let ϕ : X → Y be a map. Then the following are equivalent. i) The map ϕ is Cauchy-continuous; ii) ϕ is uniformly continuous on each totally bounded subset of X ; iii) ϕ maps all totally bounded subsets of X to totally bounded subsets of Y .Proof. i) = ⇒ ii): Suppose ϕ is Cauchy-continuous. We argue by contradiction: assume there ex-ists a totally bounded set T ⊆ X whereon ϕ is not uniformly continuous, so that inf x ∈ T ω ϕ ( x ; ε ) = 0for some ε >
0. This means that we can find a sequence ( x n ) n ⊆ T such that lim n ω ϕ ( x n ; ε ) = 0.Since T is totally bounded, we can assume (up to taking a not-relabeled subsequence) that ( x n ) n is Cauchy. Given any n ∈ N , we deduce from the very definition of ω ϕ ( x n ; ε ) that there is anelement x n +1 ∈ X such that d X ( x n , x n +1 ) < ω ϕ ( x n ; ε ) + 1 /n and d Y (cid:0) ϕ ( x n ) , ϕ ( x n +1 ) (cid:1) ≥ ε .Therefore, ( x n ) n is a Cauchy sequence but (cid:0) ϕ ( x n ) (cid:1) n is not, thus contradicting the assumptionthat ϕ is Cauchy-continuous. Then ϕ is uniformly continuous on each totally bounded set.ii) = ⇒ iii): Let ε > T ⊆ X be totally bounded. Pick δ > x ∈ T , d Y (cid:0) ϕ ( x ) , ϕ ( y ) (cid:1) < ε whenever d X ( x, y ) < δ . Pick x , . . . , x n ∈ T such that T ⊆ S ni =1 B δ ( x i ). Then ϕ ( T ) ⊆ S ni =1 B ε ( ϕ ( x i )), proving that ϕ ( T ) is totally bounded by the arbitrariness of ε .iii) = ⇒ i): The claim follows from the observation that a sequence ( x n ) n ⊆ X is Cauchy if andonly if it is totally bounded as a subset of X. (cid:3)
TERHI MOISALA AND ENRICO PASQUALETTO
Given two pseudometric spaces (X , d X ) and (Y , d Y ), we always endow their product X × Y withthe pseudodistance d X × Y , which is defined as d X × Y (cid:0) ( x, y ) , ( x ′ , y ′ ) (cid:1) := p d X ( x, x ′ ) + d Y ( y, y ′ ) for every ( x, y ) , ( x ′ , y ′ ) ∈ X × Y . If both d X and d Y are distances, then d X × Y is a distance as well. Observe also that the topologyinduced by d X × Y on X × Y coincides with the product of the topologies induced by d X and d Y .2.2. Reminder on direct and inverse limits.
Let us briefly recall the definitions of direct limitand inverse limit in an arbitrary category. For a thorough account on this topic, we refer e.g. tothe classical reference [ML98].Fix a directed set ( I, ≤ ) and an arbitrary category C . By a direct system in C over I we meana couple (cid:0) { X i } i ∈ I , { ϕ ij } i ≤ j (cid:1) , where { X i : i ∈ I } is a family of objects of C , while ϕ ij : X i → X j is a morphism for every i, j ∈ I with i ≤ j , such that the following properties hold:DS ) ϕ ii is the identity of X i for every i ∈ I .DS ) ϕ ik = ϕ jk ◦ ϕ ij holds for every i, j, k ∈ I with i ≤ j ≤ k .A given couple (cid:0) X, { ϕ i } i ∈ I (cid:1) – where X is an object of C and ϕ i is a morphism ϕ i : X i → X forevery i ∈ I – is said to be the direct limit of (cid:0) { X i } i ∈ I , { ϕ ij } i ≤ j (cid:1) provided it holds that:DL ) (cid:0) X, { ϕ i } i ∈ I (cid:1) is a target for the direct system (cid:0) { X i } i ∈ I , { ϕ ij } i ≤ j (cid:1) , i.e., the diagram X i X j X ϕ ij ϕ i ϕ j is commutative for every i, j ∈ I with i ≤ j .DL ) Given any target (cid:0) Y, { ψ i } i ∈ I (cid:1) for the direct system (cid:0) { X i } i ∈ I , { ϕ ij } i ≤ j (cid:1) , there exists aunique morphism Φ : X → Y such that the diagram X i XY ϕ i ψ i Φ commutes for every i ∈ I .The property stated in DL ) is called the universal property . An arbitrary direct system needs notadmit a direct limit, but whenever the direct limit exists, it is unique up to unique isomorphism.By abuse of notation, we sometimes refer to the object X as the direct limit of (cid:0) { X i } i ∈ I , { ϕ ij } i ≤ j (cid:1) .By an inverse system in C over I we mean a couple (cid:0) { X i } i ∈ I , { P ij } i ≤ j (cid:1) – where { X i : i ∈ I } is a family of objects of C , while P ij : X j → X i is a morphism for every i, j ∈ I with i ≤ j – suchthat the following properties hold:IS ) P ii is the identity of X i for every i ∈ I .IS ) P ik = P ij ◦ P jk holds for every i, j, k ∈ I with i ≤ j ≤ k .A couple (cid:0) X, { P i } i ∈ I (cid:1) – where X is an object of C and P i : X → X i is a morphism for any i ∈ I – is said to be the inverse limit of (cid:0) { X i } i ∈ I , { P ij } i ≤ j (cid:1) provided it holds that:IL ) The diagram XX j X iP i P j P ij IRECT LIMITS OF INFINITE-DIMENSIONAL CARNOT GROUPS 7 commutes for every i, j ∈ I with i ≤ j .IL ) Given any couple (cid:0) Y, { Q i } i ∈ I (cid:1) satisfying the property in IL ) – namely, Q i = P ij ◦ Q j forany i, j ∈ I with i ≤ j – there exists a unique morphism Π : Y → X such that the diagram Y XX i Π Q i P i commutes for every i ∈ I .The property in IL ) is referred to as the universal property . An inverse system does not necessarilyadmit an inverse limit, but when the inverse limit exists, it is unique up to unique isomorphism.Let us now spend a few words on direct and inverse limits in the category of groups, which willplay a central role in the rest of this paper. We refer the reader to [Lan84] for more on this topic. Theorem 2.5 (Direct/inverse limits of groups) . Direct limits and inverse limits always exist inthe category of groups.
For the usefulness of the reader, we also recall the explicit description of direct and inverselimits of groups; a similar construction works on many other algebraic structures, such as rings,modules, or algebras. For the rest of this section, let ( I, ≤ ) be a fixed directed set.Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) be a direct system of groups. We define an equivalence relation ∼ onthe set F i ∈ I G i : given any x ∈ G i and y ∈ G j , we declare that x ∼ y provided there exists k ∈ I with i, j ≤ k such that ϕ ik ( x ) = ϕ jk ( y ). The equivalence class of x is denoted by [ x ] ∼ . Then wedefine the group G as G := G i ∈ I G i . ∼ , (2.1)the group operation · : G × G → G being defined as follows: given x ∈ G i and y ∈ G j , we set[ x ] ∼ · [ y ] ∼ := (cid:2) ϕ ik ( x ) · ϕ jk ( y ) (cid:3) ∼ ∈ G, for some (thus any) k ∈ I with i, j ≤ k. It can be readily checked that this operation is well-defined (i.e., it does not depend on the specificchoice of the representatives x, y and of k ) and that the resulting structure ( G, · ) is a group.Moreover, for any i ∈ I we define the map ϕ i : G i → G as ϕ i ( x ) := [ x ] ∼ for all x ∈ G i . Then eachmap ϕ i is a group homomorphism and (cid:0) G, { ϕ i } i ∈ I (cid:1) is the direct limit of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) inthe category of groups. It is worth to point out that for any x ∈ G there exist i ∈ I and x i ∈ G i such that x = ϕ i ( x i ).Now let (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) be an inverse system of groups. Then we define the group G as G := n x = ( x i ) i ∈ I ∈ Y i ∈ I G i (cid:12)(cid:12)(cid:12) x i = P ij ( x j ) for every i, j ∈ I with i ≤ j o , (2.2)which is a subgroup of the direct product Q i ∈ I G i (that has a natural group structure with respectto the elementwise operation). Moreover, for any i ∈ I we define P i : G → G i as P i ( x ) := x i forevery x = ( x i ) i ∈ I ∈ G . It holds that each map P i is a group homomorphism and (cid:0) G, { P i } i ∈ I (cid:1) isthe inverse limit of (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) in the category of groups.2.3. Scalable groups.
Let G be a group. Given any element x ∈ G , we denote by L x : G → G and R x : G → G the left translation map and the right translation map at x , respectively. Namely, L x ( y ) := xy for every y ∈ G,R x ( y ) := yx for every y ∈ G. TERHI MOISALA AND ENRICO PASQUALETTO
Moreover, we shall denote by Op : G × G → G the group multiplication map ( x, y ) xy andby Inv : G → G the inversion map x x − . We now define the notions of scalable group andtopological scalable group. We stress that the definition of scalable group introduced in [LDLM19]coincides with our definition of topological scalable group. Definition 2.6 (Scalable group) . Let G be a group. Then we say that { δ λ } λ ∈ R ⊆ End( G ) is a family of dilations on G provided the following properties hold: i) δ is given by δ ( x ) := e for every x ∈ G , where e stands for the identity element of G . ii) δ λ ◦ δ µ = δ λµ for every λ, µ ∈ R . iii) δ λ ∈ Aut( G ) for every λ ∈ R \ { } .We denote by δ : R × G → G the map ( λ, x ) δ λ ( x ) . The couple ( G, δ ) is called a scalable group .If G is a topological group and the dilation δ is continuous, then ( G, δ ) is called a topologicalscalable group . Subgroups and quotients of scalable groups are defined in the natural way. A map ϕ : G → G ′ between scalable groups is a morphism of scalable groups provided it is a group homomorphismthat satisfies ϕ (cid:0) δ λ ( x ) (cid:1) = δ ′ λ (cid:0) ϕ ( x ) (cid:1) for every λ ∈ R and x ∈ G. With this notion of morphism at disposal, we can speak about the category of scalable groups.
Theorem 2.7 (Direct limits of scalable groups) . Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) be a direct system ofscalable groups. Then the direct limit exists, and it is given by the direct limit (2.1) in the categoryof groups together with a suitable dilation map.Proof. Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) be a direct system of scalable groups. In particular, it is a directsystem in the category of groups, thus call (cid:0) G, { ϕ i } i ∈ I (cid:1) its direct limit (as a group). We definea family of dilations { δ λ } λ ∈ R on G as follows: given any λ ∈ R and x ∈ G , there exist i ∈ I and x i ∈ G i such that ϕ i ( x i ) = x ; then we set δ λ ( x ) := δ iλ ( x i ), where δ i stands for the dilation on G i . Itcan be readily checked that δ λ is well-defined, that ( G, δ ) is a scalable group, and that each map ϕ i is a morphism of scalable groups. To conclude, it only remains to prove the universal property: let (cid:0) H, { ψ i } i ∈ I (cid:1) be any target of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) in the category of scalable groups. In particular,it is a target in the category of groups, thus there exists a unique group homomorphism Φ : G → H such that Φ ◦ ϕ i = ψ i for all i ∈ I . Let us show that Φ preserves the dilation: if λ ∈ R and x ∈ G ,then there exist i ∈ I and x i ∈ G i such that ϕ i ( x i ) = x , whenceΦ (cid:0) δ λ ( x ) (cid:1) = (Φ ◦ ϕ i ) (cid:0) δ iλ ( x i ) (cid:1) = ψ (cid:0) δ iλ ( x ) (cid:1) = δ ′ λ (cid:0) ψ i ( x i ) (cid:1) = δ ′ λ (cid:0) Φ( x ) (cid:1) , where δ ′ stands for the dilation on H . This proves the universal property, so that (cid:0) G, { ϕ i } i ∈ I (cid:1) isthe direct limit of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) in the category of scalable groups, as required. (cid:3) For the sake of completeness, we also give a proof of existence of inverse limits in the categoryof scalable groups.
Theorem 2.8 (Inverse limits of scalable groups) . Let (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) be an inverse systemof scalable groups. Then the inverse limit exists, and it is given by the inverse limit (2.2) in thecategory of groups together with a suitable dilation map.Proof. Let (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) be an inverse system of scalable groups. In particular, it is aninverse system in the category of groups, whose inverse limit we denote by (cid:0) G, { P i } i ∈ I (cid:1) . Wedefine the family of dilations { δ λ } λ ∈ R on G in the following way: given λ ∈ R and x ∈ G , wedefine δ λ ( x ) as the unique element of G such that P i (cid:0) δ λ ( x ) (cid:1) = δ iλ (cid:0) P i ( x ) (cid:1) for some (thus any) IRECT LIMITS OF INFINITE-DIMENSIONAL CARNOT GROUPS 9 i ∈ I , where by δ i we mean the dilation on G i . It turns out that ( G, δ ) is a scalable group andeach map P i is a scalable group morphism. Let us now prove the universal property: fix a scalablegroup H and a family { Q i } i ∈ I of scalable group morphisms Q i : H → G i such that Q i = P ij ◦ Q j for every i, j ∈ I with i ≤ j . Since each map Q i is in particular a group homomorphism, thereexists a unique group homomorphism Φ : H → G such that Q i = P i ◦ Φ for all i ∈ I . Call δ ′ thedilation on H . If λ ∈ R and y ∈ H , then P i (cid:0) δ λ (Φ( y )) (cid:1) = δ iλ (cid:0) P i (Φ( y )) (cid:1) = δ iλ (cid:0) Q i ( y ) (cid:1) = Q i (cid:0) δ ′ λ ( y ) (cid:1) for every i ∈ I, thus accordingly δ λ (cid:0) Φ( y ) (cid:1) = Φ (cid:0) δ ′ λ ( y ) (cid:1) . This shows that Φ is a morphism of scalable groups,whence the universal property is proven. Therefore, we have that (cid:0) G, { P i } i ∈ I (cid:1) is the inverse limitof (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) in the category of scalable groups, as required. (cid:3) Direct limits of (complete) metric scalable groups
Scalable groups with distances.
As a first step towards constructing direct limits ofcomplete metric scalable groups, we study scalable groups equipped with a suitable pseudometric.
Definition 3.1 (Compatible distance) . Let ( G, δ ) be a scalable group. Let d be a pseudodistanceon G . Then we say that d is compatible with ( G, δ ) , if d is left-invariant and d (cid:0) δ λ ( x ) , δ λ ( y ) (cid:1) = | λ | d ( x, y ) for every λ ∈ R and x, y ∈ G. Recall that a pseudodistance d on a group G is said to be left-invariant provided it satisfies d ( xy, xz ) = d ( y, z ) for every x, y, z ∈ G. In the following lemma we collect some technical continuity results on scalable groups with compat-ible pseudodistances. These results will be needed in the construction of direct limits of (complete)metric scalable groups. Recall Definition 2.3 for the modulus of continuity.
Lemma 3.2.
Let ( G, δ ) be a scalable group endowed with a compatible pseudodistance d . Thenfor every x, y ∈ G and ε > , it holds that a) ω R x ( e ; ε ) = ω R x ( y ; ε ) ; b) ω Inv ( x ; ε ) = ω R x − ( e ; ε ) ; c) inf y ∈ B ( x,ε/ ω R y ( e ; ε ) ≥ ω R x ( e ; ε/ .Moreover, d) if R x is continuous at e for any x ∈ G , then Op : G × G → G is Cauchy-continuous; e) if δ ( · , x ) is continuous for any x ∈ G , then δ : R × G → G is Cauchy-continuous.Proof. a) Fix any x, y, y ′ ∈ G and ε >
0. Choose any η < ω R x ( y ′ ; ε ). Then for every z ∈ B η ( y ) we have d ( y ′ y − z, y ′ ) = d ( z, y ) < η , thus d ( zx, yx ) = d ( y ′ y − zx, y ′ x ) < ε and accordingly ω R x ( y ; ε ) ≥ η .This yields ω R x ( y ; ε ) ≥ ω R x ( y ′ ; ε ), whence a) follows by the arbitrariness of x, y, y ′ ∈ G and ε > x ∈ G and ε > ω Inv ( x ; ε ) ≤ ω R x − ( e ; ε ). Pick η < ω Inv ( x ; ε ).If y ∈ G satisfies d ( y, e ) < η , then d ( xy − , x ) = d ( y − , e ) = d ( e, y ) < η and accordingly d (cid:0) R x − ( y ) , R x − ( e ) (cid:1) = d ( yx − , x − ) = d (cid:0) ( xy − ) − , x − (cid:1) = d (cid:0) Inv ( xy − ) , Inv ( x ) (cid:1) < ε. Hence, we get ω R x − ( e ; ε ) ≥ η . By arbitrariness of η , we conclude that ω Inv ( x ; ε ) ≤ ω R x − ( e ; ε ).In order to prove the converse inequality, fix any η < ω R x − ( e ; ε ). If y ∈ G satisfies d ( y, x ) < η ,then d ( y − x, e ) = d ( x, y ) < η . This implies that d (cid:0) Inv ( y ) , Inv ( x ) (cid:1) = d ( y − , x − ) = d ( y − xx − , x − ) = d (cid:0) R x − ( y − x ) , R x − ( e ) (cid:1) < ε, thus ω Inv ( x ; ε ) ≥ η . By arbitrariness of η , we conclude that ω R x − ( e ; ε ) ≤ ω Inv ( x ; ε ).c) Let us denote η := ω R x ( e ; ε/
3) and assume, without loss of generality, that η >
0. It is thenenough to observe that, given any z ∈ B η ( e ) and y ∈ B ε/ ( x ), we have d (cid:0) R y ( e ) , R y ( z ) (cid:1) = d ( y, zy ) ≤ d ( y, x ) + d ( x, zx ) + d ( zx, zy ) < ε. d) Suppose R x is continuous at e for any x ∈ G . Let T ⊆ G × G be a d G × G -totally bounded set.The projection T ′ of the set T on the second coordinate is d G -totally bounded, as a consequenceof the fact that the projection map is Lipschitz. Now fix ε >
0. Covering the set T ′ by a finitenumber of balls of radius ε/ R y at every y ∈ G that η ′ := inf y ∈ T ′ ω R y ( e ; ε ) > . Denote η := min (cid:8) ε, η ′ (cid:9) > ω Op (cid:0) ( x, y ); 2 ε (cid:1) ≥ η for every ( x, y ) ∈ T. (3.1)To prove it, fix any ( x, y ) ∈ T . Given any ( x ′ , y ′ ) ∈ B η ( x, y ), it holds that d ( xy, x ′ y ′ ) ≤ d ( xy, x ′ y ) + d ( x ′ y, x ′ y ′ ) < ε + η ≤ ε. This shows (3.1). Thanks to Lemma 2.4, we thus conclude that the group multiplication map Op : G × G → G is Cauchy-continuous, as desired.e) Suppose δ ( · , x ) is continuous for any x ∈ G . Let T ⊆ R × G be totally bounded. Fix ε > T R ⊆ R and T G ⊆ G the totally bounded projections of T on the first and the secondcomponents, respectively. Call M := sup λ ∈ T R | λ | < + ∞ and choose any x , . . . , x n ∈ T G for which T G ⊆ S ni =1 B ε/ M ( x i ). Let η ( λ ) := 12 min (cid:8) ω δ ( · ,x ) ( λ ; ε ) , . . . , ω δ ( · ,x n ) ( λ ; ε ) (cid:9) > λ ∈ R . Now pick λ , . . . , λ m ∈ T R such that T R ⊆ S mj =1 B η ( λ j ) ( λ j ). We claim that choosing η := min (cid:8) M, ε/ M, η ( λ ) , . . . , η ( λ m ) (cid:9) gives inf ( λ,x ) ∈ T ω δ (cid:0) ( λ, x ); 5 ε (cid:1) ≥ η. (3.2)To prove (3.2), fix ( λ, x ) ∈ T and ( λ ′ , x ′ ) ∈ R × G such that d R × G (cid:0) ( λ, x ) , ( λ ′ , x ′ ) (cid:1) < η . Consider i ∈ { , . . . , n } and j ∈ { , . . . , m } for which | λ − λ j | < η ( λ j ) and d ( x, x i ) < ε/ M . Observe that d (cid:0) δ ( λ, x ) , δ ( λ ′ , x ′ ) (cid:1) ≤ d (cid:0) δ λ ( x ) , δ λ ( x i ) (cid:1) + d (cid:0) δ λ ( x i ) , δ λ j ( x i ) (cid:1) + d (cid:0) δ λ j ( x i ) , δ λ ′ ( x i ) (cid:1) + d (cid:0) δ λ ′ ( x i ) , δ λ ′ ( x ) (cid:1) + d (cid:0) δ λ ′ ( x ) , δ λ ′ ( x ′ ) (cid:1) . Let us estimate the five terms appearing in the right-hand side of the previous formula. First, d (cid:0) δ λ ( x ) , δ λ ( x i ) (cid:1) = | λ | d ( x, x i ) ≤ M ε M < ε, d (cid:0) δ λ ′ ( x i ) , δ λ ′ ( x ) (cid:1) = | λ ′ | d ( x i , x ) ≤ (cid:0) η + | λ | (cid:1) d ( x i , x ) < M ε M = ε, d (cid:0) δ λ ′ ( x ) , δ λ ′ ( x ′ ) (cid:1) = | λ ′ | d ( x, x ′ ) ≤ (cid:0) η + | λ | (cid:1) η ≤ M ε M = ε. Moreover, since | λ − λ j | < η ( λ j ) < ω δ ( · ,x i ) ( λ j ; ε ), we have that d (cid:0) δ λ ( x i ) , δ λ j ( x i ) (cid:1) < ε . Finally,since | λ j − λ ′ | ≤ | λ j − λ | + η < η ( λ j ) ≤ ω δ ( · ,x i ) ( λ j ; ε ), we see that d (cid:0) δ λ j ( x i ) , δ λ ′ ( x i ) (cid:1) < ε . Allin all, we have proven that d (cid:0) δ ( λ, x ) , δ ( λ ′ , x ′ ) (cid:1) < ε , thus obtaining the claim (3.2). By takingLemma 2.4 into account, we can conclude that δ : R × G → G is Cauchy-continuous, as desired. (cid:3) Remark 3.3.
Observe that items a) – d) hold also without existence of the dilation δ . (cid:4) IRECT LIMITS OF INFINITE-DIMENSIONAL CARNOT GROUPS 11
Remark 3.4.
Note that, given a scalable group (
G, δ ) equipped with a compatible distance d , thecontinuity of the map δ ( · , x ) : R → G for x ∈ G \ { e } is not automatic even when G is an abeliantopological group and d is geodesic. Indeed, let f : (cid:0) R \ { } , · (cid:1) → ( R , +) be a non-continuous grouphomomorphism and consider the abelian group G = ( R , +). Let L θ denote the rotation in R byan angle θ ∈ R . Define the maps δ λ : R → R as δ λ = ( λ id R ) ◦ L f ( λ ) for every λ ∈ R \ { } , and δ ≡ , where id R stands for the identity map. Clearly, λ δ λ ( x ) is not continuous for any x ∈ R \ { } ,but we claim that ( R , δ ) is a scalable group and that δ is a metric dilation with respect to theEuclidean distance. It suffices to notice that δ λ is a linear bijection for each λ ∈ R \ { } satisfying δ s ◦ δ t = ( s id R ) ◦ L f ( s ) ◦ ( t id R ) ◦ L f ( t ) = ( st id R ) ◦ L f ( s )+ f ( t ) = ( st id R ) ◦ L f ( st ) = δ st and, since L θ is an isometry, (cid:12)(cid:12) δ λ ( x ) − δ λ ( y ) (cid:12)(cid:12) = (cid:12)(cid:12) ( λ id R ) ◦ L f ( λ ) ( x − y ) (cid:12)(cid:12) = | λ || x − y | . (cid:4) The well-known fact that a linear map between normed spaces is continuous (at the origin) ifand only if it is Lipschitz, can be generalized to the setting of scalable groups:
Proposition 3.5.
Let ( G, δ ) and ( G ′ , δ ′ ) be scalable groups. Let d (resp. d ′ ) be a compatiblepseudodistance on G (resp. on G ′ ). Let ϕ : G → G ′ be a morphism of scalable groups. Then ϕ iscontinuous at e ∈ G if and only if it is Lipschitz.Proof. Sufficiency is obvious. To prove necessity, suppose ϕ is continuous at e . Denote by e ′ theidentity element of G ′ . We claim that there exists a constant C > d ′ (cid:0) ϕ ( x ) , e ′ (cid:1) ≤ C d ( x, e ) for every x ∈ G. (3.3)In order to prove it, we argue by contradiction: suppose there exists a sequence ( x n ) n ⊆ G suchthat d ′ (cid:0) ϕ ( x n ) , e ′ (cid:1) > n d ( x n , e ) for all n ∈ N . Call λ n := 1 / (cid:0) n d ( x n , e ) (cid:1) > y n := δ λ n ( x n ) ∈ G for every n ∈ N . Then d ( y n , e ) = λ n d ( x n , e ) = 1 /n → n → ∞ , while d ′ (cid:0) ϕ ( y n ) , e ′ (cid:1) = d ′ (cid:0) δ ′ λ n (cid:0) ϕ ( x n ) (cid:1) , e ′ (cid:1) = λ n d ′ (cid:0) ϕ ( x n ) , e ′ (cid:1) > n ∈ N . Since ϕ ( e ) = e ′ , this contradicts the continuity of ϕ at e . Therefore, the claim (3.3) is proven. Byexploiting (3.3) and the left-invariance of d and d ′ , we finally conclude that d ′ (cid:0) ϕ ( x ) , ϕ ( y ) (cid:1) = d ′ (cid:0) ϕ ( y − x ) , e ′ (cid:1) ≤ C d ( y − x, e ) = C d ( x, y ) for every x, y ∈ G, whence the map ϕ is Lipschitz, as required. (cid:3) Metric scalable groups.
The following definition is introduced in [LDLM19]. Recall thedefinition of topological scalable group (Defintion 2.6) and of compatible distance (Definition 3.1).
Definition 3.6 (Metric scalable group) . A metric scalable group (or briefly MSG ) is a triple ( G, δ, d ) , where ( G, δ ) is a topological scalable group, while d is a compatible distance on ( G, δ ) that induces the topology of G . A complete metric scalable group is called CMSG . Let (
G, δ, d ), ( G ′ , δ ′ , d ′ ) be metric scalable groups. Then a map ϕ : G → G ′ is said to bea morphism of metric scalable groups provided it is a 1-Lipschitz morphism of scalable groups.With this notion at our disposal, we can speak about the category of MSGs. We are now goingto construct the metric limits of MSGs, as discussed in the Introduction.Given a direct system (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) of metric scalable groups, we denote by δ i and d i the dilation and the distance on G i , respectively. Since (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) is – a fortiori – adirect system also in the category of scalable groups, it admits a direct limit (cid:0) G ′ , { ϕ ′ i } i ∈ I (cid:1) in such category (recall Theorem 2.7). We define the infimum-pseudodistance d ′ on G ′ in the followingway: d ′ ( x, y ) := inf n d i ( x i , y i ) (cid:12)(cid:12)(cid:12) i ∈ I, x i , y i ∈ G i , ϕ ′ i ( x i ) = x, ϕ ′ i ( y i ) = y o for every x, y ∈ G ′ . (3.4)One can easily check that d ′ is compatible with the scalable group ( G ′ , δ ′ ). Setting Z ( d ′ ) := (cid:8) x ∈ G ′ (cid:12)(cid:12) d ′ ( x, e ′ ) = 0 (cid:9) , one may consider the induced quotient metric space (cid:0) G ′ /Z ( d ′ ) , d (cid:1) , where d is defined as d (cid:0) xZ ( d ′ ) , yZ ( d ′ ) (cid:1) := d ′ ( x, y ) for every x, y ∈ G ′ . Moreover, we may define the map δ : R × (cid:0) G ′ /Z ( d ′ ) (cid:1) → G ′ /Z ( d ′ ) as δ (cid:0) t, xZ ( d ′ ) (cid:1) = δ t (cid:0) xZ ( d ′ ) (cid:1) := δ ′ t ( z ) Z ( d ′ ) for every t ∈ R and x ∈ G ′ . (3.5)In the proof of Lemma 3.10 we show that Z ( d ′ ) is a scalable subgroup of G ′ . Hence δ is well-posed,i.e., it does not depend on the choice of representative. Definition 3.7 (Metric limit) . Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) be a direct system of metric scalablegroups. Let ( G ′ , δ ′ ) be its direct limit in the category of scalable groups and let d ′ be the infimum-pseudodistance on G ′ . The triple (cid:0) G ′ /Z ( d ′ ) , δ, d (cid:1) is called the metric limit of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) . We stress that, in general, the metric limit (cid:0) G ′ /Z ( d ′ ) , δ, d (cid:1) is not a metric scalable group.Indeed, the scalable subgroup Z ( d ′ ) might fail to be normal, so that G ′ /Z ( d ′ ) does not have anatural group structure. Moreover, even when Z ( d ′ ) is normal, the group operations on G ′ /Z ( d ′ )might fail to be continuous with respect to d ′ . We now impose conditions on the direct system (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) that will be necessary and sufficient for our constructions. In particular, theconditions will guarantee that the metric limit is a metric scalable group. Definition 3.8 (Non-degenerate direct system of MSGs) . Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) be a direct sys-tem of metric scalable groups. Equip its direct limit (cid:0) G ′ , { ϕ ′ i } i ∈ I (cid:1) in the category of scalable groupswith the infimum-pseudodistance d ′ . Then we say that (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) is non-degenerate pro-vided the following conditions are satisfied: (C1) δ ′ ( · , x ) : R → G ′ is continuous for all x ∈ G ′ , (C2) R x : G ′ → G ′ is continuous at e for all x ∈ G ′ . As we are going to show in the following result, the continuity conditions (C1) and (C2) canbe directly checked along the approximating net { G i } i ∈ I . Proposition 3.9.
Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) be a direct system of metric scalable groups. Denoteby (cid:0) G ′ , { ϕ ′ i } i ∈ I (cid:1) the direct limit of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) as scalable groups and by d ′ the infimum-pseudodistance on G ′ . Then the conditions (C1) and (C2) can be equivalently formulated as: (C1’) For all t ∈ R and x ∈ G ′ , inf η> sup s ∈ R : | s − t | <η inf n d i (cid:0) δ i ( s, x i ) , δ i ( t, y i ) (cid:1) (cid:12)(cid:12)(cid:12) x i , y i ∈ ( ϕ ′ i ) − ( x ) o = 0;(C2’) For all x ∈ G ′ , inf η> sup y i ∈ G i : d i ( e i ,y i ) <η inf n d j ( x j , y j x j ) (cid:12)(cid:12)(cid:12) x j ∈ ( ϕ ′ j ) − ( x ) , y j ∈ ( ϕ ′ j ) − (cid:0) ϕ ′ i ( y i ) (cid:1)o = 0 . IRECT LIMITS OF INFINITE-DIMENSIONAL CARNOT GROUPS 13
Proof.
We just prove the equivalence (C1) ⇔ (C1 ′ ), the argument for (C2) ⇔ (C2 ′ ) being similar.(C1) = ⇒ (C1 ′ ) Suppose (C1) holds. Fix x ∈ G ′ and t ∈ R . Given ε >
0, there exists η > d ′ (cid:0) δ ′ s ( x ) , δ ′ t ( x ) (cid:1) < ε ′ for all s ∈ R with | s − t | < η . By definition of d ′ , there exist i ∈ I andpoints x i , y i ∈ G i such that ϕ ′ i ( x i ) = ϕ ′ i ( y i ) = x and d i (cid:0) δ is ( x i ) , δ it ( y i ) (cid:1) < ε . This implies (C1’).(C1’)= ⇒ (C1) Suppose (C1’) holds. Fix ε > x ∈ G ′ , and t ∈ R . There exists η > s ∈ R with | s − t | < η there exist i ∈ I , x i , y i ∈ ( ϕ ′ i ) − ( x ) satisfying d i (cid:0) δ is ( x i ) , δ it ( y i ) (cid:1) < ε .This implies d ′ (cid:0) δ ′ s ( x ) , δ ′ t ( x ) (cid:1) < ε , which shows that δ ′ ( · , x ) is continuous at t . Thus (C1) holds. (cid:3) Lemma 3.10.
Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) be a direct system of metric scalable groups satisfying (C2) .Then its metric limit is a scalable group endowed with a compatible distance d . If also (C1) issatisfied, then the metric limit is a metric scalable group.Proof. We start by showing that Z ( d ′ ) is a normal scalable subgroup of G ′ , which ensures thatthe quotient space (cid:0) G ′ /Z ( d ′ ) , δ (cid:1) is a scalable group. Given any x, y ∈ Z ( d ′ ), we have that d ′ ( xy, e ′ ) ≤ d ′ ( xy, x ) + d ′ ( x, e ′ ) = d ′ ( y, e ′ ) + d ′ ( x, e ′ ) = 0 , which shows that xy ∈ Z ( d ′ ) as well. Moreover, we have d ′ ( x − , e ′ ) = d ′ ( xx − , x ) = d ′ ( e ′ , x ) = 0,which implies that x − ∈ Z ( d ′ ). All in all, Z ( d ′ ) is a subgroup of G ′ . Given λ ∈ R and x ∈ Z ( d ′ ),it holds that d ′ (cid:0) δ ′ λ ( x ) , e ′ (cid:1) = d ′ (cid:0) δ ′ λ ( x ) , δ ′ λ ( e ′ ) (cid:1) = | λ | d ′ ( x, e ′ ) = 0, so that δ ′ λ ( x ) ∈ Z ( d ′ ). This showsthat Z ( d ′ ) is a scalable subgroup of G ′ .To show that Z ( d ′ ) is normal, fix any x ∈ Z ( d ′ ) and y ∈ G . Since the map R y continuous at e ′ by(C2), we have that ω R y ( e ′ ; 1 /n ) > n ∈ N . Since d ′ ( x, e ′ ) = 0 < ω R y ( e ′ ; 1 /n ), wededuce that d ′ ( y − xy, e ′ ) = d ′ ( xy, y ) < /n . By letting n → ∞ , we conclude that y − xy ∈ Z ( d ′ ),thus proving that Z ( d ′ ) is a normal subgroup of G ′ , as required. Hence ( G ′ /Z ( d ′ ) , δ ) is a scalablegroup. Finally, it can be readily checked that the induced distance d on G ′ /Z ( d ′ ) is compatiblewith ( G ′ /Z ( d ′ ) , δ ), which concludes the first part of the claim.Assume then that (C1) holds and let us prove that ( G ′ /Z ( d ′ ) , δ, d ) is a metric scalable group.We need to verify that the dilation δ : R × G → G , the group operation Op : G × G → G and theinversion map Inv : G → G are continuous. Observe that a morphism fixing the subgroup Z ( d ′ )on ( G ′ , d ′ ) is continuous if and only if the induced map on (cid:0) G ′ /Z ( d ′ ) , d (cid:1) is continuous. Thenby (C1) and (C2), the continuity of δ and Op follows from Lemma 3.2 e) and Lemma 3.2 d),respectively. Moreover, the continuity of Inv is given by (C2) and Lemma 3.2 b). Hence the metriclimit ( G ′ /Z ( d ′ ) , δ, d ) is a metric scalable group, as claimed. (cid:3) We prove now the part of Theorem 1.1 concerning metric scalable groups.
Theorem 3.11 (Direct limits of MSGs) . Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) be a direct system of metricscalable groups. Then the following are equivalent: i) The direct system (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) is non-degenerate (in the sense of Definition 3.8). ii) The direct limit of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) in the category of metric scalable groups exists.Moreover, it equals the metric limit of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) as in Definition 3.7.Proof. i) = ⇒ ii) Suppose i) holds. Let (cid:0) G ′ , { ϕ ′ i } i ∈ I (cid:1) be the direct limit of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) in the category of scalable groups. Denoting G := G ′ /Z ( d ′ ), let ( G, δ, d (cid:1) be the metric limit of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) . By Lemma 3.10, ( G, δ, d (cid:1) is a metric scalable group. Let π : G ′ → G be theprojection π ( x ) = xZ ( d ′ ) and define ϕ i : G i → G for each i ∈ I as ϕ i := π ◦ ϕ ′ i . Then by definition of the infimum-distance d , each ϕ i : G i → G ′ is a 1-Lipschitz map, hence amorphism of metric scalable groups. Since ϕ i = ϕ j ◦ ϕ ij for all i, j ∈ I , i ≤ j , (cid:0) G, { ϕ i } i ∈ I (cid:1) is atarget of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) in the category of metric scalable groups.To prove that it satisfies the universal property, fix any target (cid:0) H, { ψ i } i ∈ I (cid:1) . Since (cid:0) H, { ψ i } i ∈ I (cid:1) is a target also in the category of scalable groups, there exists a unique morphism of scalablegroups Φ ′ : G ′ → H such that Φ ′ ◦ ϕ ′ i = ψ i holds for every i ∈ I . We claim then that the map Φ ′ is 1-Lipschitz. Indeed, given i ∈ I , x i ∈ G i , and y i ∈ G i with ϕ ′ i ( x i ) = x and ϕ ′ i ( y i ) = y , we have d H (cid:0) Φ ′ ( x ) , Φ ′ ( y ) (cid:1) = d H (cid:0) (Φ ′ ◦ ϕ ′ i )( x ) , (Φ ′ ◦ ϕ ′ i )( y ) (cid:1) = d H (cid:0) ψ i ( x i ) , ψ i ( y i ) (cid:1) ≤ d i ( x i , y i ) , where we denote by d H the distance on H . This proves d H (cid:0) Φ ′ ( x ) , Φ ′ ( y ) (cid:1) < d ′ ( x, y ), as wanted.It now follows that ker( π ) = Z ( d ′ ) ⊆ ker(Φ ′ ), and so by the universal property of quotients,there exists a unique map Φ : G → H such that Φ ◦ π = Φ ′ . Moreover, for any xZ ( d ′ ) , yZ ( d ′ ) ∈ G it holds that d H (cid:0) Φ( xZ ( d ′ )) , Φ( yZ ( d ′ )) (cid:1) = d H (cid:0) Φ ′ ( x ) , Φ ′ ( y ) (cid:1) ≤ d ′ ( x, y ) = d (cid:0) xZ ( d ′ ) , yZ ( d ′ ) (cid:1) , which shows that also Φ is a 1-Lipschitz map. Hence Φ is a morphism of metric scalable groups,and (cid:0) G, { ϕ i } i ∈ I (cid:1) satisfies the universal property. Then the metric limit ( G, δ, d ) is the direct limitof (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) in the category of metric scalable groups, yielding the sought conclusion.ii) = ⇒ i) Suppose ii) holds. Then, in particular, the metric limit of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) is a metricscalable group. Hence the dilation map and the right translation are continuous on G ′ /Z ( d ′ ),which implies the continuity of these operations on G ′ . Thus (C1) and (C2) are satisfied, and (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) is non-degenerate. (cid:3) Complete metric scalable groups.
We consider the category of CMSGs as a subcategoryof metric scalable groups; any CMSG is in particular a MSG, and the morphisms of CMSGs arethe morphisms of MSGs.Let (
G, δ, d ) and ( ¯ G, ¯ δ, ¯ d ) be a MSG and a CMSG, respectively. Suppose ( ¯ G, ¯ d ) is the metriccompletion of ( G, d ), with isometric embedding ι : G ֒ → ¯ G . If ι is a morphism of scalable groups,then we say that ¯ G is the completion of G as a metric scalable group . Lemma 3.12 (Extension of MSGs) . Let ( G, δ, d ) be a metric scalable group. Suppose the inversionmap Inv : G → G is Cauchy-continuous. Then the metric completion ( ¯ G, ¯ d ) of ( G, d ) can beuniquely endowed with a metric scalable group structure ( ¯ G, ¯ δ, ¯ d ) in such a way that ¯ G is thecompletion of G as a metric scalable group.Proof. First of all, observe that the assumption on
Inv and items b), d) of Lemma 3.2 grant that G × G ∋ ( x, y ) ι ( xy ) ∈ ¯ G, R × G ∋ ( λ, x ) ι (cid:0) δ λ ( x ) (cid:1) ∈ ¯ G,G ∋ x ι ( x − ) ∈ ¯ G are Cauchy-continuous, where ι : G ֒ → ¯ G is the natural isometric embedding. Therefore, we knowfrom Theorem 2.2 that the above maps can be uniquely extended to continuous operations¯ G × ¯ G ∋ ( x, y ) xy ∈ ¯ G, R × ¯ G ∋ ( λ, x ) ¯ δ λ ( x ) ∈ ¯ G, ¯ G ∋ x x − ∈ ¯ G. By a standard approximation argument, one can readily check that the resulting structure ( ¯ G, ¯ δ, ¯ d )is a complete metric scalable group and that ι is a morphism of metric scalable groups. (cid:3) IRECT LIMITS OF INFINITE-DIMENSIONAL CARNOT GROUPS 15
We now introduce a notion of non-degenerate direct system in the category of CMSGs.
Definition 3.13 (Non-degenerate direct system of CMSGs) . Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) be a directsystem of complete metric scalable groups. Equip its direct limit (cid:0) G ′ , { ϕ ′ i } i ∈ I (cid:1) in the categoryof scalable groups with the infimum-pseudodistance d ′ . Then we say that (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) is non-degenerate provided it satisfies (C1) from Definition 3.8 and the following condition: (C3) Inv : G ′ → G ′ is Cauchy-continuous. Observe that, by Lemma 3.2 b), property (C3) implies (C2). Hence, a non-degenerate directsystem in the category of CMSGs is non-degenerate also as a direct system of MSGs. However, wedo not know whether the converse implication holds (but cf. Lemma 3.17). Notice also that, thanksto Lemma 3.12, (C3) grants that the metric completion of the metric limit of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) is well-defined.In analogy with Proposition 3.9, condition (C3) can be characterized in terms of the net { G i } i ∈ I .We omit the proof of the following statement, as it is similar to the one of Proposition 3.9. Proposition 3.14.
Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) be a direct system of complete metric scalable groups.Denote by (cid:0) G ′ , { ϕ ′ i } i ∈ I (cid:1) the direct limit of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) as scalable groups and by d ′ theinfimum-pseudodistance on G ′ . Then the condition (C3) can be equivalently formulated as: (C3’) For all T ⊂ G ′ totally bounded, inf η> sup x i ,y i ∈ G i : ϕ ′ i ( x i ) ∈ T, d i ( y i ,x i ) <η inf n d j ( y − j , x − j ) (cid:12)(cid:12)(cid:12) x j ∈ ( ϕ ′ j ) − (cid:0) ϕ ′ i ( x i ) (cid:1) , y j ∈ ( ϕ ′ j ) − (cid:0) ϕ ′ i ( y i ) (cid:1)o = 0 . Together with Theorem 3.11, the following result proves Theorem 1.1.
Theorem 3.15 (Direct limits of CMSGs) . Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) be a direct system of completemetric scalable groups. Then the following conditions are equivalent: i) The direct system (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) is non-degenerate (in the sense of Definition 3.13). ii) The direct limit of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) in the category of complete metric scalable groupsexists. Moreover, it equals the metric completion of the metric limit of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) .Proof. i) = ⇒ ii) Suppose i) holds. Due to Lemma 3.2 b), (C3) implies (C2). Hence, by Theorem3.11, the direct system (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) has a direct limit ( G ′ /Z ( d ′ ) , { φ i } i ∈ I ) in the categoryof metric scalable groups. By (C3) the inversion map is Cauchy-continuous on G ′ , yielding theCauchy-continuity of the inversion on G ′ /Z ( d ′ ). By Lemma 3.12, the metric limit (cid:0) G ′ /Z ( d ′ ) , δ, d (cid:1) admits a completion ( G, δ, d ) as a metric scalable group. Denoting by ι : G ′ /Z ( d ′ ) ֒ → G theisometric embedding, we define for every i ∈ I the map ϕ i : G i → G by ϕ i := ι ◦ φ i . Since each ϕ i is a morphism of scalable groups, ( G, δ, d ) is a target of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) .To show that (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) has the universal property, let ( H, { ψ i } i ∈ I ) be any target inthe category of complete metric scalable groups. Since H is also a target in the category of metricscalable groups, there exists a unique morphism Φ ′ : G ′ /Z ( d ′ ) → H of metric scalable groups. Inparticular, Φ ′ is 1-Lipschitz, thus it can be uniquely extended to a 1-Lipschitz map Φ : G → H .By approximation, we can see that Φ is the unique morphism of metric scalable groups such that ψ i = Φ ◦ ϕ i holds for every i ∈ I . This shows that (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) satisfies the universalproperty. Therefore, ii) is achieved.ii) = ⇒ i) Suppose ii) holds. In particular, the metric completion of the metric limit G ′ /Z ( d ′ ) is a complete metric scalable group. Hence the dilation, the group operation and the inversion mapare Cauchy-continuous. Consequently, the corresponding maps on G ′ are Cauchy-continuous. Weconclude that (C1) and (C3) hold, thus i) is satisfied. (cid:3) Remark 3.16.
It is not clear if a direct system (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) of complete metric scalablegroups that does not satisfy (C1) or (C3) can have a direct limit. Indeed, Theorem 3.15 onlyimplies that for such a direct system the construction using metric limit fails. However, it is fairlystraightforward to show that, whenever there exists a direct limit (cid:0) G, { ϕ i } i ∈ I (cid:1) , then it holds that [ i ∈ I ϕ i ( G i ) is dense in G. Indeed, calling e G the closure of S i ∈ I ϕ i ( G i ) in G , it is easy to check that e G is a complete metricscalable subgroup of G and that (cid:0) e G, { ϕ i } i ∈ I (cid:1) is a target. Denote by ι : e G → G the inclusion map,which clearly is a morphism of CMSGs. Now fix any target ( H, { ψ i } i ∈ I ). Call Φ : G → H theunique morphism such that ψ i = Φ ◦ ϕ i for every i ∈ I . Then e Φ := Φ ◦ ι is the unique morphism e Φ : e G → H such that ψ i = e Φ ◦ ϕ i for every i ∈ I , which implies that actually e G = G . (cid:4) Finally, we prove that when the scalable group G ′ is nilpotent, (C2) and (C3) are equivalent. Lemma 3.17.
Let G be a nilpotent group endowed with a left-invariant distance d . If R x : G → G is continuous for every x ∈ G , then the inversion Inv : G → G is Cauchy-continuous.Proof. The proof is by induction on the nilpotency step s of G . To prove the base case, assumethat s = 1, that is, G is abelian. Then for every x, y ∈ G , d ( x − , y − ) = d ( e, xy − ) = d ( y, x ) , which proves that Inv is an isometry.Let then the nilpotency step s of G be arbitrary and assume that the claim is true for groupsof nilpotency step s −
1. Denote by G ( s ) the last element of the lower central series of G , whichis an abelian subgroup of G . Then the quotient space G/G ( s ) is a group of step s −
1. Moreover,on
G/G ( s ) one may consider (see [MZ55], p. 36) a left-invariant distance d ′ defined by d ′ ([ x ] , [ y ]) := inf h ∈ G ( s ) d ( x, hy ) for every [ x ] , [ y ] ∈ G/G ( s ) . Let T ⊆ G be totally bounded and let us show that Inv is uniformly continuous on T , which wouldprove the claim by Lemma 2.4. Since the projection map x [ x ] is Lipschitz, the set [ T ] is totallybounded. Denoting by Inv ′ the inversion on G/G ( s ) , by induction assumption and Lemma 2.4 wehave that [ Inv ( T )] = Inv ′ ([ T ]) is totally bounded as well. Fix ε > x ] , . . . , [ x n ] ∈ G/G ( s ) be such that [ Inv ( T )] ⊆ S ni =1 B ε/ ([ x i ]). Then it is easy to show that, if U := S ni =1 B ε/ ( x i ) ⊆ G ,then [ Inv ( T )] ⊆ [ U ]. Since R x is continuous for every x ∈ G , we have η := inf i ω R xi ( ε/ > . Moreover, by Lemma 3.2 c) it holds inf x ∈ U ω R x ( ε ) ≥ η. Let finally x ∈ T and let us demonstrate that ω Inv ( x ; ε ) ≥ η . Indeed, now [ x − ] ∈ [ Inv ( T )] ⊆ [ U ],which implies the existence of h ∈ G ( s ) such that hx − ∈ U . Recall that G ( s ) is an abelian subgroupof G , whence R h − is an isometry. Consequently, ω Inv ( x ; ε ) L . ω R x − ( e ; ε ) = ω R hx − ◦ R h − ( e ; ε ) ( ∗ ) = ω R hx − ( h − ; ε ) L . ω R hx − ( e ; ε ) ≥ η, where the identity ( ∗ ) is a straightforward consequence of the fact that R h − is an isometry. (cid:3) IRECT LIMITS OF INFINITE-DIMENSIONAL CARNOT GROUPS 17
Corollary 3.18 (Direct limits of CMSGs with bounded nilpotency step) . Consider a direct system (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) of complete metric scalable groups satisfying (C1) and (C2) . Let us supposethat the nilpotency step of the groups G i is uniformly bounded. Then it holds that the direct limitof (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) exists.Proof. Let s ∈ N be such that every G i is nilpotent of step less than s and fix x , . . . , x s ∈ G ′ .Then there exist k ∈ I with the following property: for every i = 1 , . . . , s , there exists y i ∈ G k with ϕ ′ k ( y i ) = x i . Since G k is nilpotent of step less than s , we have[ x , . . . , [ x s − , x s ] . . . ] = [ ϕ ′ k ( y ) , . . . , [ ϕ ′ k ( y s − ) , ϕ ′ k ( y s )] . . . ] = ϕ ′ k ([ y , . . . , [ y s − , y s ] . . . ]) = e ′ . This proves that G ′ is nilpotent. Given (C2), by Lemma 3.17 the condition (C3) holds. The claimfollows then from Theorem 3.15. (cid:3) Infinite-dimensional Carnot groups
The aim of this section is to investigate direct systems of Carnot groups and their relations tothe notion of infinite-dimensional Carnot group. We recall first the relevant concepts for infinite-dimensional Carnot groups from [LDLM19]. Direct limits of infinite-dimensional Carnot groupsare discussed in Section 4.1.A
Carnot group is, by definition, a connected and simply connected Lie group whose Lie algebraadmits a stratification. Given a stratified Lie algebra, one naturally defines a family of dilationautomorphisms on the group via the vector space scalings on the first layer of the stratification.Therefore, any Carnot group is a topological scalable group, and becomes a complete metricscalable group when equipped with a homogeneous distance. Moreover, we say that a topologicalscalable group has Carnot group structure if it is isomorphic as a topological scalable group tosome Carnot group.Separable Banach spaces have the useful property of admitting a countable, dense collectionof finite-dimensional vector subspaces. This kind of collection has a natural generalization in theCarnot setting.
Definition 4.1 (Filtration by Carnot subgroups) . We say that a topological scalable group G is filtrated by Carnot subgroups if there exists a sequence ( N m ) m , m ∈ N , of topological scalablesubgroups of G such that each N m has Carnot group structure, N m < N m +1 , and G is the closureof S m ∈ N N m . In this case, we say that the sequence ( N m ) m , m ∈ N , is a filtration by Carnotsubgroups of the topological scalable group G . We then define a non-commutative analogue of separable Banach spaces.
Definition 4.2 (Infinite-dimensional Carnot group) . A complete metric scalable group admittinga filtration by Carnot subgroups is called an infinite-dimensional Carnot group . It is often convenient to use an equivalent algebraic criterion for a topological scalable group tohave filtrations. Similarly to classical Carnot groups, those elements in which the dilation map isa one-parameter subgroup play a special role for the geometry.
Definition 4.3 (First layer) . We define for a scalable group G its first layer as V ( G ) := (cid:8) x ∈ G (cid:12)(cid:12) δ t + s ( x ) = δ t ( x ) δ s ( x ) for every t, s ∈ R (cid:9) . We say then that the map t ∈ R δ t ( x ) ∈ G is a one-parameter subgroup . If G is a group and A ⊂ G is a subset, we denote by h A i the group generated by A , i.e.,the collection of all finite products of elements of A and of their inverses. If G is a scalablegroup equipped with a topology, we denote by h A i SC the closure of the group generated by { δ t ( a ) | a ∈ A, t ∈ R } . Observe that the first layer V ( G ) of a scalable group is invariant underdilations: δ t ( V ( G )) = V ( G ) for all t ∈ R \ { } . Therefore, G = h V ( G ) i SC if and only if h V ( G ) i is dense in G .It turns out that filtrations of topological scalable groups are in close connection with thegenerating first layer. The following result is proven in [LDLM19, Proposition 2.1]. Proposition 4.4 (Alternative characterization of admitting a filtration) . Let G be a topologicalscalable group. Then the following are equivalent: i) G admits a filtration by Carnot subgroups; ii) there exists a countable set A ⊂ V ( G ) such that A generates G as a topological scalablegroup and h Ω i is nilpotent for every finite subset Ω ⊂ A . We stress that if a topological scalable group has generating first layer, then having elementsin which the dilation map fails to be a one-parameter subgroup is the characterizing feature ofnon-commutativity. We formulate this observation in the following proposition.
Proposition 4.5.
Let ( G, δ, d ) be a complete metric scalable group such that h V ( G ) i SC = G .Then the following are equivalent: i) V ( G ) = G ; ii) G is abelian; iii) G is a Banach space.Proof. We show first that i) and ii) are equivalent. We start by proving the following claim:If x ∈ V ( G ) , then δ − ( x ) = x − . (4.1)The proof is a simple computation: x δ − ( x ) = δ ( x ) δ − ( x ) = δ − ( x ) = δ ( x ) = e. Suppose now that i) holds and take x, y ∈ G . Then xy ∈ V ( G ) and ii) follows from (4.1) as[ x, y ] = xyx − y − = xyδ − ( xy ) = xy ( xy ) − = e. Assume then that ii) holds and let x = x . . . x n where each x i ∈ V ( G ). Then for every t, s ∈ R , δ t ( x ) δ s ( x ) = δ t ( x ) . . . δ t ( x n ) δ s ( x ) . . . δ s ( x n ) = δ t ( x ) δ s ( x ) . . . δ t ( x n ) δ s ( x n )= δ t + s ( x ) . . . δ t + s ( x n ) = δ t + s ( x ) . Since δ is continuous and h V ( G ) i is dense in G , we obtain i).The fact that iii) implies i) and ii) is immediate. Regarding the opposite direction, recall thata real topological vector space is an abelian topological group ( V, +) equipped with a continuousscalar multiplication R × V ∋ ( t, v ) tv ∈ V satisfying:1. 1 v = v ;2. t ( sv ) = ( ts ) v ;3. tu + tv = t ( u + v ) and4. tv + sv = ( t + s ) v .Assuming i) and ii), it is easy to check that ( G, δ ) is a topological vector space with the scalarmultiplication ( t, x ) δ t ( x ). Moreover, letting k x k := d ( e, x ) defines a norm on G . As we assumed G to be complete with respect to d , we conclude that (cid:0) G, δ, k · k (cid:1) is a Banach space. (cid:3)
IRECT LIMITS OF INFINITE-DIMENSIONAL CARNOT GROUPS 19
Notice that in Proposition 4.5 the assumption of having generating first layer cannot be removed.Indeed, consider G = ( R , +) with the dilations given by δ λ ( r ) = λ r and the distance d = | · | .Then ( G, δ, d ) is an abelian complete metric scalable group with V ( G ) = { } . The group G caneven be made geodesic, see [Moi20, p. 21].4.1. Direct limits of Carnot groups.
The following proposition shows that any complete metricscalable group admitting a filtration is obtained as a direct limit of Carnot groups. In this simplecase a direct proof of the universal property is straightforward. We nevertheless prove Proposition4.6 by applying the more general Theorem 3.15 for the sake of an example.
Proposition 4.6 (Infinite-dimensional Carnot groups as direct limits) . Let ( G, δ, d ) be a completemetric scalable group admitting a filtration ( N m ) m by Carnot groups. For any m, n ∈ N with m ≤ n , we denote by ι mn : N m ֒ → N n and ι m : N m ֒ → G the inclusion maps. Then it holds thatthe direct system of complete metric scalable groups (cid:0) { N m } m ∈ N , { ι mn } m ≤ n (cid:1) is non-degenerate andits direct limit is given by (cid:0) G, { ι m } m ∈ N (cid:1) .Proof. First of all, ι mn and ι m are morphisms of metric scalable groups satisfying ι m = ι n ◦ ι mn for all m, n ∈ N with m ≤ n , thus (cid:0) G, { ι m } m ∈ N (cid:1) is a target of (cid:0) { N m } m ∈ N , { ι mn } m ≤ n (cid:1) . Nowthe direct limit of (cid:0) { N m } m ∈ N , { ι mn } m ≤ n (cid:1) in the category of scalable groups is the union G ′ = S m ∈ N N m ⊂ G . Moreover, we have Z ( d ′ ) = { e } (recall (3.2) for the definition), so the metric limitof (cid:0) { N m } m ∈ N , { ι mn } m ≤ n (cid:1) equals ( G ′ , δ, d ).Observe that, since G is complete, the dilation and the inversion map are Cauchy-continuouson G and, therefore, on G ′ . This shows that the direct system of CMSGs (cid:0) { N m } m ∈ N , { ι mn } m ≤ n (cid:1) is non-degenerate. Hence, by Theorem 3.15, the direct limit of (cid:0) { N m } m ∈ N , { ι mn } m ≤ n (cid:1) existsin the category of complete metric scalable groups, and it equals the metric completion of G ′ .Since G ′ is dense in G by assumption, we conclude that (cid:0) G, { ι m } m ∈ N (cid:1) is the direct limit of (cid:0) { N m } m ∈ N , { ι mn } m ≤ n (cid:1) in the category of complete metric scalable groups, as claimed. (cid:3) Our next aim is to study in which circumstances a direct limit of (infinite-dimensional) Carnotgroups exists in the category of CMSGs and when the limit is an infinite-dimensional Carnotgroup. Because of the generating first layer, the condition (C1) on the continuity of the dilationis automatically satisfied, as we will show next.
Lemma 4.7.
Let ( G, δ ) be a scalable group equipped with a compatible distance d . If R x : G → G is continuous for every x ∈ G and G = h V ( G ) i SC , then δ : R × G → G is Cauchy-continuous.Proof. Call S the set of x ∈ G such that R ∋ t δ ( t, x ) ∈ G is continuous. By Lemma 3.2 e), inorder to prove the statement it suffices to show that S = G . Note first that if x ∈ V ( G ), then d ( δ t ( x ) , δ s ( x )) = d ( e, δ s − t ( x )) = | s − t | d ( e, x )and the map δ ( · , x ) is an isometric embedding of R into G , so that V ( G ) ⊆ S . Let then x, y ∈ S .Since the right translation by δ t ( y ) is continuous, we have that d (cid:0) δ s ( xy ) , δ t ( xy ) (cid:1) ≤ d (cid:0) δ s ( x ) δ s ( y ) , δ s ( x ) δ t ( y ) (cid:1) + d (cid:0) δ s ( x ) δ t ( y ) , δ t ( x ) δ t ( y ) (cid:1) = d (cid:0) δ s ( y ) , δ t ( y ) (cid:1) + d (cid:0) δ s ( x ) δ t ( y ) , δ t ( x ) δ t ( y ) (cid:1) → s → t, which proves that δ ( · , xy ) is continuous and thus xy ∈ S . A direct computation yields x − ∈ V ( G )for every x ∈ V ( G ), which shows that h V ( G ) i is the set of finite products of elements of V ( G ),whence it follows that h V ( G ) i ⊆ S . Finally, we want to exploit the density of h V ( G ) i in G toconclude that S = G . To do so, observe that for any x, y ∈ G and a ∈ (0 , + ∞ ) it holds thatsup t ∈ [ − a,a ] d (cid:0) δ t ( x ) , δ t ( y ) (cid:1) = sup t ∈ [ − a,a ] | t | d ( x, y ) ≤ a d ( x, y ) . Therefore, given x ∈ G and ( x n ) n ⊆ h V ( G ) i with lim n d ( x n , x ) = 0, we have that δ ( · , x n ) → δ ( · , x )uniformly on compact sets. Being δ ( · , x n ) continuous for every n ∈ N , we infer that δ ( · , x ) iscontinuous as well, thus proving that x ∈ S . Consequently, the statement is achieved. (cid:3) The proof of Theorem 1.2 is obtained as a corollary of the following result.
Theorem 4.8 (Direct limits of infinite-dimensional Carnot groups) . If (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) is adirect system of infinite-dimensional Carnot groups in the category of CMSGs satisfying (C3) ,then it has a direct limit. If, in addition, I is countable and each G i is nilpotent, then the limit isan infinite-dimensional Carnot group.Proof. We start with the following algebraic observation. Since each G i is an infinite-dimensionalCarnot group, by Proposition 4.4 there exists for each i ∈ I a countable set A i ⊂ V ( G i ) suchthat h A i i SC = G i and h Ω i i is nilpotent for every finite subset Ω i ⊂ A i . Consider then the directlimit G ′ of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) in the category of scalable groups equipped with the infimum-pseudodistance d ′ . Observe that ϕ ′ i ( V ( G i )) ⊂ V ( G ) for every i ∈ I , since ϕ ′ i are morphisms ofscalable groups. Hence the set A := S i ∈ I ϕ ′ i ( A i ) is contained in V ( G ′ ). Recall that the morphisms ϕ ′ i are continuous with respect to the topology induced by the infimum-pseudodistance d ′ on G ′ .Then ϕ ′ i ( h A i i SC ) ⊆ h ϕ ′ i ( A i ) i SC for every i ∈ I , and consequently G ′ ⊆ [ i ∈ I ϕ ′ i ( G i ) = [ i ∈ I ϕ ′ i ( h A i i SC ) ⊆ [ i ∈ I h ϕ ′ i ( A i ) i SC ⊆ (cid:10) [ i ∈ I ϕ ′ i ( A i ) (cid:11) SC = h A i SC . (4.2)Equation (4.2) shows that, in particular, G ′ = h V ( G ′ ) i SC . Moreover, by (C3) and Lemma3.2 b), the right translations in G ′ are continuous. Then by Lemma 3.10, the scalable group (cid:0) G ′ /Z ( d ′ ) , δ (cid:1) together with the compatible infimum-distance d is well-defined. We are now ina position to apply Lemma 4.7, which gives that the dilation map δ is Cauchy-continuous on G ′ /Z ( d ′ ). Hence (C1) is satisfied. Since also (C3) is assumed to hold, by Theorem 3.15 the directlimit (cid:0) G, { ϕ i } i ∈ I (cid:1) in the category of complete metric scalable groups exists, and it equals themetric completion of the metric limit (cid:0) G ′ /Z ( d ′ ) , δ, d (cid:1) . This proves the first part of the claim.Finally, assume that each G i is nilpotent and I is countable. Then the set A = S i ∈ I ϕ ′ i ( A i ) ⊂ V ( G ′ ) introduced above is countable. Since the image of G ′ is dense in G , (4.2) guarantees that G = h A i SC . Then, in order to prove that G is an infinite-dimensional Carnot group, by Proposition4.4 it is enough to show that h Ω i is nilpotent for every finite subset Ω ⊂ A . Now given such a setΩ, for every a ∈ Ω fix i ∈ I such that there exists a i ∈ A i with a = ϕ i ( a i ). Let ℓ ∈ I be such that,for every a ∈ Ω and i ∈ I associated to a , we have ℓ > i . Then, for every a ∈ Ω, a = ϕ i ( a i ) = ϕ ℓ ◦ ϕ iℓ ( a i ) ∈ ϕ ℓ ( G ℓ ) . Hence h Ω i ⊂ ϕ ℓ ( G ℓ ). Since each G ℓ is nilpotent by assumption, ϕ ℓ ( G ℓ ) is a nilpotent subgroupof G . We conclude that h Ω i is nilpotent and G is an infinite-dimensional Carnot group, which wewere aiming to show. (cid:3) We do not know if the assumption on nilpotency of the groups G i in Theorem 4.8 can beremoved; we do not have a counterexample.Theorem 4.8 gives, together with Corollary 3.18, the following result. Corollary 4.9.
Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) be a countable direct system of Carnot groups in thecategory of CMSGs such that the nilpotency step of the groups G i is uniformly bounded. If (C2) is satisfied, then the direct limit of (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) exists in the category of CMSGs and it isan infinite-dimensional Carnot group. IRECT LIMITS OF INFINITE-DIMENSIONAL CARNOT GROUPS 21
It would be interesting to know whether a uniform bound on the nilpotency steps of Carnotgroups G i automatically guarantees condition (C2). In Proposition 4.11 we give an example of adirect system of Carnot groups G i that does not satisfy (C2) and for which the nilpotency stepsof G i grow unlimitedly.4.2. Example of a degenerate direct system of Carnot groups.
We now give an example ofa direct system of Carnot groups that does not satisfy (C2) (see Proposition 4.11). Below we denoteby F ,k the Free Lie group of rank 2 and step k with the canonical projections π lk : F ,l → F ,k forevery l, k ∈ N with k ≤ l . For each k ∈ N , fix a basis X k , Y k for the first layer of the Lie algebrasuch that π lk ( x l ) = x k , and π lk ( y l ) = y k for every l, k ∈ N , k ≤ l , where we denote x k := exp( X k )and y k := exp( Y k ). We also fix a norm on each horizontal layer for which k X k k = k Y k k = 1 andconsider the corresponding Carnot–Carath´eodory distance d k F on F ,k .In the following lemma we put together several results in [DZ19]. Lemma 4.10.
For each k ∈ N , let F ,k be the free Lie group of rank 2 and step k with generators x k and y k as described above. Then for every ε > , it holds lim k →∞ d k F ( x k , δ ε ( y k ) x k ) > . Proof.
Fix any ε > γ : [0 , ε ] → R , γ ( t ) = ( − t, , t ∈ [0 , , ( − , t − , t ∈ [1 , ε ] , ( t − (2 + ε ) , ε ) t ∈ [1 + ε, ε ] . By [DZ19, Lemma 4.1], the curve γ admits a unique lift γ k to F ,k for every k ∈ N . It follows fromthe uniqueness of the lift that γ k (2 + ε ) = x − k δ ε ( y k ) x k for every k ∈ N . Let (cid:0) F , ∞ , { π ∞ k } k ∈ N (cid:1) be the inverse limit of the inverse system (cid:0) { F ,k } k ∈ N , { π lk } k ≤ l (cid:1) in the category of scalable groupsequipped with the supremum-metric d ′ (see (A.1) for the definition). By [DZ19, Lemma 2.7],the curve γ admits a unique lift η to the group F , ∞ satisfying π ∞ k ( η (2 + ε )) = x − k δ ε ( y k ) x k and π ∞ k ( η (0)) = e k for every k ∈ N . Moreover, the curve η has length L ( η ) = L ( γ ) = 2 + ε .According to [DZ19, Theorem 4.2], the inverse limit ( F , ∞ , d ′ ) is a metric tree. In particular, theinjective curve η is a geodesic, which implies that d ′ ( e ∞ , η (2+ ε )) = d ′ ( η (0) , η (2+ ε )) = L ( η ) = 2+ ε .Hence, by definition of d ′ , we obtainlim k →∞ d k F ( e k , x − k δ ε ( y k ) x k ) = lim k →∞ d k F (cid:0) π ∞ k ( e ∞ ) , π ∞ k ( η (2 + ε )) (cid:1) = d ′ ( e ∞ , η (2 + ε )) = 2 + ε. The claim follows then from the left-invariance of each distance d k F . (cid:3) We now define a stratified Lie algebra g i of rank i + 1 and step i for every i ∈ N as follows:we fix a basis { X i , Y i , . . . , Y ii } for the first layer of g i , where each pair { X i , Y ki } generates thefree Lie algebra of rank 2 and step k , and where all the other brackets are zero. Observe thateach g i is indeed a Lie algebra: the bracket [ Z , [ Z , Z ]] = 0 for all distinct basis elements Z , Z , Z ∈ { X i , Y i , . . . , Y ii } , and hence the Jacobi identity is satisfied. We also fix a norm onthe horizontal layer of g i that gives unit length to each basis vector.Let us denote by G i the Carnot group whose Lie algebra is g i and let d i be the Carnot–Carath´eodory distance on G i . Denote also x i := exp( X i ) and y ki := exp( Y ki ) for every i ∈ N and k = 1 , . . . , i . Observe that there exist isometric embeddings ι ij : G i → G j for i ≤ j satisfying ι ij ( x i ) = x j and ι ij ( y ki ) = y kj for every k = 1 , . . . , i . Moreover, each G i contains an isometric copyof every F ,k for k = 1 , . . . , i . Proposition 4.11 (A degenerate direct system of Carnot groups) . Let G i and ι ij : G i → G j be asabove for every i, j ∈ N , i ≤ j . Then the direct system (cid:0) { G i } i ∈ N , { ι ij } i ≤ j (cid:1) does not satisfy (C2) .Proof. Let (cid:0) G ′ , { ι i } i ∈ N (cid:1) be the direct limit of (cid:0) { G i } i ∈ N , { ι ij } i ≤ j (cid:1) in the category of scalable groups.Observe that ι i ( x i ) = ι j ( x j ) for every i, j ∈ N and ι i ( y ki ) = ι j ( y kj ) for every k ∈ N and i, j ≥ k .Let x ∈ G ′ be the element satisfying x = ι i ( x i ) for every i ∈ N . We are going to show that R x isnot continuous at the identity with respect to the pseudometric d ′ defined in (3.4), which wouldprove that (C2) is not satisfied.Let ε >
0. Since every G i contains an isometric copy of F ,i with generators x i , y ii , by Lemma4.10 there exists k ∈ N such that d k ( x k , δ ε ( y kk ) x k ) > . Moreover, we have d k ( e k , δ ε ( y kk )) = ε . Denote y ε := ι k ( δ ε ( y kk )) ∈ G ′ . Since the morphisms ι ij areisometric embeddings, it follows from the definition of d ′ that d ′ ( e ′ , y ε ) = ε and d ′ ( ι k ( x k ) , y ε ι k ( x k )) = d ′ ( x, y ε x ) > . Since ε was arbitrary, this proves that R x is not continuous at the identity, as claimed. (cid:3) A Rademacher-type theorem for direct limits of CMSGs
As already described in the Introduction, in this section we study those CMSGs satisfying asuitable form of Rademacher theorem. More specifically, we will show that such a class of CMSGsis stable under taking direct limits, thus generalizing one of the main results of [LDLM19].To begin with, we define what is a ‘null set’ in a CMSG. At this level of generality, it is notclear whether this family of null sets is actually the family of negligible sets of some measure.
Definition 5.1 (Null sets in a CMSG) . Let G be a complete metric scalable group. Then by family of null sets in G we mean a left-invariant σ -ideal N of the Borel σ -algebra B ( G ) , i.e., i) N is a subset of the Borel σ -algebra B ( G ) , containing the empty set. ii) If ( N n ) n ∈ N ⊆ N , then S n ∈ N N n ∈ N . iii) If N ∈ N and N ′ ∈ B ( G ) satisfy N ′ ⊆ N , then N ′ ∈ N . iv) gN ∈ N for every g ∈ G and N ∈ N . Let us consider a countable family { ( G i , N i ) } i ∈ I , where { G i } i ∈ I is a direct system of CMSGsadmitting a direct limit G , while the sets {N i } i ∈ I are families of null sets. Then, as we are goingto show, there is a natural way to define a family N of null sets in G . The construction is inspired(and extends) the analogous ones in [Aro76] and [LDLM19]. Lemma 5.2.
Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) be a direct system of complete metric scalable groups (wherethe directed set I is at most countable) having direct limit (cid:0) G, { ϕ i } i ∈ I (cid:1) in that category. Supposethat each space G i is endowed with a family of null sets N i . Then the set N ⊆ B ( G ) , given by N := (cid:26) [ i ∈ I N i (cid:12)(cid:12)(cid:12)(cid:12) N i ∈ B ( G ) and ϕ − i ( qN i ) ∈ N i for every i ∈ I and q ∈ G (cid:27) , (5.1) is a family of null sets in G .Proof. Let us prove that N satisfies the four conditions in Definition 5.1:i) Trivially, ∅ ∈ N ⊆ B ( G ). IRECT LIMITS OF INFINITE-DIMENSIONAL CARNOT GROUPS 23 ii) Fix any { N ni : n ∈ N , i ∈ I } ⊆ B ( G ) such that S i ∈ I N ni ∈ N for every n ∈ N . Let us definethe set N ∈ B ( G ) as N := S i ∈ I S n ∈ N N ni . Given any i ∈ I and q ∈ G , we have that ϕ − i (cid:18) q [ n ∈ N N ni (cid:19) = ϕ − i (cid:18) [ n ∈ N qN ni (cid:19) = [ n ∈ N ϕ − i ( qN ni ) ∈ N i , which proves that N ∈ N , as required.iii) Fix any N ∈ N and N ′ ∈ B ( G ) such that N ′ ⊆ N , say N = S i ∈ I N i . Given i ∈ I and q ∈ G ,we have that N i ∩ N ′ ∈ B ( G ) and ϕ − i (cid:0) q ( N i ∩ N ′ ) (cid:1) ⊆ ϕ − i ( qN i ) ∈ N i , whence N i ∩ N ′ ∈ N i aswell. This grants that N ′ = S i ∈ I N i ∩ N ′ ∈ N , as required.iv) Fix any N = S i ∈ I N i ∈ N and g ∈ G . Since ϕ − i (cid:0) q ( gN i ) (cid:1) = ϕ − i (cid:0) ( qg ) N i (cid:1) ∈ N i for every i ∈ I and q ∈ G , thus accordingly gN = S i ∈ I gN i ∈ N , as required. (cid:3) Next we introduce the notion of Gˆateaux differential in the sense of Pansu, whose formulationis taken from [LDLM19].
Definition 5.3 (Gˆateaux differentiability) . Let ( G, δ, d ) , ( G ′ , δ ′ , d ′ ) be metric scalable groups.Given f : G → G ′ and p ∈ G , we say that f is Gˆateaux differentiable at p if the following hold: i) For every element g ∈ G , we have that ∃ d p f ( g ) := lim λ → δ ′ /λ (cid:16) f ( p ) − f (cid:0) p δ λ ( g ) (cid:1)(cid:17) ∈ G ′ . ii) The resulting function d p f : G → G ′ is a continuous group morphism.We say that d p f is the Gˆateaux differential of f at p . Moreover, let us define ND( f ) := (cid:8) p ∈ G (cid:12)(cid:12) f is not Gˆateaux differentiable at p (cid:9) . We say that
ND( f ) ⊆ G is the non-differentiability set of f . Remark 5.4. If f is Gˆateaux differentiable at p , then its Gˆateaux differential d p f : G → G ′ is aLipschitz morphism of scalable groups. Indeed, it is well-known (and easy to prove) that d p f is amorphism of scalable groups, while Proposition 3.5 grants that d p f is Lipschitz. (cid:4) We say that a complete metric scalable group G together with some family N of null setshas the Rademacher property if every Lipschitz function on G is ‘ N -almost everywhere’ Gˆateauxdifferentiable. More precisely: Definition 5.5 (Rademacher property) . Let ( G, N ) be a complete metric scalable group endowedwith a family of null sets. Then ( G, N ) is said to have the Rademacher property provided it holds ND( f ) ∈ N for every Lipschitz function f : G → R . We are finally in a position to state and prove the main result of this section, which givesTheorem 1.3 as a corollary.
Theorem 5.6 (Stability of the Rademacher property) . Let (cid:0) { G i } i ∈ I , { ϕ ij } i ≤ j (cid:1) be a direct sys-tem of complete metric scalable groups (where the directed set I is countable) having direct limit (cid:0) G, { ϕ i } i ∈ I (cid:1) in that category. Let each space G i be endowed with a family of null sets N i suchthat ( G i , N i ) has the Rademacher property. Then ( G, N ) has the Rademacher property, where N is the family of null sets defined in (5.1) .Proof. Let f : G → R be a fixed Lipschitz function. We define N i := (cid:8) p ∈ G (cid:12)(cid:12) e i ∈ ND( f ◦ L p ◦ ϕ i ) (cid:9) ∈ B ( G ) for every i ∈ I. Then we claim that N := S i ∈ I N i ∈ N . To prove it, fix i ∈ I and q ∈ G . We aim to show that ϕ − i ( qN i ) ⊆ ND( f ◦ L − q ◦ ϕ i ) . (5.2)Pick any x ∈ ϕ − i ( qN i ). This means that ϕ i ( x ) = qp for some p ∈ N i . By construction, we have e i ∈ ND( f ◦ L p ◦ ϕ i ). Observe that for any g ∈ G i and λ > f ◦ L − q ◦ ϕ i ) (cid:0) x δ iλ ( g ) (cid:1) − ( f ◦ L − q ◦ ϕ i )( x ) λ = f (cid:0) q − ϕ i (cid:0) x δ iλ ( g ) (cid:1)(cid:1) − f (cid:0) q − ϕ i ( x ) (cid:1) λ = f (cid:0) p ϕ i (cid:0) δ iλ ( g ) (cid:1)(cid:1) − f ( p ) λ = ( f ◦ L p ◦ ϕ i ) (cid:0) δ iλ ( g ) (cid:1) − ( f ◦ L p ◦ ϕ i )( e i ) λ , so that e i ∈ ND( f ◦ L p ◦ ϕ i ) is equivalent to x ∈ ND( f ◦ L − q ◦ ϕ i ). This proves (5.2). Giventhat the function f ◦ L − q ◦ ϕ i is Lipschitz (as composition of Lipschitz maps) and ( G i , N i ) has theRademacher property, we conclude that ϕ − i ( qN i ) ∈ N i for all i ∈ I and accordingly N ∈ N .Let p ∈ G \ N be fixed. We aim to prove that f is Gˆateaux differentiable at p , which would beenough to achieve the statement. Let us call G ′ := S i ∈ I ϕ i ( G i ), which is a dense scalable subgroupof G (recall Remark 3.16).We define the function F p : G ′ → R as follows: given g ∈ G ′ , we set F p ( g ) := d e i ( f ◦ L p ◦ ϕ i )( g i ) for any i ∈ I and g i ∈ G i such that ϕ i ( g i ) = g. The well-posedness of F p stems from the trivial identityd e i ( f ◦ L p ◦ ϕ i )( g i ) = lim λ → f (cid:0) p ϕ i (cid:0) δ iλ ( g i ) (cid:1)(cid:1) − f ( p ) λ = lim λ → f (cid:0) p δ λ ( g ) (cid:1) − f ( p ) λ . (5.3)It can be also readily checked – by looking at (5.3) – that F p is a morphism of scalable groups.Given any λ ∈ R \ { } , we define the ‘incremental ratio’ function IR λ : G → R asIR λ ( g ) := f (cid:0) p δ λ ( g ) (cid:1) − f ( p ) λ for every g ∈ G. Observe that for any λ ∈ R \ { } and g, h ∈ G we have that (cid:12)(cid:12) IR λ ( g ) − IR λ ( h ) (cid:12)(cid:12) = (cid:12)(cid:12) f (cid:0) p δ λ ( g ) (cid:1) − f (cid:0) p δ λ ( h ) (cid:1)(cid:12)(cid:12) | λ | ≤ Lip( f ) d (cid:0) p δ λ ( g ) , p δ λ ( h ) (cid:1) | λ | = Lip( f ) d ( g, h ) , whence { IR λ } λ =0 is an equiLipschitz family of functions. As a consequence of (5.3), we see thatthe function F p : G ′ → R is Lipschitz (being the pointwise limit of an equiLipschitz family offunctions), thus it can be uniquely extended to a Lipschitz function ¯ F p : G → R . By density of G ′ in G and by continuity of the scalable group operations, we deduce that the extended function ¯ F p is a Lipschitz morphism of scalable groups. Finally, we also know that IR λ → ¯ F p in the pointwisesense, so that f is Gˆateaux differentiable at p and d p f = ¯ F p . This completes the proof. (cid:3) Appendix A. Inverse limits of complete metric scalable groups
Let (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) be an inverse system of complete metric scalable groups. We denoteby δ i and d i the dilation and the distance on G i , respectively. Since (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) is aninverse system in the category of scalable groups, it admits an inverse limit (cid:0) ( G ′ , δ ′ ) , { P ′ i } i ∈ I (cid:1) inthis category by Theorem 2.8. Set d ′ ( x, y ) := sup i ∈ I d i (cid:0) P ′ i ( x ) , P ′ i ( y ) (cid:1) = lim i ∈ I d i (cid:0) P ′ i ( x ) , P ′ i ( y ) (cid:1) for every x, y ∈ G ′ . (A.1) IRECT LIMITS OF INFINITE-DIMENSIONAL CARNOT GROUPS 25
Then d ′ is an extended distance on G ′ , meaning that it satisfies the distance axioms, but possiblytaking value + ∞ . It can be readily checked that d ′ is compatible , i.e., d ′ ( xy, xz ) = d ′ ( y, z ) , for every x, y, z ∈ G ′ , d ′ (cid:0) δ ′ λ ( x ) , δ ′ λ ( y ) (cid:1) = | λ | d ′ ( x, y ) , for every λ ∈ R and x, y ∈ G ′ , where in the second identity we are adopting the convention that 0 · ∞ = 0. Moreover, we set G := (cid:8) x ∈ G ′ (cid:12)(cid:12) d ′ ( x, e ) < + ∞ (cid:9) , δ := δ ′ | R × G : R × G → G, d := d ′ | G × G , (A.2)where e is the identity element of G ′ . Standard verifications show that G is a subgroup of G ′ and δ ′ λ ( G ) ⊆ G for every λ ∈ R , so that δ is well-defined and is a dilation on G . Moreover, therestricted distance d is compatible. Let us also define P i := P ′ i | G : G → G i for every i ∈ I. (A.3)It follows from the definition of d ′ that each scalable group morphism P i : G → G i is 1-Lipschitz. Definition A.1 (Non-degenerate inverse system of MSGs) . Let (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) be an inversesystem of metric scalable groups. Define (cid:0) ( G, δ, d ) , { P i } i ∈ I (cid:1) as in (A.2) and (A.3) . Then we saythat (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) is non-degenerate provided the following conditions are satisfied: R x is continuous at e, for every x ∈ G, (A.4a) δ ( · , x ) is continuous at t, for every x ∈ G and t ∈ R . (A.4b)Similarly to Propositions 3.9 and 3.14, we have an alternative (and more explicit) characteri-zation of non-degeneracy. We omit its standard proof. Proposition A.2.
Let (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) be an inverse system of metric scalable groups. Let usdefine (cid:0) ( G, δ, d ) , { P i } i ∈ I (cid:1) as in (A.2) and (A.3) . Then (A.4a) and (A.4b) are equivalent to inf η> lim i ∈ I sup (cid:26) d i (cid:0) P i ( yx ) , P i ( x ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) y ∈ G, lim j ∈ I d j (cid:0) P j ( y ) , e j (cid:1) < η (cid:27) = 0 , ∀ x ∈ G, (A.5a)inf η> lim i ∈ I sup s ∈ R : | s − t | <η d i (cid:0) δ is ( P i ( x )) , δ it ( P i ( x )) (cid:1) = 0 , ∀ x ∈ G, t ∈ R , (A.5b) respectively. Lemma A.3.
Let (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) be a non-degenerate inverse system of metric scalablegroups. Then ( G, δ, d ) defined in (A.2) is a metric scalable group and each map P i : G → G i as in (A.3) is a morphism of metric scalable groups. Moreover, if in addition the spaces { G i } i ∈ I are complete metric scalable groups, then G is a complete metric scalable group as well.Proof. The first part of statement immediately follows from Lemma 3.2 and the very definition ofnon-degenerate inverse system. In order to prove the second part of the statement, suppose thedistances { d i } i ∈ I are complete. We aim to show that d is a complete distance. Fix a d -Cauchysequence ( x n ) n ⊆ G . Since P i is Lipschitz, we deduce that (cid:0) P i ( x n ) (cid:1) n ⊆ G i is a d i -Cauchy sequencefor any i ∈ I . Since d i is complete, there is x i ∈ G i such that lim n P i ( x n ) = x i . Given any i, j ∈ I with i ≤ j , we see that P ij ( y j ) = y i by letting n → ∞ in P ij (cid:0) P j ( x n ) (cid:1) = P i ( x n ) (here the continuityof P ij plays a role). Consequently, there exists a unique element x ∈ G ′ such that P ′ i ( x ) = x i forall i ∈ I . We claim that x ∈ G and lim n x n = x . To prove it, fix ε > n ∈ N such that d ( x n , x m ) ≤ ε for every n, m ≥ ¯ n . This implies that d i (cid:0) P i ( x n ) , P i ( x m ) (cid:1) ≤ ε for every i ∈ I and n, m ≥ ¯ n . By letting m → ∞ we infer that d i (cid:0) P i ( x n ) , x i (cid:1) ≤ ε for every i ∈ I and n ≥ ¯ n , whence d ′ ( x n , x ) ≤ ε for all n ≥ ¯ n . This gives d ′ ( x, e ) ≤ d ′ ( x, x ¯ n ) + d ( x ¯ n , e ) < + ∞ , so accordingly x ∈ G .Moreover, it shows that lim n d ( x n , x ) = 0, as required. (cid:3) We can now state and prove our main results about inverse limits of MSGs and CMSGs.
Theorem A.4 (Inverse limits of MSGs) . Let (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) be an inverse system of metricscalable groups. Then the following conditions are equivalent: i) The inverse system (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) is non-degenerate (in the sense of Definition A.1). ii) The inverse limit of (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) in the category of metric scalable groups exists.Moreover, it coincides with the couple (cid:0) ( G, δ, d ) , { P i } i ∈ I (cid:1) given by (A.2) and (A.3) .Proof. i) = ⇒ ii) Suppose i) holds. In light of Lemma A.3, to prove that (cid:0) G, { P i } i ∈ I (cid:1) is the inverselimit of (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) , it suffices to show that (cid:0) G, { P i } i ∈ I (cid:1) satisfies the universal property.Fix a MSG H and a family { Q i } i ∈ I of MSG morphisms Q i : H → G i such that Q i = P ij ◦ Q j for every i, j ∈ I with i ≤ j . In particular, each Q i is a morphism of scalable groups, then thereexists a unique scalable group morphism Φ : H → G ′ such that Q i = P ′ i ◦ Φ for every i ∈ I . Letus prove that Φ( H ) ⊆ G : given any y ∈ H , we have that d i (cid:0) P ′ i (Φ( y )) , e (cid:1) = d i (cid:0) Q i ( y ) , e i (cid:1) ≤ d H ( y, e H ) for every i ∈ I, whence d ′ (cid:0) Φ( y ) , e (cid:1) ≤ d H ( y, e H ) < + ∞ and accordingly Φ( y ) ∈ G . The same computation showsthat Φ is 1-Lipschitz from ( H, d H ) to ( G, d ). Observe also that Q i = P i ◦ Φ for all i ∈ I . Therefore,the universal property is proven, thus obtaining ii).ii) = ⇒ i) Suppose ii) holds. Then (A.4a) and (A.4b) are satisfied, thus accordingly the inversesystem (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) is non-degenerate. This proves i). (cid:3) Corollary A.5 (Inverse limits of CMSGs) . Let (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) be an inverse system of com-plete metric scalable groups. Then the following conditions are equivalent: i) The inverse system (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) is non-degenerate (in the sense of Definition A.1). ii) The inverse limit of (cid:0) { G i } i ∈ I , { P ij } i ≤ j (cid:1) in the category of complete metric scalable groupsexists. Moreover, it coincides with the couple (cid:0) ( G, δ, d ) , { P i } i ∈ I (cid:1) given by (A.2) and (A.3) .Proof. The claim follows from Theorem A.4 and the last part of the statement of Lemma A.3. (cid:3)
Acknowledgements.
Both authors were supported by the European Research Council (ERCStarting Grant 713998 GeoMeG Geometry of Metric Groups). The first named author was alsosupported by the Academy of Finland (grant 288501 Geometry of subRiemannian groups andgrant 322898 Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory). The secondnamed author was also supported by the Academy of Finland (project number 314789) and bythe Balzan project led by Prof. Luigi Ambrosio.
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