Sequence of induced Hausdorff metrics on Lie groups
Norbil L. Cordova Neyra, Ryuichi Fukuoka, Eduardo de A. Neves
aa r X i v : . [ m a t h . DG ] N ov SEQUENCE OF INDUCED HAUSDORFF METRICS ON LIEGROUPS
NORBIL CORDOVA, RYUICHI FUKUOKA, AND EDUARDO DE A. NEVES
Abstract.
Let ϕ : G × ( M, d ) → ( M, d ) be a left action of a Lie group ona differentiable manifold endowed with a metric d , which is compatible withits topology. Let X be a compact subset of M . Then the isotropy subgroupof X is defined as H X := { g ∈ G ; gX = X } and it is closed in G . Theinduced Hausdorff metric is a metric on the left coset manifold G/H X definedas d X ( gH X , hH X ) = d H ( gX, hX ), where d H is the Hausdorff distance in M .Suppose that ϕ is transitive and that there exist p ∈ M such that H X = H p .Then gH X gp is a diffeomorphism that identifies G/H X and M . In thiswork we define a discrete dynamical system of metrics on M . Let d = ˆ d X ,where ˆ d X stands for the intrinsic metric induced by d X . We can iterate theprocess on ϕ : G × ( M ≡ G/H X , d ) → ( M ≡ G/H X , d ), in order to get d , d and so on. We study the particular case where M = G , ϕ : G × ( G, d ) → ( G, d )is the usual product, d is bounded above by a right invariant intrinsic metricon G and X is a finite subset of G containing the identity element. We provethat d i converges pointwise to a metric d ∞ . In addition, if d is complete andthe semigroup generated by X is dense in G , then d ∞ is the distance functionof a right invariant C -Carnot-Carath´eodory-Finsler metric. The case where d ∞ is C -Finsler is studied in detail. Introduction
Before talking about the subject of this paper, we present two topics related tothis work. We don’t use the second topic here. It will only illustrate our mainresult (Theorem 5.3).The first topic is invariant metrics on homogeneous spaces. Let M be a connecteddifferentiable manifold endowed with a completely nonholonomic distribution D .An absolutely continuous path γ in M is horizontal if γ ′ ( t ) ∈ D for almost every t .Chow-Rashevskii theorem states that every pair of points in M can be connectedby a horizontal curve (see [6], [15], [16]). We endow each subspace D x of D witha norm F ( x, · ) such that x F ( x, · ) is continuous, that is, given a horizontalcontinuous vector field Y on M , then x F ( x, Y ( x )) is a continuous map (see [3]).Now we proceed as in the definition of sub-Riemannian geometry: The C -Carnot-Carath´eodory-Finsler metric on M is given by d c ( x, y ) = inf γ ∈H x,y Z I F ( γ ( t ) , γ ′ ( t )) dt, (1)where H x,y is the set of horizontal curves connecting x and y (see [3], [4]). Date : November 9th, 2017.2010
Mathematics Subject Classification.
Key words and phrases.
Sequence of metrics, Induced Hausdorff metrics, Lie groups, Finslermetrics, Carnot-Caratheodory-Finsler metrics.
The following theorem due to Berestovskii is important for this work and statesthat if an intrinsic metric is homogeneous, then it has the tendency to gain extraregularity.
Theorem 1.1. [Berestovskii, [4] , Theorem 3] If ( M, d ) is locally compact, locallycontractible homogeneous space, endowed with an invariant intrinsic metric d , then ( M, d ) is isometric to the quotient space G/H of a Lie group G over a compact sub-group H of it, endowed with a G -invariant C -Carnot-Carath´eodory-Finsler metric. The second topic is geometric flows in Riemannian geometry. There are several ofthem such as the mean curvature flow (for hypersurfaces in Riemannian manifolds),the Ricci-Hamilton flow, etc. These flows, in some specific situations, converges toa more “homogeneous” Riemannian metric (eventually after some normalizationof the volume). For instance, the normalized Ricci-Hamilton flow converges toa Riemannian metric of constant sectional curvature for 3-dimensional closed Rie-mannian manifolds with strictly positive Ricci curvature (see [10]). It is an exampleof a dynamical system of metrics converging to a “well behaved” metric.Now we give an outline of this work.Let ϕ : G × M → M be a left action of a Lie group on a differentiable manifoldendowed with a metric d (distance function) compatible with the topology of M .As usual, we denote gx := ϕ ( g, x ). Let X be a compact subset of M . The isotropysubgroup of X is a closed subgroup H X of G given by H X = { g ∈ G ; gX = X } . Then there exist a unique differentiable structure on
G/H X , compatible with thequotient topology, such that the natural action φ : G × G/H X → G/H X , given by φ ( g, hH X ) = ghH X , is smooth. In [2] and [8], the authors define a metric d X on G/H X , which is called induced Haudorff metric, by d X ( gH X , hH X ) = d H ( gX, hX ) , where d H is the Hausdorff distance in ( M, d ) and gH X denote the left coset of g in G/H X . Suppose that ϕ is transitive and that there exist p ∈ M such that H p = H X .Then the map η : G/H X → M given by gH X g.p is a diffeomorphism such that ϕ ( g, η ( hH X )) = η ( φ ( g, hH X )), that is, ϕ can be identified with φ . We use theinduced Hausdorff metric in order to create a discrete dynamical system of metricson M . Define d = ˆ d X , where the hat denotes the intrinsic metric induced by d X .We can iterate this process in ϕ : G × ( M, d ) → ( M, d ) in order to obtain d ,where ( G/H X , d ) is identified with ( M, d ) via η . Through this iteration, we candefine a sequence of metrics d, d , d , . . . on M , which we call sequence of inducedHausdorff metrics .In this work we study the sequence of induced Hausdorff metrics in the followingcase: M is the Lie group G itself endowed with a metric d that is bounded aboveby a right invariant intrinsic metric d , ϕ : G × ( G, d ) → ( G, d ) is the product of G and X is a finite subset of G containing the identity element e (It is implicit herethat H X = { e } ). We prove that d i converges pointwise to a metric d ∞ . Moreoverif the semigroup S X generated by X is dense in G and d is complete, then d ∞ isthe distance function of a right invariant C -Carnot-Carath´eodory-Finsler metric.We give a necessary and sufficient condition in order to d ∞ be C -Finsler. In thiscase, an explicit formula for the Finsler metric is obtained. EQUENCE OF INDUCED HAUSDORFF METRICS ON LIE GROUPS 3
Of course the comparison of our work with the Riemannian geometric flowscan’t be taken so literally. It is given here only to illustrate the dynamics of thesequence of induced Hausdorff metrics, which in some cases converges to a more“well behaved” metric. One important restriction is that it is defined only onleft coset manifolds. On the other hand the metric d is much more general thanRiemannian metrics and the induced Hausdorff distance is the basic tool in orderto overcome the lack of differentiability.This work is organized as follows. In Section 2 we fix notations and we presentdefinitions and results that are necessary for this work. In Section 3 we study someproperties of the sequence of induced Hausdorff metrics. In particular, we prove thatit is increasing, and if d is bounded above by a right invariant intrinsic metric, thenit converges pointwise to a metric. This limit metric is intrinsic if d is complete.In Section 4, we prove that every connected Lie group G admits a finite subset X ∋ e such that H X = { e } and ¯ S X = G . In Section 5, we study the case ¯ S X = G ,where d is complete and bounded above by a intrinsic right invariant metric. Weprove that the sequence of induced Hausdorff metrics converges to a right invariant C -Carnot-Carath´eodory-Finsler metric. The case where d ∞ is C -Finsler is alsostudied here, as explained before. In Section 6, we give some additional examplesin order to illustrate better this work. Finally Section 7 is devoted to final remarks.The authors would like to thank Professor Luiz A. B. San Martin for somevaluable suggestions. 2. Preliminaries
In this section we present some definitions and results that are used in this work.They can be found in [5], [8], [11], [12], [13], [14] and [17]. For the sake of clearness,we usually don’t give the definitions and results in their most general case.Let G be a group and ( M, d ) be a metric space. Consider a left action ϕ : G × M → M of G on M . Then ex = x for every x ∈ M and ( gh ) x = g ( hx )for every ( g, h, x ) ∈ G × G × M . Every ϕ g := ϕ ( g, · ) is a bijection. We saythat a left action ϕ : G × M → M is an action by isometries if every ϕ g is anisometry. Analogously we say that ϕ is an action by homeomorphism if ϕ g is ahomeomorphism for every g ∈ G .Let ( M, d ) be a metric space. We denote the open ball with center p and radius r in ( M, d ) by B d ( p, r ). The closed ball is denoted by B d [ p, r ]. The topology inducedby d is denoted by τ d . The closure of a subset A in ( M, d ) is denoted by ¯ A . Whenmore than one metric or topology are involved, for instance we have metrics d and ρ and a topology τ on M , we use terms like d -neighborhood, τ -open subset, ρ -compact, etc.Let A, B be compact subsets of (
M, d ). The Hausdorff distance between
A, B isgiven by d H ( A, B ) = max (cid:26) sup x ∈ A inf y ∈ B d ( x, y ) , sup y ∈ B inf x ∈ A d ( x, y ) (cid:27) . It is well known that d H is a metric on the family of compact subsets of M .Let ϕ : G × M → M be a left action by homeomorphisms of a group G on ametric space ( M, d ). If X ⊂ M is a subset, then the isotropy subgroup of G withrespect to X is defined by H X = { g ∈ G ; gX = X } . Suppose that X is a compactsubset in M . In Proposition 2.1 of [8] (and in [2]), we define the induced Hausdorff NORBIL CORDOVA, RYUICHI FUKUOKA, AND EDUARDO DE A. NEVES metric on the left coset space
G/H X as d X ( gH X , hH X ) = d H ( gX, hX ), where d H is the Hausdorff distance in M .A partition P of an interval [ a, b ] is a subset { t , . . . , t n P } ⊂ [ a, b ] such that a = t < t < . . . < t n P = b . The norm of P is defined as |P| = max i =1 ,...,n P | t i − t i − | .The length of a path γ : [ a, b ] → M on a metric space ( M, d ) is given by ℓ d ( γ ) = sup P n P X i =1 d ( γ ( t i ) , γ ( t i − )) . It is well known that for every ε >
0, there exist a δ > ℓ d ( γ ) ≤ n P X i =1 d ( γ ( t i ) , γ ( t i − )) + ε for every partition P such that |P| < δ (see [5]).Given a metric space ( M, d ) we can define the extended metric (the distance canbe ∞ ) ˆ d ( x, y ) = inf γ ∈C x,y ℓ d ( γ ) , on M , where C x,y is the family of paths on ( M, d ) that connects x and y . Wedenote C dx,y instead of C x,y if there exist more than one metric defined on M . ˆ d isthe intrinsic (extended) metric induced by d , we always have that d ≤ ˆ d . We saythat the metric d is intrinsic if ˆ d = d and we have that a metric ˆ d is always intrinsic. Proposition 2.1.
Let ( M, d ) be a metric space and ˆ d be the intrinsic (extended)metric induced by d . Then ℓ ˆ d ( γ ) = ℓ d ( γ ) for every rectifiable curve γ in ( M, d ) . If (
M, d ) is a metric space, ε > x, y ∈ M , then an ε -midpoint of x and y is a point z ∈ M that satisfies | d ( x, z ) − d ( x, y ) | ≤ ε and | d ( y, z ) − d ( x, y ) | ≤ ε .We have the following results about intrinsic metrics. Their proofs can be foundin [5]. Proposition 2.2. If d is an intrinsic metric on M and ε > , then every x, y ∈ M admit an ε -midpoint. Proposition 2.3.
Let ( M, d ) be a complete metric space. If every x, y ∈ M admita ε -midpoint (for every ε > ), then d is intrinsic. C -Carnot-Carath´eodory-Finsler-metrics are intrinsic because they come froma length structure (see [5]). A particular case of C -Carnot-Carath´eodory-Finslermetrics are the C -Finsler metrics. Let M be a differentiable manifold and T M = { ( x, v ); x ∈ M, v ∈ T x M } be its tangent bundle. A C -Finsler metric on M is acontinuous function F : T M → R such that F ( x, · ) is a norm on T x M for every x ∈ M . F induces a metric d F on M given by d F ( x, y ) = inf γ ∈S x,y ℓ F ( γ ) , where ℓ F ( γ ) = Z ba F ( γ ( t ) , γ ′ ( t )) dt, is the length of γ in ( M, F ) and S x,y is the family of paths on M which are smoothby parts and connects x and y . Here the family S x,y can be replaced by the familyof absolutely continuous paths connecting x and y (see [3]). If d : M × M → R is EQUENCE OF INDUCED HAUSDORFF METRICS ON LIE GROUPS 5 a metric on M , then we say that d is C -Finsler if there exist a C -Finsler metric F on M such that d = d F . A differentiable manifold endowed with a C -Finslermetric is a C -Finsler manifold. Remark 2.4.
There are another usual (in fact more usual) definition of Finslermanifold, where F satisfies other conditions (see, for instance, [1] and [7] ): F issmooth on T M − T M , where T M = { ( p, ∈ T M ; p ∈ M } is the zero section,and F ( p, · ) is a Minkowski norm on T p M for every p ∈ M . In order to make thisdifference clear we put the prefix C before Finsler. C -Finsler metrics are studied, for instance, in [3] , [4] , [5] and [9] . Remark 2.5.
Let M be a connected differentiable manifold and D be a completelynonholonomic distribution on M . Let F as in the definition of the C -Carnot-Carath´eodory-Finsler metric in the introduction. Observe that there exist smoothssections g , g of inner products in D such that g ≤ F ≤ g . In fact, this isclear locally, and in order to see that this holds globally, just use partition of unity.This implies that balls in C -Carnot-Carath´eodory-Finsler metrics are containedand contains a ball in some Carnot-Carath´eodory metric of the same distribution.Ball-box theorem gives a qualitative behavior of balls in sub-Riemannian manifolds(see [15] ) and this behavior depends only on the distribution D . Therefore ballsin C -Carnot-Carath´eodory-Finsler metrics of the same distribution have the samequalitative behavior.Another consequence of the ball-box theorem is that a Carnot-Carath´eodory met-ric correspondent to a non-trivial completely nonholonomic distribution can’t thebounded above by the distance function of a Riemannian metric. Therefore a C -Carnot-Caratheodory-Finsler metric correspondent to a non-trivial completely non-holonomic distribution can’t be bounded above by a C -Finsler metric. The following theorem is the version of the classical Hopf-Rinow theorem forintrinsic metrics (see [5]).
Theorem 2.6 (Hopf-Rinow-Cohn-Vossen theorem) . Let ( M, d ) be a locally compactmetric space endowed with an intrinsic metric. Then the following assertions areequivalent: • ( M, d ) is complete; • B d [ p, r ] is compact for every p ∈ M and r > ; • Every geodesic (local minimizer parameterized by arclength) γ : [0 , a ) → M can be extended to a continuous path ¯ γ : [0 , a ] → M ; • There exist a p ∈ M such that every shortest path parameterized by ar-clength γ : [0 , a ) → M satisfying γ (0) = p admits a continuous extension ¯ γ : [0 , a ] → M . Lemma 2.7.
Any closed and bounded subset of a Lie group endowed with a rightinvariant intrinsic metric is compact. In particular, a right invariant intrinsicmetric on a Lie group is complete.Proof
A right invariant intrinsic metric ˜ d is a C -Carnot-Carath´eodory Finsler metricthat comes from a continuous right invariant section of norms F in a right invariantcompletely nonholonomic distribution D (see Theorem 1.1). Observe that ˜ d ≥ d g for some right invariant Riemannian manifold g . In fact, choose a right invariant NORBIL CORDOVA, RYUICHI FUKUOKA, AND EDUARDO DE A. NEVES smooth section of inner products ¯ g on D such that ¯ g ≤ F and then extend ¯ g to the whole tangent spaces resulting in a right invariant Riemannian metric g .The completeness of ( M, g ) and the classical Hopf-Rinow theorem for Riemannianmanifolds implies that any closed bounded subset of ( M, g ) is compact. Thereforeany closed bounded subset of ( M, ˜ d ) is compact because ˜ d ≥ d g .The last statement is due to Theorem 2.6. (cid:4) The rest of this section is about the orbits of a family of vector fields and it isused in Section 4 in order to prove the existence of X such that S X is dense in G (see [12]). Definition 2.8.
Let F be a family of complete vector fields on a differentiablemanifold M and x ∈ M . For X ∈ F , denote by t (exp tX )( x ) be the integralcurve of X such that (exp 0 X )( x ) = x . The orbit G ( x ) of F through x is the set ofpoints given by (exp t m X m )(exp t m − X m − ) . . . (exp t X )( x ) , with m ∈ N , t i ∈ R and X i ∈ F , i = 1 , . . . m . In the conditions of Definition 2.8, denote by
Lie ( F ) the Lie algebra generatedby F and let Lie x ( F ) be the restriction of Lie ( F ) to the tangent space T x M . Theorem 2.9 (Hermann-Nagano Theorem) . Let M be an analytic manifold and F be a family of analytic vector fields on M . Then • each orbit of F is an analytic submanifold of M ; • if N is an orbit of F , then the tangent space of N in x is given by Lie x ( F ) . Corollary 2.10.
Let G be a connected Lie group and V be a vector subspace of g such that the Lie algebra generated by V is g . Let { v , . . . , v k } be a basis of V and F be the set of left (or right) invariant vector fields with respect to { v , . . . , v k } .Then, for every x ∈ G , there exist m ∈ N , t i , . . . , t i m ∈ R and v i , . . . , v i m ∈ X such that x = exp( t i m v i m ) . . . exp( t i v i ) .Proof It is enough to observe that G is an analytic manifold, that the right (and left)invariant vector fields with respect to v , . . . , v k are analytic (see [11]) and thatLie e ( F ) = g . Then the orbit of e is an analytic submanifold of G that contains aneighborhood of e (see Theorem 2.9) and this neighborhood generates G due to theconnectedness of G . (cid:4) Convegence of the sequence of induced Hausdorff metrics on Liegroups
Let d be a metric on a Lie group G such that τ d = τ G , ϕ : G × ( G, d ) → ( G, d )be the product of G and X = { x , . . . , x k } ∋ e be a finite subset of G such that H X = { e } . In this section, we study properties of the sequence of induced Hausdorffmetrics d, d , d , . . . , d i , . . . on G .For every j = 1 , . . . , k , we define the metric d j ( p, q ) = d ( px j , qx j ) on G . τ d j = τ d because right translations are homeomorphisms on G . Define also the metric d M ( p, q ) := max j =1 ,...,k d j ( p, q ) := max j =1 ,...,k d ( px j , qx j ) on G .First of all we prove that τ d X = τ d (Proposition 3.6). Lemma 3.1. d ≤ d M and d X ≤ d M . EQUENCE OF INDUCED HAUSDORFF METRICS ON LIE GROUPS 7
Proof
The first inequality is obvious. The second inequality follows because d X ( p, q ) = max (cid:26) max i min j d ( px i , qx j ) , max j min i d ( px i , qx j ) (cid:27) ≤ max (cid:26) max i d ( px i , qx i ) , max j d ( px j , qx j ) (cid:27) = max i d ( px i , qx i ) = d M ( p, q ) . (cid:4) Lemma 3.2.
For every g ∈ G , there exist a G -neighborhood V of g ∈ G such that d X | V × V = d M | V × V .Proof For i, j = 1 , . . . , k , i = j , define ρ ij : G × G → R as ρ ij ( p, q ) = d ( px i , qx j ) − max k d ( px k , qx k ). Then for every g ∈ G , there exist a G -neighborhood V of g suchthat ρ ij ( p, q ) > p, q ∈ V and every i = j, because ρ ij ( g, g ) >
0. It implies that d X ( p, q ) = max (cid:26) max i min j d ( px i , qx j ) , max j min i d ( px i , qx j ) (cid:27) = max (cid:26) max i d ( px i , qx i ) , max j d ( px j , qx j ) (cid:27) = max i d ( px i , qx i ) = d M ( p, q ) (2)for every p, q ∈ V . (cid:4) Remark 3.3.
The formula d X ( p, q ) = max i d ( px i , qx i ) for every p, q in a suffi-ciently small neighborhood of g will be used several times in this work. Lemma 3.4. τ d = τ d M (= τ G ) on G .Proof The inequality d ≤ d M implies that τ d ⊂ τ d M .In order to see that τ d M ⊂ τ d , observe that an open ball B d M ( p, r ) can be writtenas B d M ( p, r ) = k \ j =1 B d j ( p, r ) , what implies that it is an open subset of τ d . Therefore τ d M ⊂ τ d . (cid:4) Lemma 3.5. B d X ( p, r ) is contained in a G -compact subset of G for a sufficientlysmall r > .Proof Just observe that B := k [ i =1 B d ( px i , r ) ⊃ B d X ( p, r ) . NORBIL CORDOVA, RYUICHI FUKUOKA, AND EDUARDO DE A. NEVES
In fact, if y B , then d X ( p, y ) = max (cid:26) max i min j d ( px i , yx j ) , max j min i d ( px i , yx j ) (cid:27) ≥ max (cid:26) max i min j d ( px i , yx j ) , min i d ( px i , y.e ) (cid:27) ≥ max (cid:26) max i min j d ( px i , yx j ) , r (cid:27) ≥ r, where the second inequality holds because y B . Then the result follows because B is contained in a G -compact subset of G for a sufficiently small r . Proposition 3.6. τ d = τ d X on G .Proof It is enough to prove that τ d X = τ d M due to Lemma 3.4.We know that d X ≤ d M (Lemma 3.1), what implies that τ d X ⊂ τ d M .In order to prove that τ d M ⊂ τ d X , we consider an open ball B d M ( p, r ) and weprove that there exist an ε > B d X ( p, ε ) ⊂ B d M ( p, r ). Without lossof generality, we can consider r > B d X ( p, r ) has compact closure (seeLemma 3.5). Consider a G -neighborhood V of p according to Lemma 3.2. We caneventually consider a (further) smaller r in such a way that B d M ( p, r ) ⊂ V (seeLemma 3.4). Then B d X ( p, r ) ∩ V = B d M ( p, r ) due to Lemma 3.2.If B d X ( p, r ) − V is the empty subset, then we have that B d X ( p, r ) = B d M ( p, r )and we are done. Otherwise we consider2 ε = inf x ∈ B dX ( p,r ) − V d X ( p, x ) . Observe d X is d × d -continuous because τ d X ⊂ τ d and ε is strictly positive because p is not contained in the G -compact subset B d X ( p, r ) − V . Therefore B d X ( p, ε ) = B d X ( p, ε ) ∩ V ⊂ B d M ( p, r )what settles the proposition. (cid:4) Definition 3.7.
Let X = { x , . . . , x k } ∋ e be a finite subset of a Lie group G . Thesemigroup generated by X is defined as S X = { x i . . . x i m ; m ∈ N , i j ∈ (1 , . . . , k ) , j ∈ (1 , . . . , m ) } . In what follows, if X = { x , . . . , x k } , then X − := { x − , . . . , x − k } . Proposition 3.8.
Let X be a finite subset of a Lie group G . Then S X is dense in G iff S X − is dense in G .Proof It is enough to observe that if i : G → G is the inversion map, then i ( S X ) = S X − . (cid:4) Lemma 3.9.
Let G be a Lie group and d : G × G → R be a metric on G such that τ d = τ G . Let ϕ : G × ( G, d ) → ( G, d ) be the product of G and X = { x , . . . , x k } ∋ e be a finite subset of G such that H X = { e } . Then d ≤ d and the sequence ofinduced Hausdorff metrics is increasing. EQUENCE OF INDUCED HAUSDORFF METRICS ON LIE GROUPS 9
Proof
Let x, y ∈ G and γ : [ a, b ] → G be a d X -path (which is also a G -path due toProposition 3.6) connecting x and y (If there isn’t any path connecting x and y ,then d ( x, y ) ≤ d ( x, y ) = ∞ ). Then ℓ d X ( γ ) = sup P n P X i =1 d X ( γ ( t i ) , γ ( t i − )) . Cover γ ([ a, b ]) with open subsets V of Lemma 3.2 such that (2) holds. Let ε bethe d -Lebesgue number of this covering. Let δ > | t − t | < δ , then d ( γ ( t ) , γ ( t )) < ε . Consider only partitions with norm less that δ . Then ℓ d X ( γ ) = sup P n P X i =1 max j d ( γ ( t i ) x j , γ ( t i − ) x j ) ≥ sup P n P X i =1 d ( γ ( t i ) , γ ( t i − )) = ℓ d ( γ ) (3)due to (2). Then d ( x, y ) = inf γ ∈C dXx,y ℓ d X ( γ ) ≥ inf γ ∈C dx,y ℓ d ( γ ) = ˆ d ( x, y ) ≥ d ( x, y ) . (cid:4) Theorem 3.10.
Let G be a Lie group and d : G × G → R be a metric on G such that τ d = τ G . Let ϕ : G × ( G, d ) → ( G, d ) be the product of G and X = { x , . . . , x k } ∋ e be a finite subset of G such that H X = { e } . Suppose that d is bounded above by aright invariant intrinsic metric d (what implies that G is connected). Then every d i is bounded above by d and the sequence of induced Hausdorff metrics convergespointwise to a metric d ∞ .Proof We prove that if d ≤ d , then d ≤ d . This is enough to prove that d i ≤ d forevery i : just iterate the process, replacing d by d i − .Let γ be a path defined on a closed interval. Then ℓ d X ( γ ) = sup P n P X i =1 max j d ( γ ( t i ) x j , γ ( t i − ) x j ) ≤ sup P n P X i =1 max j d ( γ ( t i ) x j , γ ( t i − ) x j )= sup P n P X i =1 d ( γ ( t i ) , γ ( t i − )) = ℓ d ( γ ) (4)because d is right invariant. From (4) and the fact that d is intrinsic, we have that d ( x, y ) = inf γ ∈C dXx,y ℓ d X ( γ ) ≤ inf γ ∈C d x,y ℓ d ( γ ) = d ( x, y )for every x, y ∈ G (Observe that C d X x,y = C d x,y due to Proposition 3.6).Observe that d i is an increasing sequence of metrics bounded above by d . Then d i converges pointwise to a function d ∞ and it is easy to prove that d ∞ is a metric. (cid:4) Observe that if d is a metric on a group G , then ¯ d : G × G → R , definedas ¯ d ( x, y ) = sup σ ∈ G d ( xσ, yσ ), is a right invariant extended metric. Its intrinsicextended metric ˆ¯ d ( x, y ) = inf C ¯ dx,y ℓ ¯ d ( γ ) is also a right invariant. Corollary 3.11.
Let G be a Lie group and d : G × G → R be a metric on G such that τ d = τ G . Let ϕ : G × ( G, d ) → ( G, d ) be the product of G and X = { x , . . . , x k } ∋ e be a finite subset of G such that H X = { e } . Suppose that ˆ¯ d is a metric on G . Thenthe sequence of induced Hausdorff metrics converges to a metric d ∞ on G .Proof It is enough to see that ˆ¯ d is a right invariant intrinsic metric such that d ≤ ˆ¯ d . (cid:4) Lemma 3.12.
Let d i be a sequence of induced Hausdorff metrics converging to d ∞ . Then d ∞ ( x, y ) ≥ d ∞ ( xσ, yσ ) for every σ ∈ ¯ S X and every x, y ∈ G . Inparticular ℓ d ∞ ( γ ) ≥ ℓ d ∞ ( γσ ) for every σ ∈ ¯ S X and every path γ . Moreover if ¯ S X is a subgroup, then d ∞ is invariant by the right action of ¯ S X . In particular, if ¯ S X = G , then d ∞ is right invariant.Proof In order to prove that d ∞ ( x, y ) ≥ d ∞ ( xσ, yσ ) for every ( x, y, σ ) ∈ G × G × ¯ S X ,it is enough to prove that d ∞ ( x, y ) ≥ d ∞ ( xx j , yx j ) for every ( x, y, x j ) ∈ G × G × X because every σ ∈ ¯ S X can be arbitrarily approximated by product of elements of X .Let d be a metric on G . Then ℓ d ( γ ) = sup P n P X i =1 d ( γ ( t i ) , γ ( t i − )) = sup P n P X i =1 d X ( γ ( t i ) , γ ( t i − ))= sup P n P X i =1 max j d ( γ ( t i ) x j , γ ( t i − ) x j ) ≥ sup P n P X i =1 d ( γ ( t i ) x j , γ ( t i − ) x j ) = ℓ d ( γx j ) (5)for every x j ∈ X . The second equality holds due to Proposition 2.1.Fix ( x, y, x j ) ∈ G × G × X . Observe that d i ( x, y ) ≥ d i − ( xx j , yx j ) for i ≥ i →∞ d i ( x, y ) = d ∞ ( x, y ) and lim i →∞ d i − ( xx j , yx j ) = d ∞ ( xx j , yx j ).Then d ∞ ( x, y ) ≥ d ∞ ( xx j , yx j ), what implies that d ∞ ( x, y ) ≥ d ∞ ( xσ, yσ ) for every σ ∈ ¯ S X .In order to prove that if ¯ S X is a subgroup, then d ∞ is invariant by the right actionof ¯ S X , it is enough to observe that d ∞ ( x, y ) ≥ d ∞ ( xσ, yσ ) ≥ d ∞ ( xσσ − , yσσ − ) = d ∞ ( x, y ). (cid:4) Now we prove that if d is complete and bounded above by a right invariantintrinsic metric, then d ∞ is intrinsic (Theorem 3.15). Lemma 3.13.
Let M be a set and suppose that d and ρ are metrics on M suchthat d ≤ ρ and τ d = τ ρ . If d is complete, then ρ is also complete.Proof Just notice that every Cauchy sequence in (
M, ρ ) is a Cauchy sequence in (
M, d ).Then the sequence converges in both metrics spaces because τ d = τ ρ . (cid:4) EQUENCE OF INDUCED HAUSDORFF METRICS ON LIE GROUPS 11
Lemma 3.14.
Let M be locally compact topological space and suppose that d i is asequence intrinsic metrics on M that converges pointwise to metric d ∞ . Supposethat there exist a complete intrinsic metric d l and a metric d h on M such that τ d l = τ d h = τ M and d l ≤ d i ≤ d h for every i . Then d ∞ is a complete intrinsicmetric.Proof First of all d ∞ is complete due to Lemma 3.13 and because d ∞ ≥ d l . Moreoverit is straightforward that τ d ∞ = τ M .Let ε > x, y ∈ M . We will prove that x and y admits an ε -midpoint z with respect to the metric d ∞ (see Proposition 2.3).Let z i be an ε/ x and y with respect to the metric d i (see Propo-sition 2.2). We claim that the sequence z i is contained in a compact subset. Infact (cid:12)(cid:12) d i ( x, z i ) − d i ( x, y ) (cid:12)(cid:12) ≤ ε d i ( x, z i ) ≤ d i ( x, y ) + ε d l ( x, z i ) ≤ d i ( x, z i ) ≤ d i ( x, y ) + ε ≤ d h ( x, y ) + ε . Then the sequence z i is bounded in ( M, d l ) and it is contained in a compact subsetdue to Theorem 2.6.Let z be an accumulation point of z i . We claim that z is an ε -midpoint withrespect to d ∞ . First of all, taking a subsequence z i coverging to z , we claim that d i ( x, z i ) converges to d ∞ ( x, z ) as i goes to infinity. In fact, for every ε >
0, thereexist N ∈ N such that | d ∞ ( x, z ) − d i ( x, z i ) | ≤ | d ∞ ( x, z ) − d i ( x, z ) + d i ( x, z ) − d i ( x, z i ) |≤ | d ∞ ( x, z ) − d i ( x, z ) | + | d h ( x, z ) − d h ( x, z i ) | < ε/ i ≥ N , and d i ( x, z i ) converges to d ∞ ( x, z ). Then for every ε >
0, thereexist a N ∈ N such that | d ∞ ( x, z ) − d ∞ ( x, y ) |≤ (cid:12)(cid:12) d ∞ ( x, z ) − d i ( x, z i ) | + | d i ( x, z i ) − d i ( x, y ) | + | d i ( x, y ) − d ∞ ( x, y ) (cid:12)(cid:12) < ε for every i ≥ N . The inequality | d ∞ ( y, z ) − d ∞ ( x, y ) | < ε is analogous. Then z is an ε -midpoint of x and y in ( M, d ∞ ) and d ∞ is intrinsicdue to Proposition 2.3. (cid:4) Theorem 3.15.
Let G be a Lie group and d : G × G → R be a complete metricon G such that τ d = τ G . Let ϕ : G × ( G, d ) → ( G, d ) be the product of G and X = { x , . . . , x k } ∋ e be a finite subset of G such that H X = { e } . Suppose thatthere exist an intrinsic right invariant metric d on G such that d ≤ d . Then thesequence of induced Hausdorff metrics converges to a complete and intrinsic metric d ∞ . Proof
Observe that τ d = τ d = τ G because d is C -Carnot-Carath´eodory-Finsler and d ≤ d i ≤ d . Notice that the sequence ( d i ) i ∈ N converges to a metric d ∞ due toTheorem 3.10 and that we are in the conditions of Lemma 3.14, with d h = d and d l = d . Therefore d ∞ is complete and intrinsic. (cid:4) Existence of dense semigroups S X Let G be a connected Lie group. In this section we prove the existence of a finitesubset X = { x , . . . , x k } ∋ e of G such that H X = { e } and ¯ S X = G .We begin with an example. Example 4.1.
Let G = R be the additive group of real numbers and consider X = {− , , √ } . We claim that S X is dense. Let q ∈ R and ε > . We will finda q ε ∈ S X such that | q − q ε | < ε .It is easy to see that there are a sequence p , p , . . . such that p i +1 − p i = − or p i +1 − p i = √ and a infinite number of p i ′ s are in [0 , . Therefore there exist p j and p k in S X , j < k , such that | p k − p j | < ε . Set p ε := p k − p j ∈ S X . In orderto fix ideas, suppose that p ε > . Then there exist an element p ∈ S X such that p < q and a l ∈ N such that q ε := p + l.p ε satisfies | q − q ε | < ε . The case p ε < isanalogous. Then S X is dense in R . We also have that S X − is dense in G due toProposition 3.8.This example is easily generalized to the case G = R n and X = {− , , √ } × . . . × {− , , √ } . (cid:4) Example 4.1 can be generalized to Lie groups in the following way:
Theorem 4.2.
Let G be a connected Lie group and g be its Lie algebra. Let V be avector subspace of g such that the Lie algebra generated by V is g . Let { v , . . . , v k } be a basis of V such that ( k X i =1 a i v i , a i ∈ [ − , , i = 1 , . . . , k ) is contained in a neighborhood A of ∈ g such that exp | A is a diffeomorphism overits image. Consider ˜ X = {− v , − v , . . . , − v k , , √ v , . . . , √ v k } . Denote x i = exp( − v i ) , y i = exp( √ v i ) and X = { e, x , . . . , x k , y , . . . , y k } . Then ¯ S X = G .Proof Let g ∈ H X . Then gX = X and e ∈ X implies that g ∈ X and g − ∈ X .If g = e , then g = exp( − v i ) or g = exp( √ v i ). In order to fix ideas, supposethat g = exp( − v i ). Then we have that g − = exp( v i ) ∈ X , which isn’t possiblebecause exp | A is a diffeomorphism over its image. The other case is analogous.Consequently H X = { e } .Let us prove that S X is dense in G . Example 4.1 states that S ˜ X ∩ R v i is densein R v i for every i = 1 , . . . , n . This implies that S X ∩ exp( R v i ) is dense in exp( R v i ).By Corollary 2.10, for every x ∈ G , there exist m ∈ N , t i , . . . , t i m ∈ R and EQUENCE OF INDUCED HAUSDORFF METRICS ON LIE GROUPS 13 v i , . . . , v i m ∈ X such that x = exp( t i m v i m ) . . . exp( t i v i ). But we know that forevery j = 1 , . . . , m , there exist a sequence of points in S X converging to exp( t i j v i j ).Therefore we have a sequence of points in S X converging to x and S X is dense in G . (cid:4) The case ¯ S X = G In this section we study the case where ¯ S X = G and d is complete. We prove thatthe sequence of induced Hausdorff metrics converges to a C -Carnot-Carath´eodory-Finsler metric d ∞ . Moreover we give a necessary and sufficient condition in orderto d ∞ be C -Finsler. In this case, an explicit formula for the Finsler metric isobtained. All these results are in Theorem 5.3. Lemma 5.1.
Let G be a connected Lie group endowed with a metric d such that τ d = τ G . Suppose that ˆ¯ d ( x, y ) = ∞ for some x, y ∈ G . Then there exist a v ∈ g such that sup σ ∈ G sup t =0 d (exp( tv ) σ, σ ) | t | = ∞ . Proof
If there exist a δ > d (exp( tv ) , e ) is finite for every v ∈ g and t ∈ ( − δ, δ ), then it is straightforward that ˆ¯ d ( x, y ) is finite for every x, y ∈ G due to theright invariance of ˆ¯ d and the connectedness of G . Then if ˆ¯ d ( x, y ) = ∞ for some x, y ∈ G , then there exist v ∈ g and t ∈ R such that ˆ¯ d (exp( tv ) , e ) = ∞ . It impliesthat for every C >
0, there exist a partition P of [0 , t ] such that n P X i =1 ¯ d (exp( t i v ) , exp( t i − v )) = n P X i =1 ¯ d (exp( t i − t i − ) v, e ) > Ct due to the right invariance of ¯ d . Then the average of¯ d (exp( t i − t i − ) v, e ) t i − t i − is greater than or equal to C . Therefore there exist t > d (exp( tv, e ) t > C. Thus sup σ ∈ G sup t =0 d (exp( tv ) σ, σ ) | t | = sup t =0 ¯ d (exp( tv ) , e ) t = ∞ . (cid:4) Lemma 5.2.
Let G be a Lie group, d be a metric on G satisfying τ d = τ G and v ∈ g . Then sup σ ∈ G lim sup t → d (exp( tv ) σ, σ ) | t | = sup σ ∈ G sup t =0 d (exp( tv ) σ, σ ) | t | . (6) Proof
The inequality ≤ follows directly from the definition of lim sup. Let us prove theinequality ≥ . If the left hand side of (6) is infinity, then there is nothing to prove.Suppose that it is equal to L ∈ R . It implies thatlim sup t → d (exp( tv ) σ, σ ) | t | ≤ L for every σ ∈ G . Then for every ε > σ ∈ G , there exist a δ > d (exp( tv ) σ, σ ) ≤ ( L + ε ) | t | whenever t ∈ ( − δ, δ ).Now fix σ ∈ G and t >
0. Define γ : [0 , t ] → G as γ ( s ) = exp( sv ) .σ . We can finda partition P = { t < t < . . . < t k = t } such that d (exp( t i v ) σ, exp( t i − v ) σ ) ≤ ( L + ε )( t i − t i − ) . (7)In fact, for every s ∈ [0 , t ], we choose δ s > d (exp( τ v ) σ, exp( sv ) σ ) ≤ ( L + ε ) | τ − s | . for every τ ∈ I s := [0 , t ] ∩ ( s − δ s , s + δ s ). If s ∈ (0 , t ), we can choose I s =( s − δ s , s + δ s ). From { I s } s ∈ [0 ,t ] , we can choose a finite subcover { I ˜ k } . From thisfinite subcover, we drop a I ˜ k , once a time, if there exist I ˜ k such that I ˜ k ⊂ I ˜ k .We end up with a family such that one element is not contained in another element.Denote this family by { [0 , δ ) , ( t − δ t , t + δ t ) , ( t − δ t , t + δ t ) , . . . , ( t − δ t , t ] } ,with t i − < t i for every i ( t i − = t i doesn’t happen). It is not difficult to see that0 < t − δ t < t − δ t < . . . < t − δ t and 0 + δ < t + δ t < t + δ t < . . . < t . Nowwe choose the “odd” points. t ∈ (0 , t ) is chosen in [0 , δ ) ∩ ( t − δ t , t + δ t ). t ∈ ( t , t ) is chosen in ( t − δ t , t + δ t ) ∩ ( t − δ t , t + δ t ) and so on. Then P = { t < t < . . . < t k = t } is such that (7) is satisfied.Therefore d (exp( tv ) σ, σ ) ≤ n P P i =1 d (exp( t i v ) σ, exp( t i − v ) σ ) ≤ ( L + ε ) | t | for every σ and t > σ ∈ G sup t> d (exp( tv ) σ, σ ) | t | ≤ L. If t <
0, thensup σ ∈ G sup t< d (exp( tv ) σ, σ ) | t | = sup σ ∈ G sup t> d (exp( t ( − v )) σ, σ ) | t |≤ sup σ ∈ G lim sup t → d (exp( t ( − v )) σ, σ ) | t | = sup σ ∈ G lim sup t → d (exp( tv ) σ, σ ) | t | = L, (8)what settles the proposition. (cid:4) In what follows we identify G × g with T G through the correspondence ( g, v ) ( g, dR g ( v )), where R g denotes the right translation by g on G . Theorem 5.3.
Let G be a connected Lie group endowed with a complete metric d such that τ d = τ G , ϕ : G × G → G be the product of G and X = { x , . . . x k } ∋ e bea finite subset of G such that H X = { e } and ¯ S X = G . EQUENCE OF INDUCED HAUSDORFF METRICS ON LIE GROUPS 15 (1) If d is bounded above by a right invariant intrinsic metric d , then the se-quence of induced Hausdorff metrics converges to a right invariant C -Carnot-Carath´eodory-Finsler metric d ∞ . In particular, if ˆ¯ d is a metric,then d ∞ is a right invariant C -Carnot-Carath´eodory-Finsler. (2) If d is bounded above by a right invariant C -Finsler metric d , then thesequence of induced Hausdorff metrics converges to a right invariant C -Finsler metric d ∞ . (3) Suppose that the sequence of induced Hausdorff metrics converges pointwiseto a metric d ∞ . Then d ∞ is C -Finsler iff ˜ F ( g, v ) := sup σ ∈ G lim sup t → d (exp( tv ) gσ, gσ ) | t | (9) is finite for every ( g, v ) ∈ T G , and in this case ˜ F is the Finsler metric on G .Proof (1) We know that the sequence of induced Hausdorff metrics converges to aintrinsic metric d ∞ due to Theorem 3.15. It is a right invariant due to Lemma 3.12.Therefore d ∞ is C -Carnot-Carath´eodory-Finsler due to Theorem 1.1.If ˆ¯ d is finite, then d = ˆ¯ d is a right invariant intrinsic bound for d and the resultfollows.(2) This is a particular case of item (1) and it is direct consequence of the secondparagraph of Remark 2.5.(3) First of all we prove that if ˜ F is finite, then d ∞ is C -Finsler.If ˜ F is finite, then ˆ¯ d is also finite due to Lemmas 5.1 and 5.2. Thus d ∞ is a rightinvariant C -Carnot-Carath´eodory-Finsler due to item (1).It is a direct consequence of the ball-box theorem that if d ∞ is a Carnot-Carath´eodory metric on a differentiable manifold M with respect to a propersmooth distribution D , then for every p ∈ M and v
6∈ D p we have thatlim t → d ∞ ( γ ( t ) , p ) | t | = ∞ , where γ : ( − ε, ε ) → M is a smooth path such that γ (0) = p and γ ′ (0) = v . Thesame conclusion holds for C -Carnot-Carath´eodory-Finsler metrics due to Remark2.5. Then d ∞ is C -Finsler whenever ˜ F is finite.Now we prove that if d ∞ is C -Finsler, then ˜ F is finite.If F : T G → R is the C -Finsler metric correspondent to d ∞ , then F ( g, v ) = lim t → d ∞ (exp( tv ) g, g ) | t | = sup σ ∈ G lim t → d ∞ (exp( tv ) gσ, gσ ) | t | ≥ ˜ F ( g, v ) , where the first equality is due to Theorem 3.7 of [8] and the second equality is dueto the right invariance of d ∞ . Therefore ˜ F is finite. Observe that this part of theproof also shows that ˜ F ≤ F if d ∞ is C -Finsler.Finally we show that if d ∞ is C -Finsler, then F ≤ ˜ F .Fix t > v ∈ g . Define γ : [0 , t ] → G by γ ( s ) = exp( sv ) and notice that if ξ ∈ G , then d (exp( tv ) ξ, ξ ) ≤ ℓ d X ( γξ ) = sup P n P X i =1 d X (exp( t i v ) ξ, exp( t i − v ) ξ )= sup P n P X i =1 max j d (exp( t i v ) ξx j , exp( t i − v ) ξx j )= sup P n P X i =1 max j d (exp(( t i − t i − ) v ) h j , h j ) ≤ sup P n P X i =1 sup σ ∈ G d (exp(( t i − t i − ) v ) σ, σ )where h j = exp( t i − v ) ξx j . Then d (exp( tv ) ξ, ξ ) ≤ sup P n P P i =1 ˜ F ( v )( t i − t i − ) = ˜ F ( v ) t, where the inequality is due to Lemma 5.2. If we iterate this process, replacing d by d , d , . . . , we get d i (exp( tv ) ξ, ξ ) ≤ ˜ F ( v ) t (10)for every i ∈ N , t > ξ ∈ G . But ˜ F ( v ) = ˜ F ( − v ) because˜ F ( v ) = sup σ ∈ G lim sup t → d (exp( tv ) σ, σ ) | t | = sup σ ∈ G lim sup t → d (exp( tv ) exp( − tv ) σ, exp( − tv ) σ ) | t | = ˜ F ( − v ) , (11)and d ∞ (exp( tv ) σ, σ ) ≤ ˜ F ( v ) | t | (12)holds for every t ∈ R and σ ∈ G due to (10). Therefore F ( v ) = lim t → d ∞ (exp( tv ) , e ) | t | ≤ ˜ F ( v ) , what settles the theorem. (cid:4) Further examples
Example 6.1.
Consider the additive group G = R with the finite metric d ( x, y ) = | arctan( x ) − arctan( y ) | . Let X = {− , , √ } . Then ¯ S X = R (see Example 4.1).Its maximum derivative is equal to one and it is not difficult to see that ˆ¯ d is ametric on G . Then d ∞ is an invariant C -Finsler metric, with F ( · ) equal to theEuclidean norm (See Theorem 5.3). Therefore d ∞ is the Euclidean metric.Now we prove that every d i of the sequence of induced Hausdorff metrics is afinite metric. First of all notice that ddt arctan t = 11 + t . It means that if < x < y , then d ( x, y ) = arctan y − arctan x ≤
11 + x ( y − x ) . EQUENCE OF INDUCED HAUSDORFF METRICS ON LIE GROUPS 17 because t arctan t has the concavity pointed downwards for t > . For < x < y ,we have that d ( x, y ) = sup P n P X i =1 max j d ( x + x j , y + x j ) ≤
11 + ( x − ( y − x ) . We can iterate this procedure and show that if i ∈ N and i < x < y , then d i ( x, y ) ≤
11 + ( x − i ) ( y − x ) . Therefore, for a fixed i ∈ N , we have that lim x →∞ d i ( i + 1 , x ) ≤ ∞ X j =1 d i ( i + j, i + j + 1) ≤ ∞ X j =1
11 + j < ∞ . The finiteness of the negative part of R follows analogously. Therefore d i is a finitemetric (and therefore it isn’t complete) for every i ∈ N despite d ∞ is complete, whatshows that the completeness of d is not necessary to make d ∞ intrinsic. Example 6.2.
Consider the additive group R endowed with the metric d ( x, y ) = | √ x − √ y | and let X = {− , , √ } . Suppose that the sequence of induced Haus-dorff metric converges to a metric. R doesn’t admit a C -Carnot-Carath´eodory-Finsler metric which isn’t a C -Finsler metric, because the only distribution thatcan generate a C -Carnot-Carath´eodory-Finsler metric is the whole tangent bun-dle. Therefore the sequence converges to a C -Finsler metric. But in this case itis straightforward to see that ˜ F in (9) is infinite (just analyze small neighborhoodsof the origin), what contradicts Theorem 5.3. Therefore the sequence of inducedHausdorff metrics doesn’t converge to a metric on R . Example 6.3.
Consider the flat torus T = S × S with the canonical groupoperation. We represent it as [0 , × [0 , endowed with the canonical metric withthe opposite sides identified and endowed with the quotient metric d . Consider X = T − Q , where Q is the equivalence class of the square (1 / , / × (1 / , / ⊂ [0 , × [0 , in T . It is not difficult to see that d X is the maximum metric whenrestricted to a small neighborhood of the identity element. Therefore d is locallyequal to the maximum metric. (Compare with Example 8.11 in [8] ). In this casewe have that d < d , a relationship that doesn’t happen in this work. This exampleshows that the sequence of induced Hausdorff metrics are not only about increasingsequences of metrics. Final remarks
Remark 7.1.
We don’t know if Theorem 5.3 holds if d isn’t complete. More ingeneral, we don’t know if the limit of an increasing sequence of intrinsic metrics isintrinsic. Remark 7.2.
From a Lie group G endowed with a metric satisfying mild condi-tions, we defined a discrete dynamical system d, d , . . . that converges to a rightinvariant C -Carnot-Carath´eodory-Finsler metric. This is the first example thatshows that a sequence of metrics defined from the Hausdorff distance can have aregularizing effect. Moreover, for a general compact subset X ⊂ G , the sequenceof induced Hausdorff metrics is not necessarily increasing, as Example 6.3 shows,what makes the dynamics more interesting. Due to these facts we think that it is worthwhile to study the properties of the sequence of induced Hausdorff metrics inother situations. References [1] D. Bao, S.-S. Chern, and Z. Shen,
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