aa r X i v : . [ m a t h . G R ] A p r ASYMPTOTIC CONES OF LIE GROUPS AND CONEEQUIVALENCES
YVES DE CORNULIER
Abstract.
We introduce cone bilipschitz equivalences between metric spaces.These are maps, more general than quasi-isometries, that induce a bilipschitzhomeomorphism between asymptotic cones. Non-trivial examples appear inthe context of Lie groups, and we thus prove that the study of asymptoticcones of connected Lie groups can be reduced to that of solvable Lie groups ofa special form. We also focus on asymptotic cones of nilpotent groups. Introduction
Let X = ( X, d ) be a metric space. The idea of defining an “asymptotic cone”for X , namely a limit when t → + ∞ for the family metric spaces t X = ( X, t d ),was brought in by Gromov [Gr1] in terms of Gromov-Hausdorff convergence. Thisdefinition was satisfactory for the purpose of groups with polynomial growth[Pa1, Bre], but to generalize the definition to arbitrary metric spaces, it wasnecessary to drop the hope of getting a limit in a reasonable topological sense,and consider ultralimits, which make use of the choice of an ultrafilter. FollowingVan der Dries and Wilkie [DW], the limit lim ω t X , formally defined in Section2, can be defined when ω is an unbounded ultrafilter on the set of positive realnumbers (by unbounded, we mean that ω does not contain any bounded subset).This is a metric space, called the asymptotic cone of X with respect to ω , anddenoted by Cone ω ( X, d ) or Cone ω ( X ) for short.Considerable progress in the study of asymptotic cones of groups was then madein Gromov’s seminal book [Gr2], in which the second chapter is entirely devoted toasymptotic cones. Since then, a vast literature appeared on the subject, includingthe papers [KL, Bri, TV, Dr, Ri, KSTT, DrS, Co, BM].The classification of groups in terms of their asymptotic cones is not as fine asthe quasi-isometry classification, but in some cases, for instance that of connectedLie groups, it looks more approachable. Here are a few facts relevant to thegeneral study of connected Lie groups up to quasi-isometry.Let ( C ) be the class of triangulable Lie groups, i.e. groups isomorphic to aclosed connected group of real upper triangular matrices.
Date : February 8, 2011.2000
Mathematics Subject Classification.
Primary 22E15; Secondary 20F65, 22E25.Supported by ANR project “QuantiT”
JC08 318197. • Every connected Lie group is quasi-isometric to a group in the class ( C )[Co, Lemma 6.7]; • open question: is it true that any two quasi-isometric groups in the class( C ), are isomorphic? In the nilpotent case, this is considered as a majoropen question in the field.In the study of the large-scale geometry of a group G in the class ( C ), afundamental role is played by the exponential radical R of G , which is defined bysaying that G/R is the largest nilpotent quotient of G .Define ( C ) as the class of groups G in ( C ) having a closed subgroup H (nec-essarily nilpotent) such that(1) G is the semidirect product R ⋊ H ;(2) the action of H on the Lie algebra of R is R -diagonalizable (in particular,[ H, H ] centralizes R ). Remark 1.1.
Let G be the (topological) unit component of G ( R ), where G isan algebraic R -subgroup of the upper triangular matrices and assume that G hasno nontrivial homomorphism to the additive one-dimensional group (so that theunipotent radical coincides with the exponential radical). Then G is in the class( C ).In general, and even if G is algebraic, the exact sequence 1 → R → G → G/R → H as above.Even when a splitting exists, Condition (2) is not satisfied in general. So the class( C ) appears as a class of groups, much smaller than the whole class ( C ), but inwhich the large-scale geometry is more likely to be understood. Our main resultis to associate, to every G in the class ( C ), a “nicer” group G ′ ∈ ( C ), of the samedimension and with the same exponential radical, such that the asymptotic conesof G and G ′ are the same. To state a precise result, we introduce the followingnew concept.We define a map between metric spaces X, Y to be a cone bilipschitz equivalence if it induces a bilipschitz homeomorphism at the level of asymptotic cones for all unbounded ultrafilters, and we say that X and Y are cone bilipschitz-equivalent ifthere exists such a map. (The precise definitions will be provided and developedin Section 2.) It is easy and standard that any quasi-isometry between metricspaces is a cone bilipschitz equivalence, but on the other hand there exist conebilipschitz-equivalent metric spaces that are not quasi-isometric (see the examplebefore Corollary 1.3). Theorem 1.2.
Let G be any connected Lie group. Then G is cone bilipschitzequivalent to a group G in the class ( C ) . More precisely, if G is triangulablewith exponential radical R , then there is a split exact sequence → R → G → G/R → , in which R embeds as the exponential radical of G . IE GROUPS AND CONE EQUIVALENCES 3 If g is a Lie algebra, denote by ( g i ) its descending central series ( g = g ; g i +1 = [ g , g i ]); we have [ g i , g j ] ⊂ g i + j for all i, j , so the bracket induces a bilinear operation[ g i / g i +1 , g j / g j +1 ] → g i + j / g i + j +1 . This defines a Lie algebra structure on g grad = M i ≥ g i / g i +1 , called the associated graded Lie algebra . The Lie algebra g is called gradable (or, more commonly but slightly ambiguously, graded ) if it is isomorphic to itsassociated graded Lie algebra. A simply connected nilpotent Lie group is calledgradable if its Lie algebra is gradable. Pansu proved in [Pa2] that any two quasi-isometric gradable Lie groups are actually isomorphic.The description of asymptotic cones of simply connected nilpotent Lie groups,due to Pansu and Breuillard [Pa1, Bre], can be understood with the help of thenotion of cone equivalences. We explain this in detail in Appendix A. In particularany simply connected nilpotent Lie group G is cone bilipschitz equivalent to itsassociated graded simply connected nilpotent Lie group G grad (on the other hand, G and G grad may be non-quasi-isometric, see Benoist-Shalom’s example in [Sh,Section 4.1]). This is explicit: if both groups are identified to their Lie algebrathrough the exponential map, the cone equivalence there is just the identity, theassociated graded Lie algebra having the same underlying subspace as the originalone. Moreover, for a suitable choice of metrics, the multiplicative constants in thedefinition of cone equivalence are equal to 1, so they induce isometries betweenthe two asymptotic cones. Corollary 1.3.
Any connected Lie group G is cone bilipschitz-equivalent to agroup in the class ( C ) for which moreover the nilpotent subgroup H (as in thedefinition of the class ( C ) ) is graded. Acknowledgments.
I thank Emmanuel Breuillard and Pierre de la Harpe foruseful comments.
Contents
1. Introduction 12. Cone maps 43. Illustration: the law in a metric group 84. Cone equivalences between Lie groups 11Appendix A. Asymptotic cones of nilpotent groups, after Pansu andBreuillard 14References 22
YVES DE CORNULIER Cone maps
In this section, (
X, d ) is a non-empty metric space and x is a given point in X .We denote, for x ∈ X , | x | = d ( x, x ). The letter t always denotes a variable realnumber, say ≥
1, and typically tending to + ∞ (e.g. f ( t ) ≪ t means f ( t ) /t → t → ∞ ).2.1. Asymptotic cones. By unbounded ultrafilter we mean an ultrafilter on theset of positive real numbers, supported at + ∞ , i.e. not containing any boundedsubset.The asymptotic cone Cone ω ( X, d ) is defined as the ultralimit with respect tothe unbounded ultrafilter ω of the family of pointed metric spaces (cid:18) X, t d, x (cid:19) for some x ∈ X . This means that Cone ω ( X, d ) is defined as follows: definePrecone(
X, d ) as the set of families ( x t ) t ≥ in X such that ( d ( x t , x ) /t ) is bounded(this set does not depend on x ). Endow it with the pseudo-distance d ω (( x t ) , ( y t )) = lim ω t d ( x t , y t ) , and obtain Cone ω ( X, d ) as the metric space obtained from the pseudo-metricspace (Precone(
X, d ) , d ω )by identifying points at distance zero. After these identifications, the constantsequence ( x ) does not depend on x ∈ X and appears as a natural base pointfor Cone ω ( X, d ). See Drutu’s article [Dr] for a more detailed construction and asurvey on asymptotic cones.
Example 2.1.
The metric space R with the Euclidean distance, is isometric toany of its asymptotic cones, through the map x ( tx ), the reciprocal map beinggiven by ( x t ) lim ω x t t .Taking asymptotic cones with respect to a given unbounded ultrafilter definesa functor from the category of non-empty metric spaces with large-scale Lipschitzmaps to the category of pointed metric spaces with Lipschitz maps preservingbase points.In the following, we extend this functor to a broader class of maps.2.2. Cone defined maps.
For real-valued functions, write u (cid:22) v if u ≤ Av + B for some constants A, B >
0. When applicable, u ≪ v means u ( g ) /v ( g ) → g → ∞ . Definition 2.2.
Let Y be another non-empty metric space, and also denote by | · | the distance to the given point of Y . A map f : X → Y is cone defined if forevery family ( x t ) in X , | x t | (cid:22) t implies | f ( x t ) | (cid:22) t , and moreover for any families( x t ) , ( x ′ t ) in X with | x t | , | x ′ t | (cid:22) t and d ( x t , x ′ t ) ≪ t , we have d ( f ( x t ) , f ( x ′ t )) ≪ t . IE GROUPS AND CONE EQUIVALENCES 5 (Note that this does not depend on the choices of the given points.) This isexactly the condition we need to define, for every unbounded ultrafilter ω , theinduced map Cone ω ( X ) → Cone ω ( Y ), by mapping the class of a sequence ( x n )to the class of the sequence ( f ( x n )). The latter map preserves base points of thecones. Example 2.3.
Let f : R → R be any function satisfyinglim | x |→∞ f ( x ) /x = 0 . Then the map x x + f ( x ) is cone defined and induces the identity map at thelevel of all asymptotic cones. Proposition 2.4.
Let
X, Y be metric spaces and f : X → Y a map. Then f iscone defined if and only if the following two conditions are satisfied • | f ( x ) | (cid:22) | x | ; • ( d ( x, y ) + 1) / ( | x | + | y | ) → implies d ( f ( x ) , f ( y )) / ( | x | + | y | ) → .Proof. The conditions are clearly sufficient. Conversely, if the first one fails, forsome sequence ( x i ) in X , we have | x i | → ∞ and | f ( x i ) | / | x i | → ∞ . If i ( t ) isthe largest i such that | x i | ≤ t , then the sequence ( x i ( t ) ) satisfies x i ( t ) (cid:22) t and f ( x i ( t ) ) (cid:14) t . The other part is similar. (cid:3) Proposition 2.5.
Let
X, Y be metric spaces and f : X → Y a cone definedmap. Then, for every unbounded ultrafilter ω , the induced map ˜ f : Cone ω ( X ) → Cone ω ( Y ) is continuous.Proof. If x is the base point of the cone, it follows from the first condition ofProposition 2.4 that d ( ˜ f ( x ) , ˜ f ( x )) ≤ Cd ( x, x ) for some constant C >
0, for all x ∈ X , so ˜ f is continuous at x .Suppose x ∈ Cone ω ( X ) − { x } and let us check that ˜ f is continuous at x .Write x = ( x t ) and let y = ( y t ). For some function u tending to 0 at 0, wehave d ( f ( x t ) , f ( y t )) / ( | x t | + | y t | ) ≤ u (( d ( x t , y t ) + 1) / ( | x t | + | y t | )) for all n . We cansuppose that u ≤ C and is continuous. We have d ( f ( x t ) , f ( y t )) t ≤ | x t | + | y t | t u (cid:18) d ( x t , y t ) + 1 | x t | + | y t | (cid:19) ;taking the limit with respect to ω , we obtain d ( ˜ f ( x ) , ˜ f ( y )) ≤ ( | x | + | y | ) u (cid:18) d ( x, y ) | x | + | y | (cid:19) , which tends to zero when y tends to x . (cid:3) Definition 2.6.
Two cone defined maps f , f : X → Y are cone equivalent ifthe induced maps Cone ω ( X ) → Cone ω ( Y ) are equal for all ω . If f is a constantmap, we then say that f is cone null . YVES DE CORNULIER
Proposition 2.7.
Let f , f be cone defined maps X → Y . Then f and f arecone equivalent if and only if for some given point x ∈ X and for some sublinearfunction q , we have, for all x ∈ Xd ( f ( x ) , f ( x )) ≤ q ( | x | ) . Proof.
The proof is very similar to the previous one. Suppose the condition issatisfied. Let ( x t ) be a linearly bounded family in X . Then d ( f ( x t ) , f ( x t )) /t ≤ q ( | x t | )) /t → t → + ∞ , so ( f ( x t )) and ( f ( x t )) coincide in any asymptoticcone of Y .Conversely, if the condition is not satisfied, there exists a sequence ( ξ i ) tendingto infinity in X and ε > d ( f ( ξ i ) , f ( ξ i )) ≥ ε | ξ i | , ∀ i. For every t , define x t = ξ i , where i is chosen maximal so that | ξ i | < t + 1 (this isvalid as ( | ξ i | ) i goes to infinity). Let I be the set of t for which x t = x t − . Thiscondition means that x t = ξ i , where t ≤ | ξ i | < n + 1; as ( | ξ i | ) i is unbounded, I isunbounded. Let ω be a unbounded ultrafilter containing I . Then for t ∈ I , wehave d ( f ( x t ) , f ( x t )) ≥ ε | x t | ≥ εt, so lim ω d ( f ( x t ) , f ( x t )) /t > (cid:3) Cone Lipschitz maps.Definition 2.8.
A cone defined map f : X → Y is called a cone C -Lipschitz map if the induced map Cone ω ( X ) → Cone ω ( Y ) is C -Lipschitz for all unboundedultrafilters ω .In particular, such a map naturally induces a C -Lipschitz map Cone ω ( X ) → Cone ω ( Y ). Say that f is a cone Lipschitz map if it is cone C -Lipschitz for some C ∈ [0 , + ∞ [. Thus Cone ω is a functor from the category of non-empty metricspaces with cone Lipschitz maps (modulo cone equivalence) to the category ofpointed metric spaces with Lipschitz maps preserving base points. Proposition 2.9.
Let f be a map from X to Y . Then f is a cone C -Lipschitzmap if and only if for some sublinear function q , we have, for all x, y ∈ Xd ( f ( x ) , f ( y )) ≤ Cd ( x, y ) + q ( | x | + | y | ) . Proof.
Suppose that the condition is satisfied. Let ( x t ), ( y t ) be linearly boundedsequences in X . Then d ( f ( x t ) , f ( y t )) /t − Cd ( x t , y t ) /t ≤ q ( | x t | + | y t | )) /t, which tends to zero. So lim ω d ( f ( x t ) , f ( y t )) /t − Cd ( x t , y t ) /t ≤ ω .Conversely, if the condition is not satisfied, there exists sequences ( ξ i ), ( υ i )with ( | ξ i | + | υ i | ) tending to infinity, and ε > d ( f ( ξ i ) , f ( υ i )) − Cd ( ξ i , υ i ) ≥ ε ( | ξ i | + | υ i | ) . IE GROUPS AND CONE EQUIVALENCES 7
For every t , define x t = ξ i and y t = υ i , where i = i ( t ) is chosen maximal so that | ξ i | + | υ i | < t + 1(this is valid as ( | ξ | + | υ i | ) i tends to infinity). It follows that ( x t ) and ( y t ) arelinearly bounded. Let I be the set of t for which i ( t ) = i ( t − i (which can be chosen as i = i ( t )), we have t ≤ | ξ i | + | υ i | 00 1 y = y x 00 1 00 0 1 is cone defined but not cone Lipschitz.2.4. Cone bilipschitz maps.Definition 2.11. A cone defined map f : X → Y is cone M - expansive if, forevery unbounded ultrafilter ω , the induced map ˜ f : Cone ω ( X ) → Cone ω ( Y ) is M -expansive, i.e. satisfies d ( ˜ f ( x ) , ˜ f ( y )) ≥ M d ( x, y )for all x, y . The map f is cone expansive if it is M -cone expansive for some M > 0. The map f is cone ( C, M )- bilipschitz if it is cone C -Lipschitz andcone M -expansive, and cone bilipschitz if this holds for some positive reals M, C .The map f is cone surjective if ˜ f is surjective for every unbounded ultrafilter,and is called a cone bilipschitz equivalence if it is both cone surjective and conebilipschitz.The following proposition is proved in the same lines as the previous ones. Proposition 2.12. Let f be a cone map from X to Y . Then f is a cone M -expansive map if and only if for some sublinear function κ , we have, for all x, y ∈ X d ( f ( x ) , f ( y )) ≥ M d ( x, y ) − κ ( | x | + | y | )) . (cid:3) Proposition 2.13. Let f be a cone bilipschitz map from X to Y and y a basepoint in Y . We have the equivalences (i) f is cone surjective; YVES DE CORNULIER (ii) for some sublinear function c , we have, for all y ∈ Y , the inequality d ( y, f ( X )) ≤ c ( | y | ) ; (iii) f is a cone bilipschitz equivalence.Proof. Clearly (iii) implies (i).Suppose (ii) and let us prove (iii). Set c ′ = c + 1. For any y ∈ Y , choose x = g ( y ) with d ( f ( x ) , y ) ≤ c ′ ( | y | ). We claim that g is a cone map. Indeed, M d ( g ( y ) , g ( y )) ≤ d ( f ◦ g ( y ) , f ◦ g ( y )) + κ ( | g ( y ) | + | g ( y ) | ) ≤ d ( y , y ) + c ′ ( | y | ) + c ′ ( | y | ) + κ ( | g ( y ) | + | g ( y ) | ) . For some constant m , κ ( t ) ≤ M t/ t . Specifying to y = y and y = y ,we get, for all y ∈ YM | g ( y ) | ≤ M | g ( y ) | + | y | + c ′ ( | y | ) + c ′ (0) + M | g ( y ) | / M | g ( y ) | / , which gives a linear control of | g ( y ) | by | y | . Thus for some suitable sublinearfunction κ ′ , we have, for all y , y M d ( g ( y ) , g ( y )) ≤ d ( f ◦ g ( y ) , f ◦ g ( y )) + κ ′ ( | y | + | y | ) , so that g is cone Lipschitz.By construction, f ◦ g and Id Y are cone equivalent. It follows that f ◦ g ◦ f and f are cone equivalent. Using that f is cone bilipschitz, it follows that g ◦ f and Id X are cone equivalent.Finally suppose (ii) does not hold and let us prove the negation of (i). Sothere exist υ i in Y and ε > d ( υ i , f ( X )) ≥ ε | y i | . We can find, by theusual argument, a sequence ( y n ) in Y with | y n | ≤ n + 1 for all n and an infinitesubset I of integers so that for n ∈ I , y n = υ i for some i and | υ i | ≥ n . Pick aunbounded ultrafilter on the integers containing I . Suppose by contradiction thatthe sequence ( y t ) is image of a sequence ( x t ) for the induced map Cone ω ( X ) → Cone ω ( Y ). Then lim ω d ( f ( x t ) , y t ) /t = 0. But for t ∈ I , we have d ( f ( x t ) , y t ) = d ( f ( x t ) , υ i ) ≥ εt and we get a contradiction. (cid:3) Remark 2.14. The cone Lipschitz (and even large scale Lipschitz map) f : R → R mapping x to x / is surjective, however the induced map on the conesis constant, so f is not cone surjective.3. Illustration: the law in a metric group Let G be a metric group, i.e. endowed with a left-invariant pseudodistance d .As usual, we write | g | = d (1 , g ). The precone Precone( G ) (see Paragraph 2.1)carries a natural group law, given by ( g n )( h n ) = ( g n h n ). The left multiplicationsare isometries of the pseudometric space Precone( G ).Fix an unbounded ultrafilter ω , and let Sublin ω ( G ) (resp. Sublin( G )) denotethe set of families ( g t ) ∈ Precone( G ) such that | g t | /t → ω (resp. IE GROUPS AND CONE EQUIVALENCES 9 | g t | /t → t → + ∞ ). This is obviously a subgroup of Precone( G ). Thenby definition Cone ω ( G ) = Precone( G ) / Sublin ω ( G ) . In particular, Cone ω ( G ) is naturally homogeneous under the action of Precone( G )by isometries. In general, Sublin ω ( G ) is not normal in Precone( G ), and thusCone ω ( G ) has no natural group structure. The following proposition gives asimple criterion. Endow G × G with the ℓ ∞ product metric (or any equivalentmetric). Proposition 3.1. Let G be a metric group. We have the equivalences (i) the group law η : G × G → G is cone defined; (ii) the group inverse map τ : G → G is cone defined; (iii) Sublin ω ( G ) is normal in Precone( G ) for every ω ; (iii’) Sublin( G ) is normal in Precone( G ) ; (iv) for every e > there exists ε ( e ) > such that for all g, h ∈ G such that | h | ≥ e | g | we have | g − hg | + 1 ≥ ε ( e ) | g | . The proof will be given at the end of this section. Lemma 3.2. Let G be a metric group with cone defined law, and H is anothermetric group with a homomorphic quasi-isometric embedding H → G . Then H also has a cone defined law. In particular, this applies if G is endowed with aword metric with respect to a compact generating subset, and H → G is a properhomomorphism with cocompact image.Proof. This immediate, e.g. from Criterion (iii’) of Proposition 3.1. (cid:3) Lemma 3.3. If G is a locally compact group with a word metric with respect toa compact generating subset and H is a quotient of G , if G has cone defined law,then so does H .Proof. We also use Criterion (iii’). We can suppose for convenience that G isendowed with the word length with respect to a compact generating set S , and H is endowed with the word length with respect to the image of S . Suppose that( g t ) ∈ Precone( H ) and ( h t ) ∈ Sublin( H ). We can lift then to elements ˜ g t and ˜ h t of G with | ˜ g t | = | g t | and | ˜ h t | = | h t | . So by (iii’) of Proposition 3.1, | ˜ g t ˜ h t ˜ g − t | ≪ t ,and therefore | g t h t g − t | ≪ t . (cid:3) Example 3.4. Here are some examples of metric groups for which the law iscone defined.(1) Arbitrary abelian metric groups are trivial examples, since Precone( G )is then abelian as well and being normal becomes an empty condition inCriterion (iii’) of Proposition 3.1.(2) Less trivial examples are nilpotent locally compact groups with the wordmetric with respect to a compact generating subset, see Corollary A.2. (3) In particular, using Lemma 3.2, it follows from Corollary A.2 that ifa finitely generated group, endowed with a word metric, is finite-by-nilpotent (i.e. has a finite normal subgroup with nilpotent quotient), thenit satisfies the conditions of Proposition 3.1. I do not know if there areany other examples among finitely generated groups (with a word metric). Example 3.5. “Most” metric groups fail to have a cone defined law, as illustratedby the following. In each of these examples, ( x n ) ∈ Precone( G ) and y a constant(hence sublinear) sequence, such that | x n y x − n | ≃ n , so clearly Condition (iii) ofProposition 3.1 (for instance) fails. • G = h a, b i , the nonabelian free group with the word metric, x n = a n , y = b ; • G = h a, b : a = b = 1 i , the infinite dihedral group with the word metric, x n = ( ab ) n , y = a , x n y x − n = ( ab ) n a ; here G has an abelian subgroupof index two, so we see that the converse of Lemma 3.2 dramatically fails; • G the Heisenberg group endowed with the metric induced by inclusion into SL ( R ) with its word metric, x n = n 00 1 00 0 1 ; y = ; x n y x − n = n . In particular we see that Example 3.4(2) is specific to the word metricand fails for general nilpotent metric groups. Proof of Proposition 3.1. • Suppose (iii) and let us prove (ii). Denote by ≡ ω the equality in Cone ω ( G ).Suppose that ( g t ) ≡ ω ( g ′ t ), i.e. g ′ t = g t s t with ( s t ) ω -sublinear. We have d ( g − t , g ′ t − ) = | g t s − t g − t | , and ( g t s − t g − t ) belongs to Sublin ω ( G ) since thelatter is normal. Thus ( g − t ) ≡ ω ( g ′ t − ) and therefore τ is cone defined. • Suppose (ii) and let us prove (i). Suppose that ( g t ) ≡ ω ( g ′ t ) and ( h t ) ≡ ω ( h ′ t ). Write g ′ t = g t s t . We have d ( g t h t , g ′ t h ′ t ) = | h − t s t h ′ t | ≤ | h − t s t h t | + | h − t h ′ t | . Since τ is cone defined and ( h − t s − t ) ≡ ω ( h − t ), we have ( s t h t ) ≡ ω ( h t ),hence by left multiplication (which is isometric) ( h − t s t h t ) ≡ ω (1), i.e.( h − t s t h t ) is ω -sublinear. Since this holds for any ω , the group law η : G × G → G is cone defined; • Suppose (i) and let us prove (iii). Suppose that ( g t ) ∈ Precone( G ) and( s t ) ∈ Sublin ω ( G ). Then, since (1) ≡ ω ( s t ) and η is cone defined, we have( η (1 , g t )) ≡ ω ( η ( s t , g t )), that is, | g − t s t g t | is ω -sublinear. So Sublin ω ( G ) isnormal for any ω . • Suppose (iii) and let us check (iii’). If Sublin ω ( G ) is normal for any ω ,then so is Sublin( G ) = T ω Sublin ω ( G ). IE GROUPS AND CONE EQUIVALENCES 11 • Suppose that (iv) fails and let us show that (iii’) fails. There exists e > g n ) , ( h n ) in G such that | h n | ≥ e | g n | and ( | g − n h n g n | +1) / | g n | → 0. Note that this forces | g n | → ∞ . Since | g − n h n g n | ≥ | h n | − | g n | , we see that | h n | ≤ | g n | for n ≥ n large enough. If r ≥ r = | g n | ,set n ( r ) = sup { n ≥ n : | g n | ≤ r } , u r = g n ( r ) and v r = h n ( r ) . For all r ≥ r we have | u r | ≤ r , e | u r | ≤ | v r | ≤ r , and the family ( u − r v r u r ) is sublinearand both families ( u r ) and ( v r ) are at most linear. Set N = { n : ∀ m >n, | g m | > | g n |} . If r belongs to the unbounded set K = {| g n | : n ∈ N } wehave | u r | = r for all r ∈ K and therefore the family ( v r ) is not sublinear. • Suppose that (iv) holds and let us prove (iii). Fix ω . Suppose that( g t ) ∈ Precone( G ) and ( s t ) ∈ Sublin ω ( G ). If ( g t s t g − t ) / ∈ Sublin ω ( G ), thenfor some I ∈ ω , some e > t ∈ I we have | g t s t g − t | ≥ e | g t | . So | s t | + 1 = | g − t ( g t s t g − t ) g t | + 1 ≥ ε ( e ) | g t | , ∀ t ∈ I, and thus | g t | ∈ Sublin ω ( G ), and in turn( g t s t g − t ) ∈ Sublin ω ( G ) , a contradiction. (cid:3) Cone equivalences between Lie groups Reduction to the split case with semisimple action. Let G be a trian-gulable group. We shamelessly use the identification between the Lie algebra andthe Lie group through the exponential map. Precisely, G has two laws, the groupmultiplication, and the addition g + h , which formally denotes exp(log( x )+log( y )),using that the exponential function is duly a homeomorphism for a triangulablegroup [Di]. Let R be the exponential radical of G , H a Cartan subgroup [Bo,Chap. 7, pp.19-20]. This means that H is nilpotent and RH = G . Set W = H ∩ R ,and let V be a complement subspace of W in H (viewed as Lie algebras).We denote the word length (with respect to some compact generating subset)in a group by | · | , by default the group is G , otherwise we make it explicit, e.g. | · | H denotes word length in H . If h ∈ H , it can be written in a unique way as rv with r ∈ W and v ∈ V , we write r = δ ( h ) and v = [ h ].The following lemma is obtained in the proof of [Co, Theorem 5.1]. Lemma 4.1. For v ∈ V we have | v | ≃ | v | G/R . (cid:3) Lemma 4.2. For v, v ′ ∈ V we have | δ ( v − v ′ ) | (cid:22) log(1 + | v | G/R ) + log(1 + | v ′ | G/R ) . Proof. As δ ( v − v ′ ) = v − v ′ [ v − v ′ ] − , | δ ( v − v ′ ) | H ≤ | v | H + | v ′ | H + | [ v − v ′ ] | H , so by Lemma 4.1 | δ ( v − v ′ ) | H (cid:22) | v | G/R + | v ′ | G/R + | v − v ′ | G/R (cid:22) | v | G/R + | v ′ | G/R . Since δ ( v − v ′ ) belongs to the exponential radical, | δ ( v − v ′ ) | (cid:22) log(1 + | δ ( v − v ′ ) | H )and we get the conclusion. (cid:3) Lemma 4.3. For r ∈ R, v ∈ V , | rv | ≃ | r | + | v | G/R . Proof. Clearly | v | G/R (cid:22) | v | , so we obtain the inequality (cid:22) .From the projection G → G/R , we get | v | G/R (cid:22) | rv | . On the other hand | r | ≤ | rv | + | v | . Now | v | (cid:22) | v | G/R by Lemma 4.1. So | rv | (cid:23) | r | + | v | G/R . (cid:3) It is wary here to distinguish the Lie group and its Lie algebra. Thus denoteby r the Lie algebra of R and consider the action of H on r , given by restrictionof the adjoint representation. It is given by a homomorphism α from H to theautomorphism group of the Lie algebra r , which is an algebraic group. Thisaction is triangulable, and we can write in a natural way, for every h ∈ H , α ( h ) = β ( h ) u ( h ) where β is the diagonal part and u ( h ) is the unipotent part;both are Lie algebra automorphisms of r (view this by taking the Zariski closureof α ( H ), which can be written in the form DU with D a maximal split torus and U the unipotent radical), and β defines a continuous action of H on r . Note thatthis action is trivial on [ H, H ] and hence on W .As H is nilpotent, we can decompose r into characteristic subspaces for the α -action; this way we see in particular that u is an action as well (this stronglyrelies on the fact that H is nilpotent and connected). In such a decomposition,we see that the matrix entries of u ( h ), for h ∈ H , are polynomially bounded interms of | h | (more precisely, ≤ C | h | k , where C is a constant (depending only on G and the choice once and for all of a basis adapted to the characteristic subspaces)and k + 1 is the dimension of r .Now define A ( h ), B ( h ), resp. U ( h ), as the automorphism of R whose tangentmap is α ( h ), β ( h ), resp. u ( h ). Note that α ( h ) is the left conjugation by h in R . Then B provides a new action of H on R . We consider the group R ⋊ H/W defined by the action B . We write it by abuse of notation R ⋊ V , identifying V ,as a group, to H/W .Consider the map ψ : G = RV → R ⋊ Vrv rv . IE GROUPS AND CONE EQUIVALENCES 13 In general ψ is not a quasi-isometry (and is even not a coarse map, i.e. thereexists a sequence of pairs of points at bounded distance mapped to points atdistance tending to infinity). Theorem 4.4. The map ψ is a cone bilipschitz equivalence. Lemma 4.5. If a, b ≥ and c ≥ , and if | log( a ) − log( b ) | ≤ log( c ) , then | log(1 + a ) − log(1 + b ) | ≤ log( c ) . (cid:3) Proof of Theorem 4.4. On both groups, the word length is equivalent by Lemma4.3 to L ( rv ) = ℓ ( r ) + | v | G/R , where ℓ is a length (subadditive and symmetric)on R with ℓ ( r ) ≃ log(1 + k r k ), and we consider the corresponding left-invariant“distances” d and d .We consider two group laws at the same time on the Cartesian product R × V ;in order not to introduce tedious notation, we go on writing both group laws bythe empty symbol; on the other hand to avoid ambiguity, we write the length L as L = L = L (and ℓ = ℓ = ℓ ) and when we compute a multiplication insidethe symbols L () or d ( , ), we mean the multiplication inside G , while in L and d we mean the new multiplication from R ⋊ V .We have, for r, r ′ ∈ R ; v, v ′ ∈ Vd ( rv, r ′ v ′ ) = L ( v − r − r ′ v ′ ) = L ( A ( v − )( r − r ′ ) v − v ′ )We have d ( rv, r ′ v ′ ) = L ( A ( v − )( r − r ′ ) δ ( v − v ′ )[ v − v ′ ])= ℓ ( A ( v − )( r − r ′ ) δ ( v − v ′ )) + | v − v ′ | G/R and d ( rv, r ′ v ′ ) = ℓ ( B ( v − )( r − r ′ )) + | v − v ′ | G/R (no δ -term for d because V is a subgroup for the second law).So d ( rv, r ′ v ′ ) − d ( rv, r ′ v ′ )= ℓ ( A ( v − )( r − r ′ ) δ ( v − v ′ )) − ℓ ( B ( v − )( r − r ′ )) . Write for short ρ = B ( v − )( r − r ′ ) (which defines the same element for the twogroup laws), so d ( rv, r ′ v ′ ) − d ( rv, r ′ v ′ ) = ℓ ( U ( v − )( ρ ) δ ( v − v ′ )) − ℓ ( ρ ) . We then get | d ( rv, r ′ v ′ ) − d ( rv, r ′ v ′ ) | = | ℓ ( U ( v − )( ρ )) − ℓ ( ρ ) | + ℓ ( δ ( v − v ′ )) . (We see that we can write ℓ because this does no longer depends on the choice ofone of the two laws.) These are not necessarily distances but are equivalent to the left-invariant word or Rie-mannian distances. Let us now work on the manifold R , endowed with a Riemannian length λ . Asthe tangent map of U ( v − ) is C | v | k -bilipschitz (i.e. both it and its inverse are C | v | k -Lipschitz), the same is true for U ( v − ) itself. In particular,1 C | v | k λ ( ρ ) ≤ λ ( U ( v − ) ρ ) ≤ C (1 + | v | ) k λ ( ρ ) . This means that | log( λ ( ρ )) − log( U ( v − ) ρ )) | ≤ log( C ) + k log(1 + | v | ) . By Lemma 4.5 (we can pick C ≥ | log(1 + λ ( ρ )) − log(1 + U ( v − ) ρ )) | ≤ log( C ) + k log(1 + | v | ) , thus | ℓ ( ρ ) − ℓ ( U ( v − ) ρ )) | (cid:22) log(1 + | v | );moreover by Lemma 4.2 δ ( v − v ′ ) (cid:22) log(1 + | v | ) + log(1 + | v ′ | ) . Accordingly, | d ( rv, r ′ v ′ ) − d ( rv, r ′ v ′ ) | (cid:22) log(1 + | v | ) + log(1 + | v ′ | ) ≪ | rv | + | r ′ v ′ | . Thus the map ψ is a cone equivalence (with constants one). This means that thecones defined from d and d are isometric; however note that these are equivalentto metrics but are not necessarily metrics (they are maybe not subadditive) sothe statement obtained for the usual cones (defined with genuine metrics) arethat they are bilipschitz. (cid:3) Appendix A. Asymptotic cones of nilpotent groups, after Pansuand Breuillard Fix a d -dimensional real s -nilpotent Lie algebra g . It is endowed with a groupstructure denoted by no sign or by a dot ( · ), defined by the Baker-Campbell-Hausdorff formula. We sometimes write G = ( g , · ), but the underlying sets arethe same.For all i , denote by v i a complement subspace of g i +1 in g i . In particular,(A.1) g = g = s M i =1 v i . Define, for any t ∈ R , the endomorphism δ t of the graded vector space g = L v i whose restriction to v i is the scalar multiplication by t i . Observe that t δ t isa semigroup action of ( R , · ). IE GROUPS AND CONE EQUIVALENCES 15 A.1. The cone in the graded case. We assume in this paragraph that ( v i ) isa grading of the Lie algebra g , i.e.[ v i , v j ] ⊂ v i + j ∀ i, j. It follows that for all t , δ t is a Lie algebra endomorphism, and therefore is alsoan endomorphism of G .Consider the subspace v of g (endowed with some norm), viewed as the tangentspace of G at 1, and translate it by left multiplication, providing for any g ∈ G a subspace D g of the tangent space T g along with a norm, in a way which iscompatible with left multiplication. The Carnot-Carath´eodory distance between x and y is the following “sub-Finslerian” distance d CC ( x, y ) = inf γ L ( γ ) , where γ ranges over regular smooth paths everywhere tangent to D and L ( γ )is the length of γ , computed by integration, using the norm on each D g . Thisdistance is finite (i.e. every two points can be joined by such a path), defines theusual topology of G [Be, Theorem 1]; moreover it is equivalent to the word length(equivalent here refers to large scale equivalence).It is immediate from the definition of d CC that d CC ( δ t ( x ) , δ t ( y )) = td CC ( x, y ) ∀ x, y ∈ G ;in particular, for any unbounded ultrafilter ω , the map j : ( G, d CC ) → Cone ω ( G, d CC )) g ( δ t ( g ))is an isometric embedding; moreover since balls in ( G, d CC ) are compact, it followsthat this is a surjective isometry, whose inverse is given by ( g t ) lim ω δ /t ( g t ).This gives in particular the following statement. Proposition A.1. Let G be any gradable simply connected nilpotent Lie group.Then (i) any asymptotic cone of G is naturally bilipschitz to ( G, d CC ) (with con-stants independent of ω ); (ii) the group law η : G × G → G is cone defined; (iii) under the identification of G with its cone, we have ˜ η = η . Corollary A.2. Let G be an arbitrary nilpotent compactly generated locally com-pact group (e.g. discrete, or connected). Then the law G × G → G is cone defined.Proof. If G is a free s -nilpotent group, this follows from Proposition A.1. ByLemma 3.3, to have cone defined law is stable under taking quotients, so if G isan arbitrary simply connected nilpotent Lie group then G has cone defined law.If G is an arbitrary nilpotent compactly generated locally compact group, then G has a unique maximal compact subgroup K ; this is a characteristic subgroup and the quotient H = G/K is a torsion-free compactly generated nilpotent Liegroup (see [GuKR, p. 104]), i.e. H is a simply connected nilpotent Lie groupand the discrete group H/H is a finitely generated torsion-free nilpotent group.By a result essentially due to Malcev (see [Wa] for the general case, when H isnot necessarily discrete) H embeds as a closed cocompact subgroup in a simplyconnected nilpotent Lie group ˜ H , which has cone defined law by the precedingcase. By Lemma 3.2, it follows that H and then G has cone defined law. (cid:3) Proof of Proposition A.1. (i) if | · | is a word length on G , then the identity map ( G, | · | ) → ( G, d CC ) isa quasi-isometry, so the induced mapCone ω ( G, | · | ) → Cone ω ( G, d CC ) = ( G, d CC )is bilipschitz, with constants depending only on the quasi-isometry con-stants, not on ω .(ii) We have to check that if ( g t ) = ( g ′ t ) and ( h t ) = ( h ′ t ) in Cone ω ( G ), then( g t h t ) = ( g ′ t h ′ t ) as well. Indeed, δ /t (( g t h t ) − g ′ t h ′ t ) = δ /t ( h t ) − δ /t ( g t ) − δ /t ( g ′ t ) δ /t ( h ′ t );since δ /t ( h t ) − and δ /t ( h ′ t ) are bounded (hence convergent for ω ), δ /t ( g t ) − δ /t ( g ′ t ) tends to zero and by compactness, the product tendsto zero by continuity of the law in G , i.e. δ /t (( g t h t ) − g ′ t h ′ t ) → 0, whichmeans that d CC (1 , g t h t ) − g ′ t h ′ t )) = td CC (1 , δ /t (( g t h t ) − g ′ t h ′ t )) ≪ t, using that d CC defines the usual topology on G .(iii) Since we know the law is well defined, it is enough to compute in on G :the new product of g and h is by definition j − ( j ( g ) j ( h )) = j − (( δ t ( g ))( δ t ( h )))= j − (( δ t ( g ) δ t ( h ))) = j − (( δ t ( gh ))) = gh. (cid:3) Remark A.3. If G is a graded simply connected nilpotent Lie group, its grouplaw is not, in general, cone Lipschitz, although it is cone defined by PropositionA.1(i). By Proposition A.1(iii), this amounts to prove that in ( G, d CC ), the grouplaw is not Lipschitz. Let us check it when G is the Heisenberg group, the argumentprobably extending to an arbitrary nonabelian G . Set x n = n 00 1 00 0 1 ; y = ; x − n y x n = − n . Defining R x : G → G by R x ( g ) = gx , we have d CC ( R x n (1) , R x n ( y )) = d CC (1 , x − n y x n ) ≃ √ n ; IE GROUPS AND CONE EQUIVALENCES 17 so the Lipschitz constant of R x n is (cid:23) √ n and since each R x n is the restrictionof the group law to a certain subset of G × G , the group law itself cannot beLipschitz.A.2. The asymptotic cone for general nilpotent Lie groups. Now we con-sider the graded Lie algebra associated to the decomposition g = L v i . The newbracket is given by [ x, y ] ∞ = X i,j [ x i , y j ] i + j , and we denote by ⊠ the corresponding group law, provided by the Baker-Campbell-Hausdorff formula [Hau]. If x = P i ≥ x i in this decomposition, define σ ( x ) = X i ≥ k x i k /i . We call the groups G = ( g , · ) and G ∞ = ( g , ⊠ ).Let d be any left-invariant pseudometric on G , quasi-isometric to the worddistance with respect to some compact generating subset S , i.e. | · | S (cid:22) | · | (cid:22) | · | S (see Paragraph 2.2).Given an unbounded ultrafilter ω on the positive real numbers, for any g, h ∈ g ,define the limit d ω ( g, h ) = lim t → ω d ( δ t ( g ) , δ t ( h )) t . Theorem A.4. The distance d ω is a continuous, ⊠ -left-invariant distance on G .Moreover, the identity map g → g is a cone 1-bilipschitz equivalence ( G ∞ , d ω ) → ( G, d ) , thus inducing a bijective isometry ( G ∞ , d ω ) → Cone ω ( G, d ) g ( δ t ( g )) . Besides, if η is the group law in G , then η and ⊠ are cone defined on ( G, d ) andare cone equivalent; in particular the above isometry G ∞ → Cone ω ( G ) is a groupisomorphism. Before proving Theorem A.4, let us give a corollary. Following Breuillard [Bre],the metric d is asymptotically geodesic if for every ε there exists s such that forall x, y ∈ G there exist n and x = x , . . . , x n = y such that d ( x i , x i +1 ) ≤ s for all i and P i d ( x i , x i +1 ) ≤ (1 + ε ) d ( x, y ). It is straightforward that if the metric d isasymptotically geodesic, then Cone ω ( g , d ) is geodesic. Corollary A.5. If d is asymptotically geodesic, then d ω is the Carnot-Carath´eodorymetric associated to a supplement subspace v ω of [ g ∞ , g ∞ ] in g ∞ . Remark A.6. If d is asymptotically geodesic, Breuillard [Bre] proves in additionthat d ω is independent of ω , or equivalently that v ω is independent of ω . On the other hand, if d is not asymptotically geodesic, then d ω may depend on ω . For instance, in the real Heisenberg group (viewed as a group of upper unipo-tent 3-matrices), consider the word metric associated with the following weightedgenerating subset: elements with coefficients of absolute value ≤ 1, with weight 1,and the elements n )! with weight (2 n )! / 2. This length is equivalentto the word metric with respect to a compact generating subset. However, thelength of n )! is (2 n )! / n + 1)! isapproximately (2 n + 1)! / √ 2, so d ω , does depend on ω . Proof of Theorem A.4. Set ρ ( g ) = d (0 , g ) and ρ ω ( g ) = d ω (0 , g ). Since d is quasi-isometric to the word distance, it is part of [Gu, Proof of Th´eor`eme II.2’] thatwe have, for suitable constants and for all g ∈ g (A.2) C σ ( g ) − C ′ ≤ ρ ( g ) ≤ C σ ( g ) + C ′ . Observe that σ ( δ t ( g )) = | t | σ ( g ). Therefore, for t > C σ ( g ) − C ′ t ≤ ρ ( δ t ( g )) t ≤ C σ ( g ) + C ′ t ;in particular we deduce(A.3) C σ ( g ) ≤ ρ ω ( g ) ≤ C σ ( g ) . Claim 1. For any ( g t ) , ( h t ) in g ( t ≥ 0) with σ ( g t ) + σ ( h t ) (cid:22) t , we havelim t → + ∞ ρ ( g − t h t ) − ρ ( g − t ⊠ h t ) t = 0 . Indeed, we have | ρ ( g − t h t ) − ρ ( g − t ⊠ h t ) | ≤ ρ (( g t ) − h t ) − ( g − t ⊠ h t ))and by Proposition A.9 (and in view of (A.2)),(A.4) ρ (( g − t h t ) − ( g − t ⊠ h t )) ≪ t, and the claim is proved.By Claim 1, we have d ω ( g, h ) = lim ω ρ ( δ t ( g ) − δ t ( h )) t = lim ω ρ ( δ t ( g ) − ⊠ δ t ( h )) t = ρ ω ( g − ⊠ h ) , IE GROUPS AND CONE EQUIVALENCES 19 so d ω is ⊠ -left-invariant. As a pointwise limit of pseudodistances, d ω is a pseu-dodistance. In particular, ρ ∞ is a length on ( g , ⊠ ). Since by (A.3), ρ ω is continu-ous at zero, it immediately follows that ρ ω (hence d ω ) is everywhere continuous.Let us turn to the part concerning asymptotic cones. First observe that themap i : ( G ∞ , d ω ) → Cone ω ( G ∞ , d ∞ )(A.5) g ( δ t ( g ))is a well-defined isometric embedding, by definition of d ω . Moreover, it is surjec-tive, its inverse being given by j : ( g t ) lim ω δ /t ( g t ). To check that i ◦ j is theidentity, compute˜ d ω (( g t ) , ( i ◦ j ( g t ))) = lim t → ω t d ω (cid:16) g t , δ t (cid:16) lim u → ω δ /u ( g u ) (cid:17)(cid:17) = lim t → ω d ω (cid:16) δ /t ( g t ) , lim u → ω δ /u ( g u ) (cid:17) = d ω (cid:16) lim t → ω δ /t ( g t ) , lim u → ω δ /u ( g u ) (cid:17) = 0 , where we use continuity of d ω in the last line.We know that η is cone-defined by Corollary A.2; η and ⊠ are cone-equivalentas a consequence of (A.4). (cid:3) Proof of Corollary A.5. Since ( G ∞ , d ω ) is geodesic, by [Be, Theorem 2], it is theCarnot-Carath´eodory metric defined by a generating subspace v ω of the Lie al-gebra g ∞ = ( g , [ · , · ] ∞ ). In particular, v ω + [ g ∞ , g ∞ ] = g ∞ . By (A.3) in the proofof Theorem A.4, ([ G ∞ , G ∞ ] , d ω ) cannot contain a bilipschitz copy of a segment,and therefore v ω ∩ [ g ∞ , g ∞ ] = { } . (cid:3) A.3. Computation in nilpotent groups. In all this paragraph, g is a finite-dimensional real s -nilpotent Lie algebra, on which a norm has been fixed; mul-tiplying the norm by a suitable scalar if necessary we suppose that k [ x, y ] k ≤k x kk y k for all x, y .The group law can be written down by the Baker-Campbell-Hausdorff formulaas(A.6) xy = x + y + s X k =2 X u ∈U ( k − λ u [ u , . . . , u k − , x, y ] , where U ( k ) = U ( k )[ x, y ] is the set (of cardinality 2 k − ) of functions { , . . . , k − } → { x, y } ( x, y denoting two distinct symbols) and λ u are rational numbers(which can be fixed once and for all, independently of G and s ).Besides, for n ≥ x , . . . , x n ∈ g , define by induction [ x , . . . , x n ] as theusual bracket for n = 2, and as [ x , [ x , . . . , x n ]] for n ≥ 3. Also by x ji wemean the i th component of x j with respect to the linear grading g = L i v i . We first need the following computational lemma, which is a formal consequence ofmultilinearity. Lemma A.7. For any x , . . . , x k ∈ g , we have [ x , . . . , x k ] ℓ = X i + ··· + i k ≤ ℓ [ x i , . . . , x ki k ] ℓ and [ x , . . . , x k ] ∞ ℓ = X i + ··· + i k = ℓ [ x i , . . . , x ki k ] ℓ . Proof. The first equality is obtained by writing x j = P i j x ji j and expanding,noting that [ x i , . . . , x ki k ] ℓ = 0 if ℓ < i + · · · + i k . For the second, it is enoughto observe that [ x i , . . . , x ki k ] ∞ is the projection of [ x i , . . . , x ki k ] on v ℓ , for ℓ = i + · · · + i k . (cid:3) Lemma A.8. We have in g , for any integer ℓ ≥ x,y k ( xy − x ⊠ y ) ℓ k max( σ ( x ) , σ ( y )) ℓ − < ∞ . Proof. Let us write the Baker-Campbell-Hausdorff formula as in (A.6). xy − x ⊠ y = s X k =2 X u ∈U ( k − λ u ([ u , . . . , u k − , x, y ] − [ u , . . . , u k − , x, y ] ∞ ) , so, using Lemma A.7,( xy − x ⊠ y ) ℓ = s X k =2 X u ∈U ( k − λ u ([ u , . . . , u k − , x, y ] − [ u , . . . , u k − , x, y ] ∞ ) ℓ = s X k =2 X u ∈U ( k − λ u X i + ...i k ≤ ℓ − [( u ) i , . . . , ( u k − ) i k − , x i k − , y i k ] ℓ . By definition of σ , we have k ( u j ) i j k ≤ M i j for all j . By the inequality [ u, v ] ≤k u kk v k , for each term we deduce k [( u ) i , . . . , ( u k − ) i k − , x i k − , y i k ] ℓ k ≤ M k ≤ M ℓ − , so for some constant C we have k ( xy − x ⊠ y ) ℓ k ≤ CM ℓ − . (cid:3) Proposition A.9. We have sup x,y σ (( xy ) − ( x ⊠ y ))max( σ ( x ) , σ ( y )) − /s < ∞ ; IE GROUPS AND CONE EQUIVALENCES 21 in particular, we have σ (( xy ) − ( x ⊠ y ))max( σ ( x ) , σ ( y )) → when max( σ ( x ) , σ ( y )) → + ∞ . Proof. Let us write the Baker-Campbell-Hausdorff formula as in (A.6).(( xy ) − ( x ⊠ y )) ℓ = ( − xy + x ⊠ y ) ℓ (A.7) + s X k =2 X u ∈U ( k − − xy,x ⊠ y ] λ u [ u , . . . , u k − , − xy, − xy + x ⊠ y ] ℓ . Fix u ∈ U ( k − − xy, x ⊠ y ] and write u k − = − xy . Expanding, we obtain[ u , . . . , u k − , − xy + x ⊠ y ] ℓ (A.8) = X i + ··· + i k = ℓ [( u ) i , . . . , ( u k − ) i k − , ( − xy + x ⊠ y ) i k ] ℓ . Since σ ( u j ) ≤ M for all j < k , we have k ( u j ) i j k ≤ M i j . Using the inequality k [ u, v ] k ≤ k u k v k we deduce k [( u ) i , . . . , ( u k − ) i k − , ( − xy + x ⊠ y ) i k ] ℓ k≤ M i + ··· + i k − k ( − xy + x ⊠ y ) i k k . By Lemma A.8, we have k ( − xy + x ⊠ y ) i k ≤ C M i − for some constant C (chosenindependent of i ), so if i + · · · + i k = ℓ , using Lemma A.8 we deduce k [( u ) i , . . . , ( u k − ) i k − , ( − xy + x ⊠ y ) i k ] ℓ k ≤ C M ℓ − . 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