aa r X i v : . [ m a t h . G R ] S e p ASSOUAD-NAGATA DIMENSION OF CONNECTED LIEGROUPS
J.HIGES AND I. PENG
Abstract.
We prove that the asymptotic Assouad-Nagata dimensionof a connected Lie group G equipped with a left-invariant Riemannianmetric coincides with its topological dimension of G/C where C is amaximal compact subgroup. To prove it we will compute the Assouad-Nagata dimension of connected solvable Lie groups and semisimple Liegroups. As a consequence we show that the asymptotic Assouad-Nagatadimension of a polycyclic group equipped with a word metric is equal toits Hirsch length and that some wreath-type finitely generated groupscan not be quasi-isometric to any cocompact lattice on a connected Liegroup. Contents
1. Introduction 12. Assouad-Nagata dimension and transformations on the metric 43. Dimension of exact sequences 74. Assouad-Nagata dimension of Nilpotent Lie groups 85. Assouad-Nagata dimension of solvable Lie groups 126. Assouad-Nagata dimension of semisimple Lie groups 147. Assouad-Nagata dimension of connected Lie groups 16References 201.
Introduction
The Assouad-Nagata dimension was introduced by Assouad in [1] inspiredfrom the ideas of Nagata. Metric spaces of finite Assouad-Nagata dimensionsatisfy interesting geometric properties. For example they admit quasisym-metric embeddings into the product of finitely many trees [21] and havenice Lipschitz extension properties (see [21] and [6]). The class of met-ric spaces with finite Nagata dimension includes in particular all doubling
Mathematics Subject Classification.
Primary: 20F69, 22E25, Secondary: 20F16.
Key words and phrases.
Asymptotic dimension, Assouad-Nagata dimension, polycyclicgroups, connected Lie groups.The first named author is supported by project MEC, MTM2006-0825 and ’contratoflechado’ i-math. spaces, metric trees, Euclidean buildings, and homogeneous or pinched neg-atively curved Hadamard manifolds.In last years a small scale version and a large scale version of the Assouad-Nagata dimension have been the focus of interesting research. The smallscale version has been studied in the framework of hyperbolic groups underthe name of capacity dimension (see [8]). The large scale version of Assouad-Nagata dimension has been referred to by several names, such as asymptoticdimension of linear type, asymptotic dimension with Higson property orasymptotic Assouad-Nagata dimension. This last name would be the onewe will use in this paper.The asymptotic Assouad-Nagata dimension is a linear version of the as-ymptotic dimension, and it is invariant under quasi-isometries. But whilethe asymptotic dimension remains invariant under coarse equivalences, theasymptotic Assouad-Nagata dimension does not. Therefore it is reasonableto expect that there would be more relationships between the asymptoticAssouad-Nagata dimension and other quasi-isometric invariants of geometricgroup theory. For example in [23] and [4] the asymptotic Assouad-Nagatadimension was related with the growth type of amenable groups and wreathproducts, in [14] it was shown that the asymptotic Assouad-Nagata dimen-sion bounds from above the topological dimension of the asymptotic cones.From the results of [15] it is easy to see that every metric space of finiteasymptotic Assoaud-Nagata dimension has Hilbert compression equals one.On the other hand, estimating the asymptotic Assouad-Nagata dimensionof a group is more difficult. Most of the methods developed to calculate theasymptotic dimension of a group behaves poorly for asymptotic Assouad-Nagata dimension. For example, in [2] Bell and Dranishnikov developed atechnique to estimate the asymptotic dimension of the fundamental group ofa graph of groups. Unfortunately such method can not be applied directlyto the asymptotic Assouad-Nagata dimension.For connected Lie groups, it is natural to study short exact sequences ofthe form: 1 → H → G → G/H → . The idea is to estimate the dimension of G from the dimensions of H and G/H . For example if G is nilpotent then H could be abelian and G/H nilpotent with lower degree of nilpotency.In [5], Brodskyi, Dydak, Levin and Mitra developed techniques to studythe asymptotic Assouad-Nagata dimension of short exact sequences of finitelygenerated groups with word metrics, and the ideas were applied successfullyin [14] to calculate the asymptotic Assouad-Nagata dimension of the Heisen-berg group. But the main difficulty in the general case is to understand thedistortion of the subgroup H in G . SSOUAD-NAGATA DIMENSION OF CONNECTED LIE GROUPS 3
In this paper we study the Assouad-Nagata dimension (at small and largescale) of connected Lie groups equipped with left invariant Riemannian met-rics. First we will analyze the Assouad-Nagata dimension of simply con-nected solvable Lie groups. The key tool for such goal is a generalizationof the results from [5] in the setting of finitely generated groups with wordmetrics, to general topological groups with left invariant metrics. Using thisand some facts from differential geometry, we will show that the Assouad-Nagata dimension of a connected solvable Lie group is equal to its topolog-ical dimension. As a consequence of this we will prove that the asymptoticAssouad-Nagata dimension of a polycyclic group is equal to its Hirsch length.This answers an open problem of the asymptotic Assouad-Nagata dimen-sion as Question 4 of [23] and Problem 8.3 of [12](notice that in [12] it issaid that Osin solved such problem but no proof has been provided). Alsoour results increase the catalogue of finitely generated groups with finiteasymptotic Assouad-Nagata dimension. So far, the classes of groups knownto have finite asymptotic Assouad-Nagata dimension are Coexeter groups,abelian groups, hyperbolic groups, free groups and some types of Baumslag-Solitar groups. Recently it was shown that the asymptotic Assouad-Nagatadimension is preserved under free products (see [7]).The next step in the proof of the main theorem of this paper will bethe study of the Assouad-Nagata dimension of semisimple Lie groups. Forsuch goal we will use the Iwasawa decomposition of a semisimple Lie group.Roughly speaking an Iwasawa decomposition will say that a semisimple Liegroup is equivalent to the product of a connected solvable group and a specialgroup K for which we can apply the methods of [5].After studying the Assouad-Nagata dimension of semisimple Lie groupswe will focus in the main result of this paper: the Assouad-Nagata dimen-sion of connected Lie groups. Notice that in [10], Carlsson and Goldfarbproved that the asymptotic dimension of a Lie group G is equal to thetopological dimension of G/C where C is its maximal compact subgroup.Therefore it is natural to see if the results of [10] can be extended to theasymptotic Assouad-Nagata dimension. Unfortunately the same techniquesof [10] cannot be applied directly for asymptotic Assouad-Nagata dimension.The main tools of [10] use strongly the invariance under coarse equivalencesof the asymptotic dimension. For example Proposition 3.4. of [10] usessuch property. We will show in Example 4.11 that also the techniques fromTheorem 3.5. of [10] can not be applied.As it was mentioned before, the asymptotic Assouad-Nagata dimensionhas relationships with the topological dimension of the asymptotic cones.In [14] it was shown that the topological dimension of the asymptotic coneis bounded from above by the asymptotic Assouad-Nagata dimension of thegroup. De Cornulier computed in [11] the topological dimension of the as-ymptotic cones of connected Lie groups. He defined the exponential radicalof a connected Lie group and used it to compute the dimension of the cones.We will also use the exponential radical to compute the Assouad-Nagata J.HIGES AND I. PENG dimension of connected Lie groups. For this final theorem the computationsmade for connected solvable Lie groups and semisimple Lie groups will playan important role.We will prove that the Assouad-Nagata dimension for a connected Liegroup G is equal to the topological dimension of G/C , where C is a maxi-mal compact subgroup. Hence our results can be seen as a bridge among theones of [10], [11] and [14]. Moreover it is not difficult to show (although te-dious) that combining the techniques of [11] and [14] with our main theoremwe can improve slightly the results of [11] to compute the Assouad-Nagatadimension of the asymptotic cones of connected Lie groups.In Section 2 we will study the behaviour of the Assouad-Nagata dimensionunder some transformations on the metric. Such results will be importantwhen we study the restriction of a metric of a group to one of its subgroups.In Section 3 we will generalize the results of [5] to groups with left invari-ant metrics. Section 4 is devoted to an intermediate step in our proof: thestudy of the Assouad-Nagata dimension of nilpotent groups. In Section 5 wewill prove that the Assouad-Nagata dimension of simply-connected solvableLie groups is equal to the topological dimension. As a consequence we willcompute the asymptotic Assouad-Nagata dimension of finitely generatedpolycyclic groups. In section 6 we will study, using Iwasawa decomposi-tions, the Assouad-Nagata dimension of semisimple Lie groups. Finally insection 7 we will prove the main result of the paper. Acknowledgements:
Both authors are very grateful to the HausdorffResearch Institute for Mathematics and specially with the organizers of the’Rigidity program’ for their hospitality and support. They also thank DavidFisher for useful conversations and U. Lang for very helpful comments andsuggestions, specially for pointing out [10] and the necessity of the example4.11.2.
Assouad-Nagata dimension and transformations on themetric
Let s be a positive real number. A s -scale chain (or s -path) betweentwo points x and y of a metric space ( X, d X ) is defined as a finite sequenceof points { x = x , x , ..., x m = y } such that d X ( x i , x i +1 ) < s for every i = 0 , ..., m −
1. A subset S of a metric space ( X, d X ) is said to be s -scale connected if every two elements of S can be connected by s -scale chaincontained in S . Definition 2.1.
A metric space (
X, d X ) is said to be of asymptotic dimen-sion at most n (notation asdim( X, d ) ≤ n ) if there is an increasing function D X : R + → R + such that for any s > U = {U , ..., U n } so that the s -scale connected components of each U i are D X ( s )-bounded i.e.the diameter of every component is bounded by D X ( s ). SSOUAD-NAGATA DIMENSION OF CONNECTED LIE GROUPS 5
The function D X is called an n -dimensional control function for X . De-pending on the type of D X one can define the following invariants: Definition 2.2.
A metric space (
X, d X ) is said to have • Assouad-Nagata dimension at most n (denoted by dim AN ( X, d ) ≤ n )if it has an n -dimensional control function D X of the form D X ( s ) = C · s with C > • asymptotic Assouad-Nagata dimension at most n (denoted byasdim AN ( X, d ) ≤ n ) if it has an n -dimensional control function D X of the form D X ( s ) = C · s + k with C > k ∈ R two fixed constants. • capacity dimension at most n (notation cdim( X, d ) ≤ n ) if it has an n -dimensional control function D X such that D X ( s ) = C · s in a neighborhood of 0 i.e. with s sufficiently small. Remark . For any metric space (
X, d ) we have dim(
X, d ) ≤ dim AN ( X, d )where dim is the topological dimension (see [1]).One map f : ( X, d ) → ( Y, D ) between metric spaces is said to be a quasi-isometric embedding if there exists two constants C ≥ , λ ≥ C · d ( x, y ) − λ ≤ D ( f ( x ) , f ( y )) ≤ C · d ( x, y ) + λ. (1)If in addiction there exists a K > D ( y, f ( X )) ≤ K for every y ∈ Y , f is said to be a quasi-isometry and the spaces ( X, d ) and (
Y, D )are said to be quasi-isometric. In (1), if λ = 0 then f is said to be a bilip-schitz equivalence and the spaces are said to be bilipschitz equivalent . Oneimportant fact about Assouad-Nagata dimension is that it is preserved un-der bi-Lipschitz equivalences. Analogously the asymptotic Assouad-Nagatadimension is preserved under quasi-isometries. The capacity dimension isinvariant under maps that are bi-Lipschitz at small scales i.e. bi-Lipschitzwhen the distances are less than some fix constant ǫ > X, d ) ≤ asdim AN ( X, d ). The relation among theAssouad-Nagata dimension, the asymptotic Assouad-Nagata dimension andthe capacity dimension in a metric space was studied in [6] via two functorsmax( d,
1) and min( d, Proposition 2.4. [6]
Let ( X, d ) be a metric space. Then: (1) cdim ( X, d ) = dim AN ( X, max( d, ǫ )) for every ǫ > . (2) asdim AN ( X, d ) = dim AN ( X, min( d, ǫ )) for every ǫ > . J.HIGES AND I. PENG (3) dim AN ( X, d ) = max { asdim AN ( X, d ) , cdim ( X, d ) } It was shown in [21] that the Assouad-Nagata dimension is also invariantunder quasisymmetric embeddings as for example snow-flake transforma-tions. Recall that a snow flake transformation of a metric space (
X, d ) is ofthe form (
X, d α ) where 0 < α ≤
1. We will need the following lemma thatis in some sense a generalization of this fact for geodesic spaces:
Lemma 2.5.
Let ( X, d ) be a (non bounded) geodesic metric space. Suppose f : R + → R + is a surjectively non decreasing function such that ( X, f ( d )) is a metric space. Then dim AN ( X, f ( d )) ≤ dim AN ( X, d ) .Proof. Notice that f ( x + y ) ≤ f ( x ) + f ( y ) for every x, y ∈ R + . This followsfrom the fact that ( X, d ) is geodesic, unbounded, and (
X, f ( d )) is a metric.Let us show it. Pick a, b ∈ X such that d ( a, b ) = x + y . As ( X, d ) is geodesicthere exists a c ∈ X such that d ( a, c ) = x and d ( b, c ) = y . Hence by thetriangle inequality in ( X, f ( d )) we get f ( d ( a, b )) ≤ f ( d ( a, c )) + f ( d ( b, c )).Suppose dim AN ( X, d ) ≤ n . Let D ( s ) = C · s be an n -dimensional controlfunction of ( X, d ). Without loss of generality we can assume C ∈ N . Let s >
0. Take some inverse f − of f . Hence there exists a covering of ( X, d ) ofthe form U = S ni =0 U i such that the f − ( s )-scale components of each U i are C · f − ( s )-bounded for every i ∈ { , · · · , n } . This implies that if x, y ∈ X are in different f − ( s )-scale components of U i then s ≤ f ( d ( x, y )). Thereforethe s -scale components of U i in ( X, f ( d )) are contained in the f − ( s )-scalecomponents of U i in ( X, d ). Let x, y ∈ U i be two elements that belongs tothe same f − ( s )-scale component of U i . We have d ( x, y ) ≤ C · f − ( s ). Thisimplies f ( d ( x, y )) ≤ f ( C · f − ( s )). On the other hand by the subadditivityof f and C ∈ N we get f ( C · f − ( s ) ≤ C · f ( f − ( s )) = C · s as desired. (cid:3) From the previous lemma we have the following result that can be appliedto many remarkable cases as for example unbounded trees or Cayley graphsof finitely generated groups such that the asymptotic dimension coincideswith the asymptotic Assouad-Nagata dimension.
Corollary 2.6.
Let ( X, d ) be a geodesic space and let f : R + → R + as in theprevious lemma such that lim x →∞ f ( x ) = ∞ . If asdim ( X, d ) = dim AN ( X, d ) then dim AN ( X, f ( d )) = dim AN ( X, d ) Proof.
Just notice that asdim(
X, d ) = asdim(
X, f ( d )) as f induces a coarseequivalence. (cid:3) The condition on the equality of dimensions can not be dropped as it isshown in the following example.
Example . In [4] it was shown that the Cayley graph Γ of the group Z ≀ Z has infinite Assouad-Nagata dimension but asymptotic dimensionequals two. Moreover from Theorem 5.5 of [4] it follows that the Cayleygraph has a 2-dimensional control function of polynomial type. Let d be SSOUAD-NAGATA DIMENSION OF CONNECTED LIE GROUPS 7 the metric of the Cayley graph Γ. It is not hard to check (or you can applydirectly proposition 2.2 of [16]) that dim AN (Γ , log ( d + 1)) = 2.3. Dimension of exact sequences
In this section we extend a result of [5] about dimension of exact sequencesof finitely generated groups equipped with word metrics to dimension ofexact sequences of general groups(not necessarily countable) equipped withleft invariant metrics. First we will recall some definitions and results from[5].
Definition 3.1.
Given a function between metric spaces f : X → Y andgiven m ∈ N . An m -dimensional control function of f is a function D f : R + × R + → R + such that for all r X > R Y >
0, every set A ⊂ X such that diam( f ( A )) ≤ R Y can be expressed as the union of m + 1 subsets U = S ni =0 U i such that the r x -scale components of each U i are D f ( r X , R Y )-bounded.As in the previous section depending on the type of D f we could havedifferent notion of dimension of functions (see [5]). We will use just thefollowing: Definition 3.2.
A function between metric spaces f : X → Y is said tohave Assouad-Nagata dimension at most n (notation dim AN ( f ) ≤ n ) ifthere exists an n -dimensional control function D f of the form D f ( r x , R Y ) = a · r x + b · R Y . If n is the minimum number such that f satisfies this propertythen f is said to have Assouad-Nagata dimension exactly n . Proposition 3.3. [5, Theorem 7.2] If f : X → Y is a Lipschitz functionbetween metric spaces then:dim AN ( X, d X ) ≤ dim AN ( f ) + dim AN ( Y, d Y ) . (2) Proposition 3.4. [5, Proposition 3.7]
Suppose A is a subset of a metricspace ( X, d ) , m ≥ , R > . If D A is an m -dimensional control function of A then D B ( s ) := D A ( s + 2 R ) + 2 R is an m -dimensional control function ofthe R -neighborhood B ( A, R ) . Let 1 → K → G → H → d G is a left invariant metric. This metric induces in H a natural metric calledthe Hausdorff metric defined by the norm k g · K k H = inf {k g · k k G ; k ∈ K } .Notice that when G is a finitely generated group and d G is a word metricthen the corresponding induced metric d H is also a word metric.The induce Hausdorff metric allows us to extend Corollary 8.5 of [5] togeneral groups as follows. Proposition 3.5.
Let → K → G → H → be an exact sequence ofgroups. Then (1) dim AN ( G, d G ) ≤ dim AN ( K, d G | K ) + dim AN ( H, d H );(2) asdim AN ( G, d G ) ≤ asdim AN ( K, d G | K ) + asdim AN ( H, d H ); J.HIGES AND I. PENG (3) cdim ( G, d G ) ≤ cdim ( K, d G | K ) + cdim ( H, d H ) , where d G is a left invariant metric of G and d H is the induced (left invariant)Hausdorff metric on H .Proof. We will prove only the first inequality. The other two follow fromthe first one and Proposition 2.4 (notice that the induced Hausdorff metricsare preserved by the two functors).The proof is close to that of Corollary 8.5 in [5]. First, it is clear that if d H is the Hausdorff metric, then the projection map f : ( G, d G ) → ( H, d H )is 1-Lipschitz. Let B = B (1 H , R H ) and let A = f − ( B ). Take a ∈ A ,then k f ( a ) k H < R H . As f is an homomorphism and by the symmetryof the norm we get k f ( a − k H = k f ( a ) − k H = k f ( a ) k H < R H . But onthe other hand we have k f ( a − ) k H = inf {k a − · k k G ; k ∈ K } . Therefore k f ( a − ) k H = d G ( a, K ) < R H , and we have shown A ⊂ B G ( K, R H ). ByProposition 3.4 we have that D f ( r G , r H ) := D K ( r G + 2 · R H ) + 2 · R H is an m -dimensional control function of B ( K, R H ), provided D K is an m -dimensional control function of K . Now if dim AN ( K, d G | K ) ≤ m then thereexists an C > D K ( s ) = C · s . Since all f − ( B ( y, R H )) areisometric we can apply Proposition 3.3 to get the inequality: dim AN ( G, d G ) ≤ dim AN ( K, d G | K ) + dim AN ( H, d H ) . (cid:3) Assouad-Nagata dimension of Nilpotent Lie groups
As an intermediate step we study in this section the Assouad-Nagatadimension of Nilpotent Lie groups. From Proposition 2.4 we would needto calculate the capacity dimension and the asymptotic Assouad-Nagatadimension. For all connected Lie groups (not necessarily solvable) we willshow first that the capacity dimension is less or equal than the topologicalone. We will use the ideas of Buyalo and Lebedeva. Following definitionsand theorem 4.3 come from [9]:
Definition 4.1.
A map f : Z → Y between metric spaces is said to be quasi-homothetic with coefficient R >
0, if for some λ ≥ z , z ′ ∈ Z , we have R · d Z ( z, z ′ ) /λ ≤ d Y ( f ( z ) , f ( z ′ )) ≤ λ · R · d Z ( z, z ′ ) . In this case, it is also said that f is λ -quasi-homothetic with coefficient R . Definition 4.2.
A metric space Z is locally similar to a metric space Y ,if there is λ ≥ R > A ⊂ Z with diam A ≤ Λ /R , where Λ = min { , diam Y /λ } , there is a λ -quasi-homothetic map f : A → Y with coefficient R . Theorem 4.3. [9, Theorem 1.1]
Assume that a metric space Z is locallysimilar to a compact metric space Y . Then cdim ( Z ) < ∞ and cdim ( Z ) ≤ dim ( Y ) . SSOUAD-NAGATA DIMENSION OF CONNECTED LIE GROUPS 9
Lemma 4.4.
Let G be a connected Lie group of dimension n with a left-invariant Riemannian metric d G . Then it is locally similar to a closed ballin R n .Proof. Note that on a smooth manifold M , the exponential map Exp : T g M → M is a diffeomorphism between a closed ball of radius ǫ (whichmight depend on the point p ) and a convex neighborhood of p . Now, anydiffeomorphism between two compact Riemannian metric spaces is bilips-chitz, so there exists a constant λ (which might also depend on the point p ) such that the exponential map restricted to the ball of radius ǫ is λ -bilipschitz for some λ ≥
1. If M is a Lie group, by left translations neitherthe ǫ nor the λ depends on the point p , so we can assume p = 1 G . Letlog : ¯ B (1 G , ǫ ) → ¯ B (0 , K ) be the inverse of the exponential map where¯ B (0 , K ) is some ball of R n that contains the image of log( ¯ B (1 G , ǫ )). Themetric d of R n that we are considering is the usual one. Without loss ofgenerality let us suppose K > · λ . In such case Λ := min { , · Kλ } = 1and R = ǫ . Given R > R , let A be a closed ball of radius at most R .By left translation we can assume A is centered in 1 G . Notice log(1 G ) = 0.Define the natural dilatation g : ¯ B (0 , K ) → ¯ B (0 , K · R ). By the assumptionof K > · λ the image of the composition of g ◦ log restricted to A lies in¯ B (0 , K ). Moreover it satisfies: λ − · d G ( x, y ) ≤ /R · d ( g (log( x )) , g (log( y ))) ≤ λ · d G ( x, y ) . (cid:3) Combining this lemma with Theorem 4.3 we get:
Corollary 4.5.
Let G be a connected Lie group equipped with a left-invariantRiemannian metric d G . Then cdim ( G, d G ) ≤ dim ( G ) . Now we study the asymptotic Assouad-Nagata dimension of a nilpotentLie group G . For such purpose we will apply Proposition 3.5 to the exactsequence 1 → N r → G → G/N r → N r is the last term in the lowercentral series. Notice that G/N r is also a nilpotent Lie group. The metricsconsidered will be a Riemannian metric in G , d G , and the correspondinginduced Hausdorff metric in G/N r . We will need the following. Lemma 4.6.
Let G be a connected Lie group and H ✁ G a normal subgroupsuch that G/H is a connected Lie group. Then there are Riemannian metrics ρ G and ρ G/H on G and G/H such that the (left-invariant) Hausdorff metricon
G/H induced by ρ G agrees with the path metric induced by ρ G/H .Proof.
Write π : G → G/H for the canonical projection. In G , choose acomplement V to T e H in T e G . Define ρ G at T e G by choosing inner productson T e H and V , and define ρ G/H at T e ( G/H ) to be the inner product chosenon V . By left translating V we obtain a distribution △ V on G , so everydifferentiable path γ ∈ G/H has a isometric lift ˜ γ lying entirely in △ V ,which is unique up to the choice of the starting point. Fix a coset pH ∈ G/H . Since both metrics are left invariant, it sufficesto show that d ρ G/H ( H, pH ) is the same as d H ( H, pH ). Let K be the set ofisometric lifts of differentiable paths connecting H with Hp in G/H thatstart at the identity, and B be the set of differentiable paths starting at theidentity and end at some point in pH . That is, K = { ˜ γ : ˜ γ (0) = e, π (˜ γ ) = γ, γ : [0 , L ] → G/H, γ (0) =
H, γ ( L ) = Hp } Clearly
K ⊂ B . However if η ∈ B , g π ( η ), the lift of its projection, is also in B ,with length no bigger than η , so min {k η k : η ∈ B} = min {k g π ( η ) k : η ∈ B} . But { g π ( η ) : η ∈ B} = K . The claim now follows since d H ( H, pH ) = min {k η k : η ∈ B} = min {k ζ k : ζ ∈ K} = d ρ G/H ( H, Hp ) (cid:3) Let N be a connected, simply connected nilpotent Lie group, and let N i be the i -th term in its lower central series. That is, N = [ N, N ], and N i +1 = [ N, N i ]. By construction, each quotient N i /N i +1 is an abelianLie group of dimension n i , so by fixing a subset K i ⊂ N i , a set of N i +1 coset representatives in N i , we have a bijection φ i : R n i → K i , and a map φ : N → ⊕ i R n i defined as p φ −→ ( p , p · · · ) , p i ∈ R n i where φ ( p ) N = pN , and φ i ( p i ) N i +1 = ( φ i − ( p i − )) − · · · ( φ ( p )) − ( φ ( p )) − pN i +1 .One can check that φ is a bijection with the inverse given by( p , p , · · · ) φ − −→ φ ( p ) φ ( p ) φ ( p ) · · · Note that if the degree of nilpotency of N is r , then N r , the last non-trivialsatisfies φ ( N r ) = { (0 , , · · · , p r ) : p r ∈ R n r } . With this coordinate system we define D : N × N → R as D ( p, q ) = D (1 , p − q ) = r X i k ~x i k /i , where φ ( p − q ) = ( ~x , · · · ~x r ), and kk is the standard Euclidean norm on R n .The main result from [19] is the following. Theorem 4.7. [19, Theorem 4.2]
Let N be a connected, simply connectednilpotent Lie group. Then there is a Riemannian metric d N on N andconstant κ such that for any point p with d N ( e, p ) > , /κD (1 , p ) ≤ d N (1 , p ) ≤ κD (1 , p ) . Equivalently, ( N, min(1 , D )) is bilipschitz to ( N, min(1 , d N )) . SSOUAD-NAGATA DIMENSION OF CONNECTED LIE GROUPS 11
An understanding of distance distortions in nilpotent Lie groups can nowbe obtained.
Lemma 4.8.
Let N be a connected, simply connected nilpotent Lie group,and H the last non-trivial term in its lower central series. Then there areRiemannian metrics d N , d H on N and H such that ( H, min(1 , d N | H )) isbilipschitz to ( H, min(1 , ( d H ) /r )) where r is the degree of nilpotency of N .Proof. Note that H = N r . Equip H with the Riemannian metric induced by φ r : R n r → H , while putting on G the Riemannian metric given by Theorem4.7. With these choices we have D | H = ( d H ) /r , and the claim follows sinceTheorem 4.7 says that ( H, min(1 , D | H )) is bilipschitz to ( H, min(1 , d N | H )). (cid:3) Lemma 4.9.
Let N be a nilpotent Lie group and let d N be a left invariantRiemannian metric. Then asdim AN ( N, d N ) ≤ dim ( N ) Proof.
Recall the topological dimension of N is the same as the sum ofthe topological dimensions of factors in its lower central series. We willprove the lemma by induction on the degree of nilpotency. The base caseis when N is an abelian Lie group, and in this case we have the equality asdim AN ( N, d N ) = dim ( N ). In general we consider the following shortexact sequence 1 → H → N → N/H → H is the last term in the lower central series of N . Let d N/H de-notes the induced Hausdorff metric on
N/H from d N and d H denotes theRiemannian metric on H induced by d N . By Proposition 3.5 we have:asdim AN ( N, d N ) ≤ asdim AN ( H, d N | H ) + asdim AN ( N/H, d
N/H ) . But asdim AN ( H, d N | H ) = asdim AN ( H, d H ) because of Lemma 4.8 and thefact that the Assouad-Nagata dimension is invariant under quasi-isometriesand snowflake transformations. Moreover asdim AN ( H, d H ) = dim ( H ) since H is abelian. On the other hand, Lemma 4.6 says that d N/H is a Riemannianmetric in
N/H , and since
N/H is nilpotent with one less degree of nilpo-tency, induction hypothesis yields asdim AN ( N/h, d
N/H ) ≤ dim ( N/H ), towhich the desired claim now follows. (cid:3)
Theorem 4.10.
Let N be a nilpotent Lie group and let d G be some Rie-mannian metric. Then dim AN ( N, d N ) = dim ( N ) Proof.
By Proposition 2.4 the Assouad-Nagata dimension is equal the maxi-mum of the capacity dimension and the asymptotic Assouad-Nagata dimen-sion. Hence applying corollary 4.5 and lemma 4.9 we get dim AN ( N, d N ) ≤ dim ( N ). The other inequality follows from the fact that the Assouad-Nagatadimension is always greater or equal to the topological one (see [1]). (cid:3) Next example will show that the proof of Theorem 3.5 of [10] can notbe applied to compute the Assouad-Nagata dimension of nilpotent groups.
We recommend the reading of such proof in order to understand better theexample.
Example . Consider the 4-dimensional nilpotent Lie group determinedby the following Lie algebra.[ e , e ] = e , [ e , e ] = e It has a group structure given by x x x x • y y y y = x + y x + y x + y + ( − x y + x y ) x + y + ( x − y )( − x y + x y ) + ( − x y + x y ) A left-invariant Riemannian metric is given by (cid:18) dx + 12 x dx − x x dx (cid:19) + (cid:18) dx − x dx + 16 x dx (cid:19) + (cid:18) − x dx (cid:19) + dx Following the proof of Theorem 3.5 in [10], we express this 4-dimensionalnilpotent groups as a semidirect product T ⋉ N , where N is the subgroupgenerated by e , e , e , and T is the subgroup generated by e . Then themetric on N ( x e ) is (cid:0) dx + x dx + x dx (cid:1) + ( dx + x dx ) + ( dx ) Note the role of e coordinate, x , plays in the metric. For example if I isan interval of size c in the e direction, then the diameter of I ( x e ), righttranslate of I by x e ∈ T , is x c . Similarly if J is an interval of size c inthe e direction then the diameter of J ( x e ) is x c . In this way we see fora subset W ⊂ N of bounded diameter, the diameter W ( x e ) depends onthe value of x . Since the metric is left-invariant, the diameter of W ( x e )is the same as the diameter of ( x ′ e ) W ( x e ) for any x ′ e ∈ T , the same istrue for conjugates of W by x e .Now to see that the diameter of S li ( U ) is about D times the diameter of U , we observe that S li ( U ) consists of right translates of U by elements of T .Express U as U = t i W for some W ⊂ N , t i ∈ T , we have that S li ( U ) = [ t ∈ D U t = [ t ∈ D t i W t = [ t ∈ D t i t ( t − W t ) , which makes the diameter of S li ( U ) the maximum of 2 D and the diameterof t − W t . For
D <
1, the diameter of S li ( U ) varies linearly with D (since D > D for D <
D >
1, the discussion above shows that thediameter depends on D .5. Assouad-Nagata dimension of solvable Lie groups
Definition 5.1.
Let G be a connected, simply connected solvable Lie group.The exponential radical, denoted as Exp ( G ), is a closed normal subgroupsuch that G/Exp ( G ) is the biggest quotient with polynomial growth. SSOUAD-NAGATA DIMENSION OF CONNECTED LIE GROUPS 13
Osin shown in [24] that given a Riemannian metric d G on G and a Rie-mannian metric on the exponential radical d Exp ( G ) then there exists twoconstants C > ǫ ≥ h ∈ Exp ( G ):1 C log ( k h k Exp ( G ) + 1) − ǫ ≤ k h k G ≤ C · log ( k h k Exp ( G ) + 1) + ǫ (3)Also he proved that the exponential radical is contained in the nilrad-ical of G , so it is nilpotent and we have asdim AN ( Exp ( G ) , d Exp ( G ) ) ≤ dim ( Exp ( G ))) by Lemma 4.9. But if we apply Lemma 2.5 to the particularcase f ( s ) = log ( s + 1) and the invariance of the asymptotic Assouad-Nagatadimension under quasi-isometries we get asdim AN ( Exp ( G ) , d G | Exp ( G ) ) ≤ dim ( Exp ( G )). On the other hand by the definition of exponential radicalthere is an exact sequence:1 → Exp ( G ) → G → S → , where S is a solvable Lie group with polynomial growth, and quasi-isometricto a nilpotent group of the same topological dimension [3]. Hence by Lemma4.9 and Lemma 4.6 we have asdim AN ( S, d S ) ≤ dim ( S ) where d S is theRiemannian metric induced by d G . Therefore applying Proposition 3.5 weget: Proposition 5.2.
Let G be a connected solvable Lie group then:asdim AN ( G, d G ) ≤ asdim AN ( Exp ( G ) , d )+ asdim AN ( G/Exp ( G ) , d ) ≤ dim ( G ) where d and d are two Riemannian metrics defined in Exp ( G ) and G/Exp ( G ) respectively. Now we can get the main result of this paper:
Theorem 5.3.
Let G be a connected solvable Lie group then:dim AN ( G, d G ) = dim ( G ) . Proof.
The proof is analogous to the that of Theorem 4.10. On the one handwe have dim AN ( G, d G ) ≥ dim ( G ) by [1]. By Proposition 2.4 the Assouad-Nagata dimension is equal the maximum of the capacity dimension and theasymptotic Assouad-Nagata dimension. Hence the other inequality followsfrom Proposition 5.2 and Corollary 4.5. (cid:3) As a consequence of this theorem and Theorem 1.3 of [21] we get thefollowing interesting property of connected solvable Lie groups:
Corollary 5.4.
Let G be a connected solvable Lie group equipped with aRiemannian metric d G . Then there exists a < p ≤ such that ( G, d pG ) canbe bilipschitz embedded into a product of dim ( G ) + 1 many trees. Definition 5.5.
Let Γ be a finitely generated solvable group. Then thehirsch length is defined as: h (Γ) = X dim Q (Γ i / Γ i +1 ⊗ Q )where Γ = [Γ , Γ] and Γ i +1 = [Γ i , Γ i ] Next Corollary answers Question 4 of [23] and problem 8.3 of [12]:
Corollary 5.6.
Let (Γ , d w ) be a polycyclic group equipped with a word metric d w . Then asdim AN (Γ , d w ) = h (Γ) Proof.
In [13] it was proved that asdim(Γ , d w ) ≥ h (Γ), so the unique thingwe need to show is that asdim AN (Γ , d w ) ≤ h (Γ). It is known that everypolycyclic group is a cocompact lattice in a connected, simply connectedsolvable Lie group H after modding out a finite torsion subgroup. Moreover h (Γ) = dim( H ) and by Theorem 5.3 we have dim( H ) = dim AN ( H ). As Γ isa lattice we have asdim AN (Γ) = asdim AN ( H ) ≤ dim AN ( H ), where the lastinequality follows from Proposition 2.4. (cid:3) Remark . The condition of word metrics can not be relaxed as it is shownin [17], that a countable nilpotent group G can always be equipped with aproper left invariant metric d G , such that asdim AN ( G, d G ) is infinite.6. Assouad-Nagata dimension of semisimple Lie groups
In this section we will compute the asymptotic Assouad-Nagata dimensionof semisimple Lie groups. The idea will be to study the Iwasawa decompo-sitions of such groups. Theorem 6.3 from [20] and [18] gives the structuraldescription of a semisimple Lie group that we will need.
Definition 6.1.
A Lie algebra is semisimple if it does no have a non-trivialsolvable ideal.
Definition 6.2.
A Lie group is semisimple if its Lie algebra is semisimple.
Theorem 6.3.
Theorem 6.31, 6.46 from [20] , Theorem 3.1, Lemma 3.3,Chap XV from [18]
Let G be a semisimple Lie group with finitely many component. Then thereexist subgroups K , A and N such that the multiplication map A × N × K → G given by ( a, n, k ) ank is a diffeomorphism. The groups A and N aresimply connected abelian and simply connected nilpotent respectively, and A normalizes N .Furthermore, Z ( G ) , the center of G , is contained in K , and there isan isometrically embedded connected, simply connected abelian Lie group V = Z ( K ) , a compact subgroup T < K such that K is the semidirect product V ⋊ T , and that Z ( G ) /V is compact.Finally, the group T is maximal compact in G and any compact subgroupcan be conjugated into T .Remark . This decomposition of G = AN K is called an
Iwasawa decom-position . SSOUAD-NAGATA DIMENSION OF CONNECTED LIE GROUPS 15
Remark . By definition, a semisimple Lie algebra has no center. Since asemisimple Lie group is one for which its Lie algebra is semisimple, it followsthat the center of a semisimple Lie group is necessarily discrete, and thatthe K in a Iwasawa decomposition is compact if and only if the center isfinite. Corollary 6.6.
Let G be a semisimple Lie group with finite center, and AN K be a Iwasawa decomposition. Then G is bilipschitz diffeomorphic to AN × K .Proof. The map G → AN × K sends an element p = ank ( an, k ),and we only need to check that the ratio between d ( a n k , a n k ) and d (( a n , k ) , ( a n , k )) is bounded. This will follow from the following cal-culation. Take k i ∈ K , g i ∈ AN , d ( g k , g k ) = d ( e, k − g − g k ) = d ( e, k − k (cid:0) k − g − g k (cid:1) )As K is compact,max { d ( e, g ) d ( e, kgk − ) , d ( e, kgk − ) d ( e, g ) , g ∈ G, k ∈ K } < ∞ . (cid:3) Notice that when the center of G is finite then it is easy to compute theAssouad-Nagata dimension applying the previous corollary and the fact that K is compact in such case. The idea of the proof for semisimple Lie groups(not necessarily with finite center) will be to reduce the general case tothe finite center case using universal covers. So we have to focus in thefundamental group of a Lie group and its universal cover. Proposition 6.7. [22] (Proposition 1.6.4) The fundamental group of anyLie group is a subgroup of its center.Proof.
Let G be a Lie group and ˜ G its universal cover. Then π ( G ) is anormal discrete subgroup of ˜ G and G = ˜ G/π ( G ). Since π ( G ) is normal,for a fixed d ∈ π ( G ) we can define a map α : ˜ G → π ( G ) by g gdg − .Since this is a continuous map and π ( G ) is discrete, it follows that theimage must be a point, namely d . So this shows that π G lies in the centerof ˜ G . (cid:3) Lemma 6.8.
Let G be a semisimple Lie group, and AN K a Iwasawa decom-position. Then ˜ G , the universal cover of G , is also semisimple and AN ˜ K isa Iwasawa decomposition of ˜ G where ˜ K is the universal cover of K .Proof. That ˜ G is semisimple follows from the fact that a covering map is alocal diffeomorphism and a Lie group is defined to be semisimple if its Liealgebra is semisimple. Since A and N are simply connected, it follows that π ( G ) = π ( K ), so the second claim follows. (cid:3) The next lemma will be used also in the next section. It is just a technicallemma about abelian Lie groups. It will help to compute the Assouad-Nagata dimension of a group of the form G/ Γ with G an abelian Lie groupand Γ a discrete subgroup. Lemma 6.9.
Let Γ be a discrete subgroup of Z k × R l . Then (cid:0) Z k × R l (cid:1) / Γ is quasi-isometric to Z j for some j .Proof. As Z k × R l is q.i. to Z k × Z l , suffice to describe (cid:0) Z k × Z l (cid:1) / Γ. SinceΓ is a graph of a homomorphism from Z min { k,l } → Z max { k,l } , by expressing (cid:0) Z k × Z l (cid:1) in terms of a basis that contains a basis representing Γ as a graph,the quotient (cid:0) Z k × Z l (cid:1) / Γ = Z j for some j . (cid:3) Now we provide the main theorem of this section.
Theorem 6.10.
Let G ba a semisimple Lie group with a Riemannian met-ric d . Then asdim AN ( G, d ) = dim ( G/T ) where T is a maximal compactsubgroup of G Proof.
By corollary 3.6 of [10] we have dim ( G/T ) = asdim ( G ) ≤ asdim AN ( G ).So the unique thing we have to prove is asdim AN ( G ) ≤ dim ( G/T ).By Theorem 6.3, G = AN K , where K = V ⋊ T , and V is connected,simply connected abelian Lie group and T is a maximal compact subgroup.We cannot apply Proposition 3.5 because AN is not normal in G . Instead,we observe that G/Z ( G ) is a semisimple Lie group with trivial center, andsince Z ( G ) < K , it follows that ˆ G = AN ( K/Z ( G )) is a Iwasawa decom-position for ˆ G , where K/Z ( G ) is compact. By Corollary 6.6, we have abilipschitz diffeomorphism ˆ φ between ˆ G and AN × K/Z ( G ). Lifting ˆ φ up to G we have a bilipschitz diffeomorphism φ between G and AN × K followingcommutative diagram. G φ −−−→ AN × K y y ˆ G ˆ φ −−−→ AN × K/Z ( G )So now asdim AN ( G ) ≤ asdim AN ( AN ) + asdim AN ( K ). But Theorem 6.3says that K/V is compact and V is isometrically embedded in K , it followsthat asdim AN ( K ) = asdim AN ( V ), and so asdim AN ( G ) ≤ asdim AN ( AN ) + asdim AN ( V ) = dim ( AN ) + dim ( V ), since AN and V are both connected,simply connected solvable Lie groups. But dim ( AN )+ dim ( V ) = dim ( G/T ). (cid:3) Assouad-Nagata dimension of connected Lie groups
In this section we will prove the main result of this paper. We will re-duce the general case to the computations made for nilpotent, solvable andsemisimple Lie groups in the previous sections. In some sense the proof is
SSOUAD-NAGATA DIMENSION OF CONNECTED LIE GROUPS 17 similar to the one of solvable Lie group. We will need to study an exactsequence generated by some subgroup that is exponentially distorted in ourLie group. Hence we will need the following concept from [11]:
Definition 7.1.
Two Lie groups are said to be locally isomorphic if theyhave isomorphic Lie algebras.
Remark . If G , G are two Lie groups with isomorphic Lie algebras, thenthere is a simply connected Lie group ˜ G and discrete subgroups Γ , Γ Let G be a connected Lie group and let R exp ( G ) be its expo-nential radical. Then R exp ( G ) is strictly exponentially distorted in G and itis contained in the nilpotent radical of G .Proof. See Theorem 6.3. of [11] (cid:3) So now we need to study the quotient G/R exp ( G ). The key will be tounderstand what means to be locally isomorphic from a large scale point ofview. For such purpose we need the following: Lemma 7.5. Let G be simply connected semisimple Lie group. Then G isquasi-isometric to G/Z ( G ) × Z ( G ) .Proof. We already know that G is quasi-isometric to G/ ˜ K × ˜ K where ˜ K isthe appropriate factor in the Iwasawa decomposition of G . Since Z ( G ) < ˜ K is a co-compact subgroup, the following sequence1 → G/Z ( G ) → G/ ˜ K → ˜ K/Z ( G )shows that G/Z ( G ) is quasi-isometric to G/ ˜ K , to which the desired claimfollows. (cid:3) Theorem 7.6. ( Theorem 2.3 from [18] )Leg G be a topological group containing a vector group V as a closed normalsubgroup. If G/V is compact, then V is a semidirect product of G . Corollary 7.7. Let G be a Lie group and S < G a normal solvable Liegroup. If G/S is compact, then G is a semidirect product between a solvablegroup with the same topological dimension as S and a compact group.Proof. We induct on the length of the commutator series of S . The basecase is when S is abelian and this is given by Theorem 7.6. Now supposethis is true for solvable groups of length j − 1. Since S is normal in G , itfollows that Z ( S ), is also normal in G . Therefore we can apply the inductivehypothesis to the pair G/Z ( S ) with subgroup S/Z ( S ) and conclude that G/Z ( S ) = ˆ S ⋊ ˆ T , where ˆ S is a solvable group and ˆ T a compact group. Write ˆ S = ˜ S/Z ( S ), and ˆ T = A/Z ( S ) for subgroups ˜ S, A < G . Then G = ˜ S ⋊ A .We can apply Theorem 7.6 to A and Z ( S ) to conclude that A = Z ( S ) ⋊ T for some compact subgroup T . Therefore G = ˜ S ⋊ ( Z ( S ) ⋊ T ). (cid:3) Remark . Note that the center of a connected, simply connected solvableLie group is at most exponentially distorted. To justify this claim, we needtwo observations. First, for such a solvable Lie group S , the exponential mapis a diffeomorphism and we can coordinatize S by its Lie algebra. We nowdescribe the second observation which is taken from Section 2 of [3]. Given s a solvable Lie algebra, there exist a vector subspace v and a nilpotentideal n (the nilpotent radical) such that as a vector space, s = v ⊕ n , andfor each x ∈ v , v lies in the zero generalized eigenspace of ad ( x ). So if theLie algebra of a subgroup lies in v , then the distance of the subgroup isat most polynomially distorted. On the other hand, since n is a nilpotentideal, whenever the Lie algebra of a subgroup lies in n , the distance of thesubgroup is at most exponential-polynomially distorted.The claim now follows since we can always split the center of S into adirect sum of subgroups, whose Lie algebras lie in v and n respectively. Theorem 7.9. Let G be a connected Lie group. Then asdim AN ( G ) = dim ( G/C ′ ) , where C ′ is a maximal compact subgroup of G .Proof. By Corollary 3.6 of [10] we have dim ( G/C ) = asdim ( G ) ≤ asdim AN ( G ),so suffice to show that asdim AN ( G ) ≤ dim ( G/T ).Let R exp ( G ) be the exponential radical of G . Then we have the followingshort exact sequence.1 → R exp ( G ) → G → G/R exp ( G ) → , (4)where quotient U = G/R exp ( G ) is locally isomorphic to a direct productof a semisimple Lie group and a solvable Lie group. By Remark 7.2 thereare simply connected semisimple Lie group L and a simply connected solv-able Lie group S such that U is isomorphic to ( L × S ) / Γ for some discretesubgroup Γ < Z ( L × S ) = Z ( L ) × Z ( S ). By Lemma 7.5 we have U q.i. ∼ L/Z ( L ) × ( Z ( L ) × S ) / Γ . (5)In addition, we also have1 → ( Z ( L ) × Z ( S )) / Γ → ( Z ( L ) × S ) / Γ → S/Z ( S ) → , (6)where we note that Z ( S ) is at most exponentially distorted in S by Remark7.8, and the same is true for ( Z ( L ) × Z ( S )) / Γ in ( Z ( L ) × S ) / Γ. SSOUAD-NAGATA DIMENSION OF CONNECTED LIE GROUPS 19 Putting (4), (5) and (6) together we see that asdim AN ( G ) ≤ asdim AN ( R exp ( G )) + asdim AN ( G/R exp ( G ))= dim ( R exp ( G )) + asdim AN ( U ) ≤ dim ( R exp ( G )) + asdim AN ( L/Z ( L )) + asdim AN (( Z ( L ) × S ) / Γ)= dim ( R exp ( G )) + dim ( L/Z ( L )) + asdim AN (( Z ( L ) × S ) / Γ) ≤ dim ( R exp ( G )) + dim ( L/Z ( L ))+ asdim AN (( Z ( L ) × Z ( S )) / Γ) + asdim AN ( S/Z ( S ))= dim ( R exp ( G )) + dim ( L/Z ( L )) + dim (( Z ( L ) × Z ( S )) / Γ) + dim ( S/Z ( S ))Now let L = AN K be a Iwasawa decomposition, and K = V ⋊ T as givenby Theorem 6.3. Then dim ( AN ) = dim ( L/Z ( L )), and U = AN (( V × S ) / Γ ⋊ T ) . But ( V × S ) / Γ is diffeomorphic to a product E × T where T is a maximalsubgroup in ( V × S ) / Γ, and E is a manifold diffeomorphic to R n , where n = dim ( E ) = dim ( S/Z ( S )) + dim (( Z ( L ) × Z ( S )) / Γ). In other words, U = G/R exp ( G ) is diffeomorphic to AN × E × C where C is the compactsubgroup T ⋊ T . By Corollary 7.7 we see that the subgroup ˜ T < G suchthat ˜ T /R exp ( G ) = C is a semidirect product between a solvable group S ofthe same dimension as R exp ( G ) and a maximal subgroup C ′ < G . Therefore dim ( G/C ′ ) = dim ( AN ) + dim ( E ) + dim ( R exp ( G ))= dim ( L/Z ( L )) + dim ( S/Z ( S )) + dim (( Z ( L ) × Z ( S )) / Γ) + dim ( R exp ( G )) (cid:3) We now have an analogous result to Corollary 3.6 of [10]. Corollary 7.10. Let Γ be a cocompact lattice in a connected Lie group G .Then asdim AN (Γ) = dim ( G/K ) , where K is a maximal compact subgroup.Proof. Since Γ is quasi-isometric to G , it follows from Theorem 6 that asdim AN (Γ) = asdim AN ( G ) = dim ( G/K ) (cid:3) For the global Assouad-Nagata dimension the result is the following: Corollary 7.11. Let G be a connected Lie group equipped with a Riemann-ian metric. Then dim AN ( G ) = dim ( G ) .Proof. On one hand we have dim AN ( G ) = max { asdim AN ( G ) , cdim ( G ) } , bycorollary 4.5 we get cdim ( G ) ≤ dim ( G ). Combining this inequality with theprevious lemma we get dim AN ( G ) ≤ dim ( G ).On the other hand dim ( G ) ≤ dim AN ( G ) (see the original paper of As-souad [1]). (cid:3) The asymptotic Assouad-Nagata dimension gives us an obstruction for afinitely generated group to be quasi-isometrically embeddable in a connectedLie group in particular quasi-isometric to a lattice. For example for certainclasses of wreath products we have the following: Corollary 7.12. Let H be a finite group ( H = 1 ) and let G be a finitely gen-erated group such that the growth is not bounded by a linear function. Then H ≀ G equipped with a word metric can not be embedded quasi-isometricallyin any connected Lie group.Proof. In corollary 5.2 of [4] it was proved that asdim AN ( H ≀ G ) = ∞ . ByTheorem 7.9 the asymptotic Assouad-Nagata dimension of any connectedLie group is finite. 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