Assuad-Nagata dimension of nilpotent groups with arbitrary left invariant metrics

ASSOUAD-NA GA T A DIMENSION OF NILP OTENT GR OUPS WITH ARBITRAR Y LEFT INV ARIANT M ETRICS J. HIGES Abstract. Suppose G is a coun table, not necessarily finitely generated, group. W e sho w G admits a proper, left-inv ariant metric d G suc h that the Assouad- Nagata dimension of ( G, d G ) is infinite, provide d the cen ter of G is not lo cally finite. As a corollary we solv e tw o problems of A.D r anishniko v. Contents 1. Int ro duction 1 2. Preliminarie s 3 3. Low er bounds for Asso uad-Nagata dimension 4 4. Main results 6 References 9 1. In troduction The asymptotic dimension was in tro duced b y Gro mov in [13] as a coarse in- v ar iant to study the geometric structure of fin itely generated groups. W e refer to [2] for a s urvey abo ut this topic. Closely related with the asymptotic dimension is the asymptotic dimension of linear type. It is a lso called asymptotic Assouad- Nagata dimension in hono r of Patrice Asso ua d who introduced it in [1] from the ideas o f Nagata. Such dimension can b e co nsidered a s the linear version of the asymptotic dimension. In recent years a part of the resear ch activity was fo cused on this dim ension and its relationship wit h the asymptotic dimension (see for ex- ample [16], [9 ], [10], [3], [4], [6], [5], [17] [12], [1 5]). One of the main pr oblems of int erest consists in s tudying the differences betw een the asy mptotic dimension a nd the asymptotic Asso uad-Nagata dimension in the context of the g eometric gr oup theory . In particular there are t wo main questions: (1) Given a finitely generated group G with a word metr ic d G . Ar e the as y mp- totic dimensio n and the asymptotic Assouad-Nagata dimension of ( G, d G ) equal? (2) In the case the t wo dimensions differ, m ust their difference b e infinite? Date : April 23, 2008. 2000 Mathema tics Subje ct Classific ation. Pri mary 54F45; Secondary 55M10, 54C65. Key wor ds and phr ases. Assouad-Nagata dimension, asymptotic dimension, nilp oten t groups. The author i s supp orted by Grant AP2004-2494 from the Ministerio de Educaci´ on y Ciencia, Spain. He also thanks Jerzy Dydak and N . Brodskyi for helpful comment s and supp ort. 1 2 J. HIGES It is k nown that the first ques tio n has an affirmative answer for ab elian gro ups , finitely presen ted g roups o f a symptotic d imension one, and for h yp erb olic groups. But in general the answer is neg ative. Now ak ([17]) fo und for ev ery n ≥ 1 a finitely generated gr oup of asymptotic dimension n but of infinite asymptotic Assouad- Nagata dimension. As far as the a uthor knows the second questio n is still o pen. There is no example of a finitely generated group such that the asymp otic dimensio n is stric tly sma ller than the a s ymptotic Assoua d-Nagata dimension but bo th a re finite. In [15] the second question was solved in a more general context. It was prov ed tha t for every n and m there exists a c o unt able ab elian gr oup(non finitely genera ted) with a pro per left inv aria n t metric such that the gr oup is of asymptotic dimension n but of asympotic Assouad-Naga ta dimension equa l to n + m . Pro pe r left inv a riant metrics are natural generalizations o f word metr ic s . Therefore it is natural to ask the sa me tw o problems in the case of finitely gener ated gro ups equipp ed with prop er left inv ar iant metrics. The aim of this no te is to study the b ehaviour of the asymptotic Ass ouad-Naga ta dimension in nilpotent groups with prop er left inv ar iant metrics. It is highly lik ely that both dim ensions co incide in nilpotent groups with w ord metrics (see [7]). F or example in [12] it was prov ed their coincidence fo r the Heisenberg g roup. W e will show that for e very nilpo ten t group it is p ossible to find a prop er left inv ariant metric such that b oth dimensio ns are different. Dranishniko v in [7] asked a bo ut what can be considered a spec ia l case: Problem 1. 1. (Dranishniko v [7]) Do es dim AN ( Z , d ) = 1 for every left inv ariant metric on Z ? In relation to the a bove pro blem he asked the following: Problem 1.2. (Dranishnikov [7 ]) Does dim AN (Γ × Z ) = dim AN (Γ) + 1 for any left inv ar iant metric o n Z ? Notice that in [9], Dranishniko v a nd Smith prov ed that asdim AN ( G × Z ) = asdim AN ( G ) + 1 for every finitely generated group G but in this case the metrics consideres were t he word metrics. Our main theor em deals with a large r class of g roups than nilp otent ones: Theorem 1.3. If G is a gr oup such that its c enter is not lo c al ly finite then ther e exists a pr op er left invariant metric d G such that asdim AN ( G, d G ) = ∞ . It is clea r that if we apply the pre v ious theorem to G = Z , the t wo questions of Dranishniko v are solved in neg a tive. The key ingr edient of the pro o f was int ro duced by the a utho r in [1 5]. In such pap er it was shown that if ther e exis ts a seq ue nc e of isometric embeddings (up to dilatation) of ba lls { B (0 , k i ) } i ∈ N of Z m int o a metric space X where the sequence of radious k i tends to infinity then the a symptotic Ass ouad-Naga ta dimension of X is grea ter than m . Tha t result should b e viewed a s a pplying the philoso phy of Wh yte [19] in a rather restricted for m. Instead of lo o king for s ubsets of a gr oup G that a re bi-Lipschitz equiv alent to Z n , we are c o nstructing groups that c ontain rescaled copies of larg e ba lls in Z n . Section 3 is devoted to a key ing redient used in Section 4 to prese n t pro ofs of main results. ASSUAD-NAGA T A DIM ENSION OF NILPOTENT GR OUPS WITH ARBITRAR Y LE FT INV ARIANT ME TRICS 3 2. P r el iminaries Let s b e a p ositive real nu mber. An s -sc ale chain (or s -pa th) betw een tw o po int s x and y of a metric spa ce ( X , d X ) is defined as a finite sequence points { x = x 0 , x 1 , ..., 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 b e s -sc ale c onne cte d if there exists an s -scale chain con tained in S for every t wo elemen ts of S . Definition 2.1. A metric space ( X , d X ) is said to b e of asymptotic dimension at most n (notation asdim( X , d ) ≤ n ) if there is an incr easing function D X : R + → R + such that for a ll s > 0 there is a cov er U = {U 0 , ..., U n } so that the s -scale connected comp onents of eac h U i are D X ( s )-bo unded i.e. the diameter o f suc h comp onents is bo unded by D X ( s ). The function D X is called an n -dimensional c ontr ol funct ion for X . Depe nding on the type of D X one can define the following tw o inv ariants: A metric space ( X , d X ) is said to be o f Assouad-Nagata dimension at most n (notation dim AN ( X, d ) ≤ n ) if it has a n n -dimensiona l cont rol function D X of the form D X ( s ) = C · s with C > 0 so me fixed constant. A metric space ( X , d X ) is said to b e of asymptotic Assouad-Nagata d imension a t most n (notatio n asdim 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 > 0 and k ∈ R t wo fixed constan ts. It is clear from the definition that for every metr ic s pace ( X , d X ), asdim ( X , d X ) ≤ asdim AN ( X, d X ). One important fact about the as y mptotic dimensio n is that it is in v aria n t under coarse equiv alences. G iven a map f : ( X, d X ) → ( Y , d Y ) betw een t wo metrics spaces it is said to b e a c o arse emb e dding if there exis t tw o increa s ing functions ρ + : R + → R + and ρ − : R + → R + with lim x →∞ ρ − ( x ) = ∞ such that: ρ − ( d X ( x, y )) ≤ d Y ( f ( x ) , f ( y )) ≤ ρ + ( d X ( x, y )) for every x, y ∈ X . Now a c o arse e quivalenc e betw e en tw o metrics spaces ( X, d X ) and ( Y , d Y ) is defined as a coar se embedding f : ( X , d X ) → ( Y , d Y ) for which there exis ts a co nstant K > 0 such that d Y ( y , f ( X )) ≤ K for every y ∈ Y . If ther e exis ts a coar se e q uiv alence betw een X and Y bo th spaces ar e said to be c o arsely e quivalent . The metrics spaces in which w e are in ter ested are coun table groups with prop er left inv ar iant metrics. Definition 2.2. A metric d G defined in a gro up G is s aid to be a pr op er left invariant m et ric if it satisfies the following conditions: (1) d G ( g 1 · g 2 , g 1 · g 3 ) = d G ( g 2 , g 3 ) for every g 1 , g 2 , g 3 ∈ G . (2) F or every K > 0 the num ber of elements g of G such that d (1 G , g ) G ≤ K is finite. The following basic result of Smith will b e us ed to get the main theo r em. Theorem 2.3. (S mith [18] ) Two pr op er left invariant metrics define d in a c ountable gr oup ar e c o arsely e quivalent. One wa y of c o nstructing prop er left inv aria nt metrics in a co untable gro up is via prop er norms. Definition 2 .4. A map k · k G : G → R + is c alled t o be a pr op er norm if it sa tisfies the following conditions: 4 J. HIGES (1) k g k G = 0 if and only if g = 1 G . (2) k g k G = k g − 1 k G for every g ∈ G . (3) k g · h k G ≤ k g k G + k h k G for every g, h ∈ G . (4) F or every K > 0 the n umber of element s o f G such that k g k G ≤ K is finite. It is clear that there is a o ne-to-one cor resp ondence b etw een pro per norms and prop er left inv ariant m etrics. Now we are interested in metho ds to get prop er norms with some sp ecia l prop er- ties. In ge neral this tas k is not eas y . In this pap er we will use the metho d of weigh ts describ ed by Smith in [18]. Let S be a symmetric s ystem of generato rs(p ossibly infinite) of a c o unt able g roup G and let ω : L → R + be a function( weight function or system of weights ) that satis fie s : (1) ω ( s ) = 0 if a nd only if s = 1 G (2) ω ( s ) = w ( s − 1 ). (3) ω − 1 [0 , N ] is finite for every N . Then the function k · k w : G → R + defined by: k g k w = min { n X i =1 w ( s i ) | x = Π n i =1 s i , s i ∈ S } is a pr o pe r norm. Suc h nor m will b e called the prop er norm gener ate d by the system of weights ω and the as so ciated left inv a riant metric will b e the left invaria nt metric gener ate d by the system of weights ω . R emark 2.5 . (1) If we define w ( g ) = 1 for all the elements g ∈ S of a finite generating system S ⊂ G ( G a finitely generated group) w e will obtain the usual word metric. (2) Notice that if we ha v e a prop er norm k · k G in a countable group G and we take the system of weiths defined by ω ( g ) = k g k G then the pr op er norm k · k ω generated by this system of weigh ts coincides with k · k G . (3) W e can constr uct easily in teger v alued prop er left inv ariant metrics by getting weigh t functions with integer range. 3. Lower bounds for Asso uad-Naga t a dimension The a im o f this section is to give a sufficien t co ndition in a metric space ( X , d X ) that implies asdim AN ( X, d X ) ≥ n for so me n . T o get that conditio n we will use the notion of asymptotic cone. Let ( X , d X ) b e a metric space. Given a non-pr incipal ultrafilter ω of N and a sequence { x n } n ∈ N of po ints of X , the ω -limit o f { x n } n ∈ N (notation: lim ω x n = y ) is an element y of X such that fo r every neighborho o d U y of y the s et F U y = { n | x n ∈ U y } belo ngs to ω . It can b e proved easily that the ω -limit of a seq ue nc e alwa ys ex is ts in a compact space. Assume ω is a non p rincipal ultrafilter of N . Let d = { d n } n ∈ N be an ω -divergen t sequence of p ositive real n umbers a nd let c = { c n } n ∈ N be any sequence of elements of X . No w we can cons truct the asymptotic c one (notatio n: C one ω ( X, c, d )) o f X as follows: ASSUAD-NAGA T A DIM ENSION OF NILPOTENT GR OUPS WITH ARBITRAR Y LE FT INV ARIANT ME TRICS 5 Firstly define the set of all s e quences { x n } n ∈ N of elements of X suc h that lim ω d X ( x n ,c n ) d n is bo unded. In such set take t he pseudo metr ic given b y: D ( { x n } n ∈ N , { y n } n ∈ N ) = lim ω d X ( x n , y n ) d n . By ident ifying sequences whos e dista nces is 0 we get the metric space C one ω ( X, c, d ). Asymptotic cones were firstly intro duced by Gro mov in [13]. There has b e e n a lot o f resea rch r elating proper ties o f groups with top olog ical pro per ties o f its asymptotic cone s . F or exa mple a finitely genera ted group is virtually nilp otent if and only if all its a s ymptotic cones ar e lo cally finite [14] or a g r oup is hyperb olic if and only if all o f its a symptotic cones are R -tr ees ([13] and [11]). In [12] it w as s hown the following rela tionship b etw een the topo logical dimensio n of an asymptotic cone and th e asymptotic Assouad-Na gata dimension of t he space: Theorem 3.1. [Dydak, H iges [12] ] dim ( C one ω ( X, c, d ) ≤ dim AN ( C one ω ( X, c, d ) ≤ asdim AN ( X, d X ) for any m et ric sp ac e ( X, d X ) and every asymptotic c one C one ω ( X, c, d ) . W e r ecall now the follo wing: Definition 3.2 . A function f : ( X , d X ) → ( Y , d Y ) be t ween metric spaces is said to b e a dilatation if there exists a constant C ≥ 1 such that d Y ( f ( x ) , f ( y )) = C · d X ( x, y ) for every x, y ∈ X . The num ber C will b e called the dilatation c onstant . In the following prop osition dilatatio ns will be fr om ba lls o f Z n with the l 1 -metric to general metric spac e s. Prop ositio n 3. 3. L et ( X, d X ) b e a m et ric sp ac e and let { k m } m ∈ N b e an incr e asing se qu enc e of natur al n u mb ers. If for some n ∈ N ther e is a se quenc e of dilatations { f m } ∞ m =1 of the form f m : B n (0 , k m ) → ( X , d X ) with B n (0 , k m ) ⊂ Z n the b al l of r adious k m then ther e exists an asymptotic c one C one ω ( X, c, d ) of ( X , d X ) su ch that [ − 1 , 1 ] n ⊂ C one ω ( X, c, d ) . Pro of. Suppose given ( X , d X ) and { k m } m ∈ N as in the hypo thesis. Let us prov e firstly the ca se n = 1. Assume that { C m } m ∈ N is the sequence of dilata tion constants of { f m } m ∈ N . T ake ω some ultrafilter of N and define c = { f m (0) } ∞ m =1 and d = { d m } ∞ m =1 with d m = C m · k m . W e will prove that C one ω ( X, c, d ) c ontains [ − 1 , 1]. F or each t ∈ [ − 1 , 1 ] let A t m be the subse t {− k m , ..., k m } such that x ∈ A t m if and only if the distance betw een C m · x d m and t is minimum . Notice that this implies that the distance b etw een C m · x and d m · t is less thatn C m . T ake now the sequence { r t m } ∞ m =1 where r t m is the infimum of A t m . Define the map g : [ − 1 , 1 ] → C one ω ( X, c, d ) by g ( t ) = x t if the sequence { f m ( r t m ) } ∞ m =1 is in the cla ss x t . As: lim ω d ( f m (0) , f m ( r t m )) d m = lim ω C m · | r t m | d m ≤ lim ω C m · k m d m = 1 the ma p is well defined. Let us prove it is an isometr y . F r om the definition of r t m we get that if t 1 < t 2 then r t 1 m ≤ r t 2 m what implies lim ω d ( f m ( r t 1 m ) ,f m ( r t 2 m )) d m = lim ω C m ( r t 2 m − r t 1 m ) d m . So the unique thing we need to show is that lim ω C m · r t m d m = t for every t . Notice that we hav e lim ω C m d m = 0 as lim ω k m = ∞ but lim ω C m · k m d m = 1. This 6 J. HIGES implies that g iven ǫ > 0 ther e exis ts G ǫ ∈ ω such that C m d m < ǫ for e very m ∈ G ǫ . Therefore b y the c hoice of r t m if m ∈ G ǫ we have | C m · r t m − d m · t | < C m and then | C m · r t m d m − t | < C m d m ≤ ǫ . Now let us do the general case. Let ( s 1 , ..., s n ) ∈ [ − 1 , 1] n . By the previous case we get that for every j = 1 , ..., n there ex ists a sequence { r s j m } m ∈ N with r s j m ∈ {− k m , ..., k m } s uc h that lim ω C m · r t m d m = s j . In a similar way as b efore we construct a map g : [ − 1 , 1] n → C one ω ( X, c, d ) by defining g ( s 1 , ..., s m ) as the class tha t contains the s equence { f m ( r s 1 m , ..., r s n m ) } ∞ m =1 . T o finish the pr o of it will b e enoug h to check that for every s, t ∈ [ − 1 , 1 ] n with s = ( s 1 , ..., s n ) and t = ( t 1 , ..., t n ), the following eq uality h olds: lim ω d X ( f m ( r s 1 m , ..., r s n m ) , f m ( r t 1 m , ..., r t n m )) d m = n X i =1 | s i − t i | As f m is a dilatation of cons ta n t C m we c a n write: lim ω d X ( f m ( r s 1 m , ..., r s n m ) , f m ( r t 1 m , ..., r t n m )) d m = n X i =1 lim ω C m · | r s i m − r t i m | d m And a gain b y the case n = 1 we can deduce that the last term satisfies the equality: n X i =1 lim ω C m · | r s i m − r t i m | d m = n X i =1 | s i − t i | .  F ro m the pr evious prop osition we can get the follo wing result that is o ne of the ingredients o f the main theorem. Corollary 3. 4. If for met ric sp ac e ( X, d X ) and fo rm some n ∈ N ther e exists a se quenc e of dilatations f m : B n (0 , k m ) → ( X, d X ) with lim m →∞ k m = ∞ and B n (0 , k m ) ⊂ Z n the b al l of r adious k m then asdim AN ( X, d X ) ≥ n . Pro of. By prop osition 3.3 we g e t that there ex is ts a n as y mptotic cone of X such that [ − 1 , 1] n ⊂ C one ω ( X, c, d ). Applying theorem 3.1 we obtain immediately: n ≤ di m ( C one ω ( X, c, d )) ≤ di m AN ( C one ω ( X, c, d )) ≤ asdim AN ( X, d X )  4. Main resul ts The idea of the pro of of the main theor em consists in crea ting a metric induc- tively . In e ach step we will cons truct a new metr ic that satisfies tw o c onditions. First condtion says that the new metric do er not change a sufficiently large ball of the old metric. Second condtion implies there is a dilatation f rom some s ufficie ntly large ball of Z n int o the gr oup with the new metric. In fact that dila tation will be the restriction of s ome ho momorphism f : Z n → G . Then we will apply cor ollary 3.4. The following lemma could b e considere d as the induction step. Lemma 4. 1. L et G b e a finitely gener ate d gr oup such that its c enter is not lo- c al ly fin ite. L et d G b e a pr op er left invariant metric. In su ch c ondtions for every k , m, R ∈ N ther e exists a pr op er left invariant metric d ω that satisfies the fol lowing c onditions: ASSUAD-NAGA T A DIM ENSION OF NILPOTENT GR OUPS WITH ARBITRAR Y LE FT INV ARIANT ME TRICS 7 (1) k g k ω ≤ k g k G . (2) k g k G = k g k ω if k g k ω ≤ R . (3) Ther e is an homomorphi sm f : Z m → G su ch t hat t he r estriction f | B (0 ,k ) of f to t he b al l r adious k is a dilatation in ( G, d ω ) . Pro of. Suppose k , m and R given and let a and C b e tw o natura l num ber s that satisfy: R < C < a 2 · k · m 2 . As the center of G is not lo cally finite there exists an element g in the cent er of infinite o r der. The restriction of the metr ic d G to the subg roup g enerated by g defines a pr op er left inv ariant metric in Z = < g > . By theorem 2 .3 we know that t wo prop er left inv aria nt metrics defined in a gr oup ar e coar sely equiv a le n t, hence we ca n find an integer num ber h 1 ∈ Z such that if | h | ≥ | h 1 | then k g h k G ≥ a . Let p 1 = 1 and for every j = 2 ...m we define p j as a sufficiently la rge num ber that sa tisfies P j − 1 i =1 (2 · k · m )2 p i < 2 p j . T ake now the finite set of integer num bers { h 1 , ..., h m } with h j = 2 p j · h 1 for every j = 2 , ..., m . In this s ituation we c r eate the norm k · k ω generated by the following system of w eights: ω ( z ) = ( k z k G if z 6 = g ± h i for every i = 1 ...m C o therwise By the choice of C and { h 1 , ..., h m } it is clear that the two first conditions of the lemma ar e satisfied. T o pr ov e the third condition we define the homomorphism f : Z m → G a s f ( x 1 , ..., x m ) = g h with h = P m i =1 x i · h i . Let us show that the restriction f | B (0 ,k ) : B (0 , k ) → G to the ball of radious k is a dilatation of constant C . It will b e enough to chec k that: k g h k ω = m X i =1 | x i | · C if h = m X i =1 x i · h i and | x i | ≤ k The r e a soning will be by c ontradiction. Supp ose there exis ts an element o f the form g h with h = P m i =1 x i · h i and | x i | ≤ k suc h that k g h k ω < P m i =1 | x i | · C . This implies that there exist and r = P m i =1 y i · h i and an s ∈ G such that g h = g r · s and: k g h k ω = m X i =1 | y i | · C + k s k G . Notice that | y i | ≤ k · m . Ther e are now t w o pos sible cases : Case s = 1 G : In this situation we hav e P m i =1 | y i | · C < P m i =1 | x i | · C so there exists an i such that x i 6 = y i . L et j = max { i | x i 6 = y i } . F ro m the fact g h = g r we can deduce ( x j − y j ) · h j = P j − 1 i =1 ( y i − x i ) · h i it means ( x j − y j ) · 2 p j · h 1 = P j − 1 i =1 ( y i − x i ) · 2 p i · h 1 and then: 2 p j ≤ | x j − y j | · 2 p j ≤ j − 1 X i =1 | y i − x i | · 2 p i ≤ j − 1 X i =1 | y i | + | x i | · 2 p i ≤ j − 1 X i =1 2 p i · ( k · m + k ) < 2 p j A contradiction. Therefor e the first case is not p oss ible. Case 2: s 6 = 1 G . In this case we have g ( h − r ) = s and h − r 6 = 0 what implies | h − r | ≥ | h 1 | and hence k s k G ≥ a . But from the fact k g h k ω = P m i =1 | y i | · C + k s k G < 8 J. HIGES P m i =1 | x i | · C we c an deduce: a ≤ k s k G < m X i =1 ( | x i | − | y i | ) · C ≤ m X i =1 ( k + k · m ) · C ≤ 2 · k · m 2 · C . Therefore C ≥ a 2 · k · m 2 and this contradicts the choice o f a and C .  Here is the main theorem. Theorem 4.2. If G is a gr oup such that its c enter is not lo c al ly finite then ther e exists a pr op er left invariant m etric d G such that asdim AN ( G, d G ) = ∞ . Pro of. W e will use cor o llary 3.4 and the pre vious lemma. T ake any in teger v alued prop er left in v ar iant metric d in G (see rema rk 2.5) a nd some incr easing sequences { k i } i ∈ N and { M i } i ∈ N of natur a l num b ers. Let us co nstruct the metric d G of the theorem by an inductiv e pro cess. Step 1: Apply the previous lemma to d with k = k 1 , m = 1 a nd R = M 1 . W e obtain a pro pe r left in v ar iant metric d ω 1 such that the ball B ω 1 (1 G , R 1 ) is equa l to the ball of ra dious R 1 of d . Also ther e exists a dilatation f : B (0 , k 1 ) → G from the ball or ra dio us k 1 of Z to G . Induction Step: Supp ose now that we hav e co n tructed a finite sequence of prop er left inv ar iant metrics L = { d ω 1 , ..., d ω n } a nd a finite sequence of na tur al num b ers R 1 < R 2 < ... < R n that satisfy the following conditions: (1) k g k ω i ≤ k g k ω i − 1 . (2) k g k ω i = k g k ω i − 1 if k g k ω i ≤ R i (3) There exists an homomorphism f i : Z i → G such that the restr iction f i | B (0 ,k i ) is a dilatation in ( G, d ω i ) for every i = 1 , ..., n (4) diam ( f i ( B (0 , k i )) < R i +1 for every i = 1 , ..., n − 1 In these conditions define R n +1 = max { M n +1 , R n + 1 , diam ( f n ( B (0 , k n )) } and apply the previo us lemma to d ω n with k = k n +1 , m = n + 1 and R = R n +1 . W e hav e now a new pr op er left inv aria nt metric d ω n +1 . It is clear that the new finite sequence of pr op er left inv ar iant metrics { d ω n +1 } ∪ L and the ne w finite sequence of n umbers R 1 < R 2 < ... < R n < R n +1 satisfy the same four co nditio ns . Repe a ting this pro cedur e we construct a sequence o f integer v a lue d prop er left inv ar iant metrics { d ω i } ∞ i =1 and an increasing s e quence of natural num b ers { R i } ∞ i =1 . By the first tw o prop erties a nd the fact lim i →∞ R i = ∞ we deduce that for every g ∈ G the sequence { k g k ω i } ∞ i =1 is asy mptotica lly co nstant. Define no w the function k · k G : G → N b y k g k G = lim i →∞ k g k ω i . Aga in by the first t wo prop erties we can chec k k · k G is a prop er nor m. So it defines a prop er left inv aria nt metric d G . Using the third and fourth pro p er ties we hav e that for every i ∈ N there exists an homomor phism f i : Z i → G such that the res tr iction to the ball B (0 , k i ) is a dilatation in ( G, d G ). As in each step w e are increasing the dimension of the balls, we get that for every m ∈ N the metric space ( G, d G ) satisfies the co nditions of corolla r y 3.4 so we get asdi m AN ( G, d G ) ≥ m . Therefore asdim AN ( G, d G ) = ∞ .  Recall that in [8] Dranishniko v a nd Smith s how ed that the asy mpto tic dimension of finitely genera ted nilpo ten t groups is equal to the hirsch length of the gr oup. Hence the asymptotic dimension of a nilpo tent gr oup is alwa ys finite for ev ery prop er left in v ariant metric. Next trivial coro llary shows that the unique nilpotent groups that satisfy the sa me pro pe r ty for the asymptotic Asso uad-Nagata dimens io n are finite. ASSUAD-NAGA T A DIM ENSION OF NILPOTENT GR OUPS WITH ARBITRAR Y LE FT INV ARIANT ME TRICS 9 Corollary 4.3. L et G b e a fi nitely gener ate d nilp otent gr oup. G is non fi n ite if and only if ther e exists a pr op er left invariant metric d ω define d in G such that asdim AN ( G, d ω ) = ∞ . Pro of. F or o ne implication just use that the a symptotic Assouad- Nagata dimen- sion is zer o for all b ounded spaces. The other imp lication is a particular case of the main theorem.  References [1] P . 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[16] U. Lang, T. Schlic henmaier, Nagata dimension, quasisymmetric emb e ddings, and L ipschitz extensions , IMRN Inte rnational Mat hematics Research Notices (2005), no.58, 3625–3655. [17] P .W. Now ak, On exactness and i sop erimetric pr ofiles of discr e te g r oups , Journal of f unctional analysis. 243 (2007), no.1, 323-344. [18] J. Smith, On Asymptotic dimension of Countable Ab elian Gr oups , T opology A ppl . 153 (2006), no. 12, 2047–2054 [19] K. Wh yte, Amenability, Bilipschitz Equivalenc e, and the V on Neumann Conje ctur e , Duke Journal of Mathematics.(1999), 93–112. Dep a r t amento de Geometr ´ ıa y Topolog ´ ıa, F acul t ad de CC.Ma tem ´ aticas. Universidad Complutense de Madrid. Madrid, 28040 Sp ain E-mail a ddr e ss : josemhiges@y ahoo.es