On asymptotic dimension and a property of Nagata

ON ASYMPTOTIC DIMENSION AND A PR OPER TY OF NA GA T A J. HIGES AND A. M ITRA Abstract. In this note we pro ve that ev ery metric spa ce ( X , d ) of asymptotic dimmension at most n is coarsely equiv alen t to a metric space ( Y , D ) that satisfies the foll o wing prop ert y of Nagata: F or every x, y 1 , · · · , y n +2 ∈ Y ther e exist i, j ∈ { 1 , · · · , n + 2 } with i 6 = j , such that D ( y i , y j ) ≤ D ( x, y i ) . This solves problem 1400 of [1]. Contents 1. Int ro duction 1 2. Main theor em 2 References 3 1. Introduction Nagata in tro duced tw o prop erties to characterize metric spaces with finite top o- logical dimension (see [5], [6] and [7]). Such proper ties are generaliza tions to higher dimensions of the notion of ultrametric space. The definitio n of the prop erties are the following o nes: Definition 1.1. A metr ic space ( X , d ) is s aid to satisfy the prop erty ( N 1) n if for every r > 0 and every x, y 1 , · · · , y n +2 ∈ X s uch that d ( y i , B ( x, r )) < 2 · r then there exists i , j ∈ { 1 , · · · , n + 2 } with i 6 = j such that d ( y i , y j ) < 2 · r . Definition 1.2. A met ric space ( X , d ) is said to satis fy prop erty ( N 2) n if f or every x, y 1 , · · · , y n +2 ∈ X there ex ists i, j ∈ { 1 , · · · , n + 2 } with i 6 = j , such that d ( y i , y j ) ≤ d ( x, y i ). In [3] Dr anishniko v and Zarichn yi show ed that every pr op er metric s pace ( X , d ) of asymptotic dimension a t most n is coa rsely equiv alen t to a pro p er metric s pace that satisfies ( N 1) n . So a natural que s tion is if the same statement is tr ue for the second prop er t y . Suc h pr oblem app eared in [1] as problem 1 400. In this pa pe r we solve it for g e ne r al metric spa ces using a technique of [2]. Date : Decem b er 7, 2008. 1991 Mathematics Subje c t Classific ation. Primary: 54F45, 54C55, Secondary: 54E35, 18B30, 20H15. The first named author i s s upp orted by grant 2004-2494 from Ministerio Educacion y Ciencia, Spain and pro ject MEC, MTM2006-0825. He thanks Univ ersity of T ennessee for hospitality . Special thanks to J. Dydak and N. Brodskyi f or helpful suggestions. 1 2 J. HIGES AND A. MITRA 2. Main theorem The no tion of a symptotic dimension was in tro duced by Gromov in [4] a nd it has bee n the fo cus of intense resear ch in r e cent years. T o give a definition we need to recall that a family of subsets U of a metric space ( X , d ) is sa id to be r -disjoint if for e very tw o diferent U ∈ U a nd V ∈ U then d ( U, V ) > r . Definition 2 .1. A metric spa ce ( X , d ) is said to b e of asymptotic dimension at most n ( asdim ( X , d ) ≤ n ) if for every r > 0 there exists a unifor mly bo unded cov ering U of subsets of X such that U splits in a unio n of the form U = S n +1 i =1 U i where e ach U i is an r -disjoint fa mily of subse ts. T o simplify we will say that a covering U of a metric s pace ( X , d ) has L eb esgue numb er at least r ( L ( U ) ≥ r ) if for every x ∈ X the ball B ( x, r ) is contained in one set of the cov ering. Next lemma characterizes asymptotic dimension in a nice wa y . Lemma 2 .2. A metric sp ac e ( X, d ) is of asymptotic dimension at most n if and only if for every r > 0 ther e e xists a un iformly b ounde d c overing U wher e U = S n +1 i =1 U i such that e ach U i is an r -disjoint family and the L eb esgue numb er of U is at le ast r . Pro of. Only one implication is not trivial. Suppo se asdi m ( X , d ) ≤ n and let r be a p os itive num b er. As asdi m ( X, d ) ≤ n there exists a uniformly b ounded covering V ′ = S n +1 i =1 V i such that each V i is a 3 · r - disjoint family . Let U i be the family of subsets given by U i = { N ( V , r ) | V ∈ V i } where N ( V , r ) = { x | d ( x, V ) < r } . No w if U , U ′ ∈ U i are different elements of U i then cle a rly d ( U, U ′ ) > r . Therefore the family U = S n +1 i =1 U i is a covering of mesh at most mesh ( V ) + 2 · r s o it satis fie s the conditions of the lemma.  Next pro p o s ition is a version of prop osition 3.6. of [3]. W e give a pro o f to make the pap er s elf-contained. Prop ositio n 2 .3. L et ( X , d ) b e a metric sp ac e su ch that asdim ( X , d ) ≤ n . Th en ther e exist a se quenc e of un iformly b ounde d c overings {U k } ∞ k =1 and an incr e asi ng se quenc es of numb ers { d k } ∞ k =1 such that: (1) F or every k U k = S n +1 i =1 U i k and e ach U i k is a d k -disjoint family. (2) L ( U k ) ≥ d k . (3) d k +1 > 2 · m k wher e m k = mesh ( U k ) . (4) F or every i, k , l with k < l and every U ∈ U i k , V ∈ U i l , if U ∩ V 6 = ∅ , then U ⊂ V . Pro of. Let us cons truct the se q uence by induction on k . F or k = 1 the res ult is just an application of lemma 2 .2 for d 1 = 1. Suppo se we have a finite sequence of unifor mly b ounded coverings {U k } t k =1 and a finite seq ue nce of num b ers { d k } t k =1 that sa tisfy pr op erties (1 )-(4). Let d t +1 > 0 be a p ositive num ber such that d t +1 > 2 · m t and de fine D t +1 = d t +1 + 2 · m t . Hence by Lemma 2.2 there exists an uniformly b ounded cov ering V t +1 = S n +1 i =1 V i t +1 such that ea ch V i t +1 is D t +1 -disjoint and L ( V t +1 ) ≥ D t +1 . Now for every i ∈ { 1 , ..., n + 1 } we define the family of subsets U i t +1 = { U V | V ∈ V i t +1 } where U V is defined as the union of V with a ll the subsets W ∈ U i k with k ∈ { 1 , ..., t } such tha t W ∩ V 6 = ∅ . W e claim that the covering given by U t +1 = S n +1 i =1 U t +1 satisfies the required conditions. Clearly the mesh of U t +1 is b ounded by 2 · m t + mesh ( V t +1 ). Also we hav e L ( U t +1 ) ≥ L ( V t +1 ) ≥ ON ASYMPTOTIC DIMENS ION AND A PR OPER TY OF NAGA T A 3 D t +1 ≥ d t +1 . Let i ∈ { 1 , ..., n + 1 } b e a fix num b er. F o r every U V , U W ∈ U i t +1 we hav e d ( U V , U W ) > d ( V , W ) − 2 · m t = d t +1 so ea ch U i t +1 is d t +1 -disjoint. The unique prop er t y that we need to check is (4 ). Let k , l b e tw o num be r s such that k < l ≤ t + 1. If l < t + 1 the fourth condition fo llows fro m the induction hypothesis. Let us s upp os e l = t + 1. Let W ∈ U i k and U V ∈ U i t +1 such that W ∩ U V 6 = ∅ . This implies there ex ist an s ≤ t a nd a V ′ ∈ U i s such that V ′ ∩ V 6 = ∅ and W ∩ V ′ 6 = ∅ . By the induction hyp o thesis the last co ditio n implies W ⊂ V ′ or V ′ ⊂ W . In b oth cases we can co nclude W ⊂ U V .  A map f : ( X , d X ) → ( Y , d Y ) betw een metric spaces is said to b e a c o arse map if for every δ > 0 there is an ǫ > 0 such that for every tw o po ints a, b ∈ X that s atisfy d X ( a, b ) ≤ δ then d Y ( f ( a ) , f ( b )) ≤ ǫ . If ther e exist also a coarse map g : ( Y , d Y ) → ( X , d X ) and a constant C > 0 such that for every x ∈ X and every y ∈ Y , d X ( x, g ( f ( x ))) ≤ C and d Y ( y , f ( g ( y ))) ≤ C then f is said to b e a c o arse e quivale nc e and the metric s paces X a nd Y are said to b e c o arse e quivalent . Theorem 2.4. Every metric sp ac e ( X , d ) with asdim X ≤ n is c o arsely e quivalent to a metric sp ac e that satisfies pr op erty ( N 2) n . Pro of. Let {U k } ∞ k =1 be a seq uence of coverings as in pr op osition 2.3. F o r every x and y in X we define: D ( x, y ) = min { k | there ex ists U ∈ U k such tha t x, y ∈ U } Notice that by the second and third prop erties o f U k we c a n easily deduce that for every x, y , z ∈ X : D ( x, y ) ≤ max { D ( x, z ) , D ( y , z ) } + 1 This implies ( X , D ) is a metr ic space. Moreover if D ( x, y ) ≤ δ and k is the minimum natural num ber such tha t δ ≤ k then d ( x, y ) ≤ m k . In a similar wa y let us asume d ( x, y ) ≤ δ . Let d r be the minim um num ber of the se quence { d k } ∞ k =1 such tha t δ ≤ d r . W e hav e L ( U r ) > d r , this implies there exists an U ∈ U r such that x ∈ U and y ∈ U what means D ( x, y ) ≤ r . W e hav e shown ( X , D ) is coar sely equiv alen t to ( X , d ). Now le t y 1 , · · · , y n +2 , x ∈ X and define for every i ∈ { 1 , · · · , n + 2 } the num ber r ( i ) = D ( y i , x ). By definition of D we g et that for every i there exists a V i ∈ U r ( i ) such that y i ∈ V i and x ∈ V i . F or every i there ex is ts k ( i ) such that V i ∈ U k ( i ) r ( i ) . By the pigeon principle there a re tw o i, j with i 6 = j suc h that k ( i ) = k ( j ). As x ∈ V i ∩ V j by the fourth prop er t y o f prop osition 2.3 we co nclude V i ⊆ V j or V j ⊆ V i . The first case means D ( y i , y j ) ≤ D ( x, y j ) and the seco nd one mea ns D ( y j , y i ) ≤ D ( x, y i ). Therefore ( X , D ) satis fies ( N 2) n .  References [1] T. Banakh, B.B ok alo, I. Guran, T. Radul, M. Zarichn yi Pr oblems fr om the Lviv top olo gic al seminar , in Op en problems in T opology I I.(E. Pearl ed.) Elseiver, 2007. P . 655–667. [2] N. Bro dskiy , J. Dydak, J. Higes, A. Mitra, Assouad-Nagata dimension via Lipschitz exten- sions , to app ear in Israel Journal of M athematics. [3] A. Dranishniko v, M. Zarichn yi, Universal sp ac es for asymptotic dimension , T opology Appl. 140 (2004) , no.2-3, 203–225. [4] M. Gromov , Asymptotic invariants for infinite gr oups , in Geometric Group Theory , vol. 2, 1–295, G.Niblo and M.Roller , eds., Cambridge Universit y Press, 1993. [5] J. Nagata, Note on dimension the ory for metric sp ac es , F und M ath. 45 (1958), 143-181. 4 J. HIGES AND A. MITRA [6] J. Nagata , On a sp e cial metric char acterizing a metric sp ac e of dim ≤ n , Pro c. Jap an Acad. 39 (1963), 278-282. [7] J. Nagata, On a sp e cial metric and dimension , F und Math. 55 (1964), 181-194. Dep ar t amen to de Geometr ´ ıa y Topolog ´ ıa, F acul t ad de CC.Ma tem ´ aticas. Universidad Complutense de Ma drid. M a drid, 28 040 S p ain E-mail addr ess : josemhiges@y ahoo.es University of South Florida, St. Petersburg, FL 33701, USA E-mail addr ess : atish@stpt.u sf.edu