Property A and asymptotic dimension

PR OPER TY A AND ASYM PTOTIC DIMENSION M. CENCELJ, J. D YDAK, AND A. V A VPETI ˇ C Abstract. The purpose of this note i s to c haracterize the asymptotic dimen- sion asdim( X ) of metric spaces X in terms si milar to Prop erty A of Y u [6]: Theorem 0.1. If ( X, d ) is a metri c sp ac e and n ≥ 0 , then the following c onditions ar e e quivalent: a. asdim( X, d ) ≤ n , b. F or e ach R, ǫ > 0 ther e is S > 0 and finite non-empty subsets A x ⊂ B ( x, S ) × N , x ∈ X , such that | A x ∆ A y | | A x ∩ A y | < ǫ if d ( x, y ) < R and the pr oje ct ion of A x onto X c ontains at most n + 1 elements for al l x ∈ X , c. F or e ach R > 0 ther e is S > 0 and finite non-e mpty subsets A x ⊂ B ( x, S ) × N , x ∈ X , such t hat | A x ∆ A y | | A x ∩ A y | < 1 n +1 if d ( x, y ) < R and the pr oje ct ion of A x onto X c ontains at most n + 1 elements for al l x ∈ X . Date : October 31, 2018. 2000 M athematics Subje ct Classific ation. Primary 54F45; Secondary 55M10. Key wor ds and phr ases. asymptotic dimension, Prop ert y A, coarse geometry . Supported in part by the Slo v enian-USA r esearc h grant BI–US/05-06/002 and the ARRS researc h pro ject No. J1–6 128–0101–0 4. The second-named author was partially supp orted by MEC, M TM2006-0825. 1 2 Contents 1. Int ro duction 2 2. Main theore m 2 References 3 1. Introduction Prop erty A was introduced by G.Y u in [6] in order to prove a sp ecial case of the Novik ov Co njectur e . W e adopt the following definition from [3] (se e also [5]): Definition 1.1. A discr ete metric space ( X, d ) has proper t y A if for all R, ǫ > 0, there exists a family { A x } x ∈ X of finite, non-empty subsets of X × N s uch that: • for all x, y ∈ X with d ( x, y ) ≤ R w e ha ve | A x ∆ A y | | A x ∩ A y | < ǫ • there exists S > 0 such that for ea ch x ∈ X , if ( y , n ) ∈ A x , then d ( x, y ) ≤ S Asymptotic dimens ion was introduced b y M. Gromov in [1] (see s ection 1.E ) as a large-sc ale ana logue of the classical notion of top ologica l covering dimensio n. It is a co a rse inv ariant that has b een extensively inv estig ated (see c hapter 9 of [4] for some results and further references). Definition 1.2. A metric space X is said to hav e finite asy mpto tic dimension if there exists k ≥ 0 suc h that for all L > 0 t here exis ts a uniformly bounded cover of X (that means the existence of S > 0 such that all elemen ts of the cover ar e of diameter at most S ) of Leb esgue n umber at least L (that mea ns every R - ball B ( x, R ) is contained in some element of the co ver) and multiplicit y at most k + 1 (i.e. each p oint o f X belo ngs to at most k + 1 elements o f the cov er). The least po ssible such k is the asymptotic dimension of X . One of the bas ic r esults is that spaces of finite asymptotic dimension have prop- erty A and known pro ofs o f it use Higson-Ro e characteriza tion of Pro p e rty A (see [2] and [5]). The purp ose of this note is to provide a simple pr o of o f that res ult a nd prov e a characterization of asymptotic dimension in terms simila r to Pro p erty A. 2. Main theorem Theorem 2. 1. If ( X , d ) is a metric sp ac e and n ≥ 0 , then t he fol lowing c onditions ar e e quivalent: a. asdim( X , d ) ≤ n , b. F or e ach R , ǫ > 0 t her e is S > 0 and finite non-empty subsets A x ⊂ B ( x, S ) × N , x ∈ X , such that | A x ∆ A y | | A x ∩ A y | < ǫ if d ( x, y ) < R and the pr oje ction of A x onto X c ontains at most n + 1 elements for al l x ∈ X , c. F or e ach R > 0 ther e is S > 0 and fin ite non-empty su bset s A x ⊂ B ( x, S ) × N , x ∈ X , such that | A x ∆ A y | | A x ∩ A y | < 1 n +1 if d ( x, y ) < R and the pr oje ction of A x onto X c ontains at most n + 1 elements for al l x ∈ X . Pro of. a) = ⇒ b). Suppos e asdim ( X , d ) ≤ n and R, ǫ > 0. Pick a uniformly bo unded co ver U of X of m ultiplicity at most n + 1 and Leb egue num b er at least L = 2 R + 2 R · n ǫ . Let S b e a num b er such that diam( U ) < S f or each U ∈ U . Pick a U ∈ U for each U ∈ U a nd define A x as the union of s ets a U × { 1 , . . . l U ( x ) } , PROPER TY A AND ASYMP TOTIC DIM ENSION 3 where x ∈ U and l U ( x ) is the length o f the shortest R -chain joining x and a point outside of U (if there is no such chain, w e put l U ( x ) equal to the int eger part of L R + 1). If d ( x, y ) < R , then | l U ( x ) − l U ( y ) | ≤ 1, so | A x ∆ A y | ≤ 2 n (a s the total nu mber of elements of U containing exactly o ne of x o r y is at most 2 n ), and | A x ∩ A y | > L − R R − 1 (choo se U con ta ining B ( x, L ) and notice every R -chain joining x or y to X \ U m ust hav e at least L − R R elements), yielding | A x ∆ A y | | A x ∩ A y | < 2 n · R L − 2 R ≤ ǫ . c) = ⇒ a). Given R > 0 pick S > 0 and finite subsets A x ⊂ B ( x, S ) × N , x ∈ X , such that | A x ∆ A y | | A x ∩ A y | < 1 n +1 if d ( x, y ) < R and the pro jection of A x onto X con tains at mos t n + 1 elements f or all x ∈ X . Define sets U x as consisting precisely of y ∈ X such that ( { x } × N ) ∩ A y 6 = ∅ . The m ultiplicit y of the cov er { U x } x ∈ X of X is a t most n + 1 as z ∈ k T i =1 U x i implies x i belo ngs to the pro jection o f A z , so k ≤ n + 1. Given x ∈ X cho ose z ∈ X so that | ( { z } × N ) ∩ A x | maximizes all | ( { y } × N ) ∩ A x | . In particular | ( { z } × N ) ∩ A x | ≥ | A x | n +1 . If d ( x, y ) < R we claim y ∈ U z which pr ov es that the Lebeg ue num b er of { U x } x ∈ X is at least R . Indeed, y / ∈ U z implies | A x ∆ A y | ≥ | A x | n +1 , s o | A x ∆ A y | | A x | ≥ 1 n +1 , a contradiction.  References [1] M . Gromov , Asy mptotic i nv ariants for infinite gr oups , in Geometric Group Theory , v ol. 2, 1–295, G. Niblo and M. Roller, eds., Cam bridge U ni v ersity Press, 1993. [2] N. Hi gson and J. Roe, Amenable gr oup actions and the Novikov c onje ctur e , J. Reine Agnew. Math. 519 (2000). [3] Pi otr Now ak and Gu oliang Y u, Wh at is ... Pr op erty A? , Notices of the AMS V olume 55, Number 4, pp.474–475. [4] J. Roe, L ectur es on co arse ge ometry , Universit y Lecture Series, 31. American Mathematical Society , Providenc e, RI, 2003. [5] Ruf us Willett, Some note s on Pr op erty A , arXi v:math/06124 92 v2 [ m ath.O A] [6] G. Y u, The c o arse Baum-Connes co nje c tur e for sp ac es which admit a uniform emb ed ding into Hilb ert sp ac e , Inv entiones 139 (2000), pp. 201–240. IMFM, Univerza v Ljubljani, Jadranska ulica 19, S I-1111 Ljubljana, Slovenija E-mail addr ess : matija.cence lj@guest.a rnes.si University of Tennessee, Knoxville, TN 37996, USA E-mail addr ess : dydak@math.u tk.edu F akul tet a za Mat ema tiko in Fiziko, Univ erza v Ljubljani, Jadranska ulica 1 9, SI-1111 Ljubljana, Slovenija E-mail addr ess : ales.vavpeti c@fmf.uni- lj.si