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Nagata Dimension and Quasi-M¨Obius Maps CONFORMAL GEOMETRY AND DYNAMICS An Electronic Journal of the American Mathematical Society Volume 12, Pages 1–9 (January 22, 2008) S 1088-4173(08)00173-2 NAGATA DIMENSION AND QUASI-MOBIUS¨ MAPS XIANGDONG XIE Abstract. We show that quasi-M¨obius maps preserve the Nagata dimension of metric spaces, generalizing a result of U. Lang and T. Schlichenmaier (Int. Math. Res. Not. 2005, no. 58, 3625–3655). 1. Introduction In this note we show that a certain covering dimension of metric spaces – the Nagata dimension – is invariant under quasi-M¨obius maps. Nagata dimension was introduced and studied by Assouad ([A]). It captures both macroscopic and microscopic structures of metric spaces. Hence Nagata dimension has potential applications in both large scale geometry of metric spaces and analysis on metric spaces. S. Buyalo and V. Schroeder ([BS]) have used Nagata dimension to prove that any Gromov hyperbolic group whose Gromov boundary has topological dimension n admits a quasi-isometric embedding into the product of (n+1) trivalent trees. On the other hand, U. Lang and T. Schlichenmaier ([LS]) showed that Nagata dimension is preserved by quasisymmetric maps. Notice that it is clear from the definition (see Section 2) that Nagata dimension is invariant under bi- Lipschitz maps. The purpose of this note is to extend the result of U. Lang and T. Schlichenmaier to quasi-M¨obius maps: Theorem 1.1. Let f :(X1,d1) → (X2,d2) be a quasi-M¨obius map. Then (X1,d1) and (X2,d2) have the same Nagata dimension. The proof of Theorem 1.1 is not difficult. We first observe that each quasi-M¨obius map can be written as a composition of a quasisymmetric map and at most two metric inversions (see Section 3 for the definition). Hence by the above mentioned result of U. Lang and T. Schlichenmaier it suffices to show that metric inversions preserve Nagata dimension. This statement is proved by modifying the argument of U. Lang and T. Schlichenmaier (see the proof of Theorem 1.2 in [LS]). As an immediate application of Theorem 1.1, consider X = {1/n : n =1, 2, ···} ⊂ R2 and Y = {(m, n) ∈ R2 : m, n ∈ Z}⊂R2 with the metric induced from R2. Since X has Nagada dimension 1 and Y has Nagata dimension 2, they are not quasi-M¨obius equivalent. The author is not aware of a direct proof of this fact. There are two other dimensions closely related to Nagata dimension: capac- ity dimension ([B]) and asymptotic dimension of linear type. Capacity dimen- sion concerns the microscopic structure of a metric space and is also invariant Received by the editors January 23, 2007. 2000 Mathematics Subject Classification. Primary 54F45, 30C65. Key words and phrases. Nagata dimension, quasi-M¨obius maps, metric inversion. c 2008 American Mathematical Society Reverts to public domain 28 years from publication 1 2 XIANGDONG XIE under quasisymmetric maps. Asymptotic dimension of linear type captures the large scale structure of a metric space and is invariant under quasi-isometries. No- tice that capacity dimension is not invariant under quasi-M¨obius maps: consider X = {1, 1/2, ··· , 1/n, ···} ⊂ R2 and Y = {1, 2, ··· ,n,···} ⊂ R2 with the metric induced from R2. The inversion about the unit circle in R2 sends X to Y ,soX and Y are M¨obius equivalent; in particular, they are quasi-M¨obius equivalent. However, the capacity dimension of X is 1 while that of Y is 0. 2. Nagata dimension In this section we recall some basic facts about Nagata dimension (see [LS], [BL] and [BDLM] for more details). Let (X, d)beametricspaceandB = {Bi}i∈I a family of subsets of X.The family B is called D-bounded, D ≥ 0, if diam(Bi):=sup{d(x, x ):x, x ∈ Bi}≤D for all i ∈ I.Fors>0, the s-multiplicity of B is the infimum of all integers n such that every subset of X with diameter ≤ s meets at most n members of the family. Definition 2.1. Let (X, d) be a metric space. The Nagata dimension of X, denoted by dimN (X, d), is the infimum of all integers n with the following property: There exists a constant c ≥ 1 such that for all s>0, X has a cs-bounded covering with s-multiplicity at most n +1. Recall that two metric spaces (X1,d1)and(X2,d2)arebi-Lipschitz equivalent if there is a bijection f :(X1,d1) → (X2,d2)andanL ≥ 1 such that d1(x, y)/L ≤ d2(f(x),f(y)) ≤ Ld1(x, y)holdsforallx, y ∈ X1. It is clear from the definitions that two bi-Lipschitz equivalent metric spaces have the same Nagata dimension. Suppose s>0andB is a covering of X.IfB can be written as a union B n B B = k=0 k such that each k has s-multiplicity at most 1, then it is clear that B has s-multiplicity at most n +1. If X = A ∪ B,thendimN (X, d)=max{dimN (A, d), dimN (B,d)} (Theorem 2.7 of [LS]). In particular, if X contains at least two points and p ∈ X,then dimN (X, d)=dimN (X\{p},d) (since the Nagata dimension of a singleton is 0). A key ingredient in the proof of Theorem 1.1 is the following existence result of a sequence of coverings. There are similar results for asymptotic dimension ([D], Proposition 1) and capacity dimension ([B], Proposition 4.4). Notice that the existence of a sequence of coverings as in Proposition 2.2 implies dimN (X, d) ≤ n. We shall only use properties (i)–(iii). Proposition 2.2 (Proposition 4.1 of [LS]). Suppose (X, d) is a metric space with dimN (X, d) ≤ n<∞. Then there is a constant c ≥ 1 such that for all sufficiently large r>1, there exists a sequence of coverings Bj of X, j ∈ Z, with the following properties: ∈ Z Bj n Bj Bj j (i) For every j , we have = k=0 k where each k is a cr -bounded family with rj-multiplicity at most 1. (ii) For all j ∈ Z and x ∈ X,thereexistsaC ∈Bj that contains the closed ball B(x, rj). ∈Bi ∈Bj ⊂ (iii) Whenever B k and C k for some k and i<j, then either B C or d(x, y) ≥ ri for every pair of points x ∈ B, y ∈ C. ∈{ ··· } ⊂ ∈ (iv)For every k 0, 1, ,n and every bounded set B X,thereisaC B Bj ⊂ k := j∈Z k such that B C. NAGATA DIMENSION AND QUASI-MOBIUS¨ MAPS 3 3. Quasi-Mobius¨ maps and metric inversions In this Section we recall the notions of quasisymmetric maps, quasi-M¨obius maps and metric inversions, and discuss their relations (see [BHX], [V], [H] and [V2] for more details). Let η :[0, ∞) → [0, ∞) be a homeomorphism. A homeomorphism between metric spaces f :(X, dX ) → (Y,dY )isη-quasisymmetric if for all pairwise distinct points x, y, z ∈ X,wehave d (f(x),f(y)) d (x, y) Y ≤ η X . dY (f(x),f(z)) dX (x, z) A homeomorphism f :(X, dX )→(Y,dY ) is quasisymmetric if it is η-quasisymmetric for some η. Quasisymmetric maps send bounded metric spaces to bounded metric spaces (Proposition 10.8 of [H]). Let Q =(x1,x2,x3,x4) be a quadruple of pairwise distinct points in (X, d). The cross ratio of Q with respect to the metric d is: d(x ,x )d(x ,x ) cr(Q, d)= 1 3 2 4 . d(x1,x4)d(x2,x3) Let η :[0, ∞) → [0, ∞) be a homeomorphism. A homeomorphism between metric spaces f :(X, dX ) → (Y,dY )isanη-quasi-M¨obius map if cr(f(Q),dY ) ≤ η(cr(Q, dX )) for all quadruple Q of distinct points in X,wheref(Q)=(f(x1),f(x2),f(x3),f(x4)). A homeomorphism f :(X, dX ) → (Y,dY )isquasi-M¨obius if it is η-quasi-M¨obius for some η. The inverse of a quasi-M¨obius map is quasi-M¨obius, and the composition of two quasi-M¨obius maps is also quasi-M¨obius (see [V]). Quasisymmetric maps are quasi-M¨obius. In general quasi-M¨obius maps are not quasisymmetric; for example, the inversions about spheres in Euclidean spaces are M¨obius (hence quasi-M¨obius) but are not quasisymmetric. However, quasi-M¨obius maps between bounded metric spaces are quasisymmetric (see [V]). Below we will find further connection between quasisymmetric maps and quasi-M¨obius maps (Proposition 3.6). We first recall the notion of metric inversion, which is a general- ization of inversions about unit spheres in Euclidean spaces. Let (X, d)beametricspaceandp ∈ X.SetIp(X)=X\{p} if X is bounded and Ip(X)=(X\{p})∪{∞} if X is unbounded, where ∞ is a point not in X.Then there is a metric dp on Ip(X) such that the following holds for all x1,x2 ∈ X\{p}: d(x1,x2) d(x1,x2) (3.1) ≤ dp(x1,x2) ≤ . 4 d(x1,p)d(x2,p) d(x1,p)d(x2,p) When X is unbounded, the following also holds for all x ∈ X\{p}: 1 1 ≤ d (x, ∞) ≤ . 4 d(x, p) p d(x, p) Furthermore, the identity map id : (X\{p},d) → (X\{p},dp)isanη-quasi-M¨obius homeomorphism with η(t)=16t (see [BHX] for a proof of the above statements). For any metric space (X, d)andp ∈ X,wecalltheidentitymapid:(X\{p},d) → (X\{p},dp)ametric inversion. 4 XIANGDONG XIE Lemma 3.2. Let (X, d) be a metric space, p ∈ X and A ⊂ X\{p} asubset.
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