aa r X i v : . [ m a t h . M G ] F e b COARSE MEDIANS AND PROPERTY A
J ´AN ˇSPAKULA AND NICK WRIGHT
Abstract.
We prove that uniformly locally finite quasigeodesic coarse medianspaces of finite rank and at most exponential growth have Property A. Thisoffers an alternative proof of the fact that mapping class groups have PropertyA. Introduction
Coarse median spaces and groups were invented by Bowditch [Bow13a, Bow13b,Bow14] as (we are guessing here) a device offering a unified approach to hyperbolicgroups and mapping class groups.Indeed, hyperbolic groups are precisely coarse median groups of rank 1 [Bow13a,Theorem 2.1], and mapping class groups are instances of coarse median groups offinite rank [Bow13a, Theorem 2.5].Furthermore, groups that are relatively hyperbolic with respect to a collectionof coarse median groups are again coarse median [Bow13b]. This provides moreexamples of coarse median groups, for instance geometrically finite Kleinian groupsand Sela’s limit groups.Coarse median approach to these classes of groups is quite powerful: In this seriesof papers, Bowditch uses it to give unified proofs of some properties, for instancethe Rapid Decay property, quadratic isoperimetric inequality and a computing thedimension of asymptotic cones.Intuitively, a coarse median space is a metric space endowed with a ternarystructure (a map assigning a point to every triple of points), which is metrically acontrolled amount away from being an actual median structure. (Finite) set withan actual median structure are just (vertex sets of) CAT(0) cube complexes. Henceone may loosely regard coarse median structures as coarse versions of (metrized)CAT(0) cube complexes. This analogy works exactly in the “rank one” situation,where the CAT(0) cube complexes are trees, and hyperbolic groups are “coarselytree-like”. For the actual definitions, see Section 2.The main result of this piece is that quasigeodesic coarse median spaces of finiterank, which are uniformly locally finite and have at most exponential growth,have Yu’s Property A. For proving Property A we use a criterion which is anadaption Brown and Ozawa’s proof [BO08] that hyperbolic groups act amenably
Mathematics Subject Classification.
Key words and phrases. coarse median, Yu’s Property A. on the boundary. As a side-effect, we obtain a quick proof of Property A for finitedimensional CAT(0) cube complexes, a fact originally established in [BCG +
09] bya different, more combinatorial, method. Our proof for coarse median spaces is acoarsification of this short argument.As a consequence, we obtain an alternative proof of the result that mappingclass groups have Property A (i.e. are exact), originally proved by Hamenst¨adt[Ham09] and Kida [Kid06].Finally, we would like to mention a related notion of hierarchically hyperbolicspaces (and groups), developed recently in [BHS14, BHS15b]. While this propertyis stronger (see [BHS15b, Section 7]), and somewhat more involved than coarsemedians it is also substantially more powerful: it implies even finite asymptoticdimension [BHS15a]. Having finite asymptotic dimension is a strictly strongerproperty than Property A. We close off with a question: do coarse median groupsof finite rank have finite asymptotic dimension?The structure of the paper is as follows: in Section 2 we recall the relevant defi-nitions and facts. Section 3 explains Brown and Ozawa’s criterion for Property A.In Section 4 we outline the quick proof of Property A for CAT(0) cube complexes.In Section 5 we establish some facts about (metric) median algebras; and finallySection 6 contains the proof of the main result.1.1.
Acknowledgements.
The first author thanks Goulnara Arzhantseva for herencouragement, continuing support, and the initial impetus for this work.2.
Preliminaries
CAT(0) cube complexes.
We recall the notions related to CAT(0) cubecomplexes. For details, please consult [BH99, NR98].A cube complex is a polyhedral complex in which the cells are Euclidean cubesof side length one, the attaching maps are isometries identifying the faces of agiven cube with cubes of lower dimension and the intersection of two cubes is acommon face of each. One-dimensional cubes are called edges ; and the complex is finite-dimensional if there is a bound on the dimension of its cubes.Recall that we can endow a cube complex with a naturally defined geodesicmetric . Furthermore, we can endow the set of vertices of a cube complex with an edge-path metric; in the finite-dimensional case, this metric is coarsely equivalentto (the restriction of) the geodesic metric [BCG +
09, Proposition 1.7].A cube complex is a
CAT(0) cube complex , if the underlying topological spaceis simply connected and the complex satisfies Gromov’s link condition [Gro87]. Inthe finite-dimensional case, this is equivalent to asking that the geodesic metricsatisfies the CAT(0) inequality [BH99].A hyperplane H (or a wall ) is a geometric hyperplane, which cuts each cubethat it intersects exactly in half. Such an H divides the vertex set into two path-connected subspaces which are referred to as half-spaces . Two hyperplanes cross , OARSE MEDIANS AND PROPERTY A 3 if each of the four possible intersections of the associated half-spaces is non-empty.We say that H separates two vertices, if every edge-path connecting them crosses H . For two sets for vertices A and B , we shall write A | H B if H separates everyvertex in A from every vertex in B , i.e. A and B are in different half-spacesdetermined by H . The interval [ x, y ] between two vertices x, y is the intersectionof all half-spaces containing both vertices.Every n -dimensional cube in a CAT(0) cube complex defines n pairwise inter-secting hyperplanes (whose it crosses ), and conversely, a collection of n pairwiseintersecting hyperplanes define a unique n -cube (which crosses exactly these hy-perplanes).Note that the set of vertices of a CAT(0) cube complex is a median algebra inthe sense defined below — the median of three points x, y, z is the unique vertexin the intersection [ x, y ] ∩ [ y, z ] ∩ [ z, x ] [Rol98]. Equivalently the median of x, y, z is the unique point lying on a geodesic between x and y , on a geodesic between y and z and a geodesic between z and x . Furthermore, the notions of an interval,wall, etc. . . are the same whether defined as here, or using the median structure(below).In a CAT(0) cube complex, each collection of pairwise intersecting hyperplanesdetermines a unique cube, and conversely, each cube (of dimension k ) provides k pairwise intersecting hyperplanes. A cube path from a vertex x to a vertex y in aCAT(0) cube complex X is a sequence of cubes C , . . . , C n , such that x is a vertexof C , y is a vertex of C n , and every two consecutive cubes intersect in exactly onevertex. A normal cube path from x to y is a cube path from x to y , such that everyhyperplane separating x and y is crossed exactly once, with the maximal numberof hyperplanes crossed at each step [NR98]. Note that if X is finite-dimensional,then d ρ ( x, y ) ≤ n ≤ ρ ( x, y ), where d is the dimension of X and ρ denotes theedge-path distance. We also refer to the sequence of the common vertices betweenthe consecutive cubes on the normal cube path as the normal cube path.2.2. Metric median algebras.
We summarise the notions that we need for thispaper. For a more thorough account on median structures, we refer to [Bow14,Bow13a]. The median algebras can be thought of an abstraction of CAT(0) cubecomplexes — every finite median algebra is actually the vertex set of a finiteCAT(0) cube complex. While in one direction of this link works in general, medianalgebras can be “larger” (for instance R -trees are also median algebras).A median algebra is a set, Φ, equipped with a ternary operation, µ : Φ → Φ,such that for all a, b, c, d, e ∈ Φ, we have:(M1): µ ( a, b, c ) = µ ( b, c, a ) = µ ( b, a, c ),(M2): µ ( a, a, b ) = a , and(M3): µ ( a, b, µ ( c, d, e )) = µ ( µ ( a, b, c ) , µ ( a, b, d ) , e ). OARSE MEDIANS AND PROPERTY A 4
While this is the formal definition, we prefer to think about finite median alge-bras as the vertex sets of finite CAT(0) cube complexes (with the natural medianstructure).Given a, b ∈ Φ, the interval [ a, b ] is defined to by [ a, b ] = { c ∈ Φ | µ ( a, b, c ) = c } .A subset H ⊂ Φ is convex if [ a, b ] ⊂ H for all a, b ∈ H .For A ⊂ Φ, define the convex hull , denoted hull( A ), to be the smallest convexsubset of Φ containing A . Note that hull( { a, b } ) = [ a, b ] for a, b ∈ Φ. Furthermore,for A ⊂ Φ, define the join , J ( A ) = S a,b ∈ A [ a, b ]. Continuing inductively, we put J ( A ) = J ( A ) and J i ( A ) = J ( J i − ( A )). In general, there always exists some p ∈ N such that J p ( A ) = hull( A ), and moreover we know that p can be taken to be nolarger than the rank of Φ [Bow13a, Lemma 5.5].A wall , W , is a partition { H − ( W ) , H + ( W ) } of Φ into two non-empty convexsubsets. We say that two walls W, W ′ cross if each of the sets H − ( W ) ∩ H − ( W ′ ), H − ( W ) ∩ H + ( W ′ ), H + ( W ) ∩ H − ( W ′ ) and H + ( W ) ∩ H + ( W ′ ) is non-empty.We say that Φ has rank at most d if there is no collection of d + 1 pairwisecrossing walls of Φ.By a topological median algebra we mean a topological space Φ endowed witha structure of a median algebra µ : Φ → Φ, such that µ is continuous in theinduced topology. When the topology on Φ comes from a metric ρ , we say that Φis a metric median algebra .Let Φ be a metric median algebra as above. We also recall one of the conditionsused in [Bow14] to obtain the embedding result:(L2) : There exists K ≥ a, b, c, d ∈ Φ, ρ ( µ ( a, b, c ) , µ ( a, b, d )) ≤ Kρ ( c, d ).Let us mention that the main embedding result of [Bow14] states that if a metricmedian algebra Φ satisfies (L2), is Lipschitz path connected and is ν -colourable ,then it bilipschitzly embeds into a product of ν R -trees.2.3. Coarse median spaces.
In this subsection, we recall the definitions andfacts related to coarse medians. For more details, we refer to [Bow14, Bow13a].Let (
X, ρ ) be a metric space, and let µ : X → X be a ternary operation. Wesay that µ is a coarse median and that ( X, ρ, µ ) is a coarse median space , if thefollowing conditions hold:(C1): There are constants K ≥ , H (0) ≥
0, such that for all a, b, c, a ′ , b ′ , c ′ ∈ X we have ρ ( µ ( a, b, c ) , µ ( a ′ , b ′ , c ′ )) ≤ K ( ρ ( a, a ′ ) + ρ ( b, b ′ ) + ρ ( c, c ′ )) + H (0) . (C2): There is a function H : N → [0 , ∞ ) with the following property. Supposethat A ⊆ X with 1 ≤ | A | ≤ p < ∞ . Then there is a finite median algebra This is a more restrictive version of rank, which is equivalent to rank for intervals.
OARSE MEDIANS AND PROPERTY A 5 (Π , µ Π ) and maps π : A → Π and σ : Π → X such that for all x, y, z ∈ Πwe have ρ ( σµ Π ( x, y, z ) , µ ( σx, σy, σz )) ≤ H ( p )and ρ ( a, σπa ) ≤ H ( p )for all a ∈ A .We refer to K, H as the parameters of (
X, ρ, µ ).Without loss of generality, we may assume that µ satisfies the axioms (M1) and(M2) by [Bow13a, page 22].We say that X has rank at most d if we can always choose Π to have rank atmost d .Let us recall the asymptotic cones from [Bow13a, Section 9] and [Bow14, Section8]: Let ( X, ρ, µ ) be a coarse median space, let ( r n ) be a sequence of positive realssuch that r n → ∞ , ( x n ) ⊂ X be a sequence of points in X , and finally fixa non-principal ultrafilter on N . With this data, we can construct an ultralimit( X ∞ , ρ ∞ , µ ∞ ) of pointed coarse median spaces (( X, ρ/r n , µ ) , x n ). This ultralimit isreferred to as an asymptotic cone of X (with the given data), and it is a complete metric median algebra satisfying (L2) (with the constant K being the same as inthe definition of coarse median). Moreover if X has rank at most d , then X ∞ alsohas a rank at most d .2.4. Property A.
Property A is coarse geometric property of metric spaces (ormore generally coarse spaces), first defined by G. Yu [Yu00] as a criterion that (fordiscrete countable groups) implies the Coarse Baum-Connes conjecture, and hencethe Novikov conjecture. The catchphrase here is “non-equivariant amenability” or“coarse amenability”. Since its inception, many equivalent formulations were dis-covered, including analytic (exactness of the reduced group C*-algebra, nuclearityof the uniform Roe algebra [GK02, Oza00]) and dynamical (admitting an amenableaction on a compact topological space [HR00]).We shall recall one of the possible definitions (the one used in Proposition 3.1)for completeness; we refer to [Wil09] for the whole spectrum.Let (
X, ρ ) be a uniformly locally finite discrete metric space. We say that X has Property A , if for all
R, ε > ξ : X → ℓ ( X ) from X intothe Banach space ℓ ( X ), such that • k ξ ( x ) k = 1 for all x ∈ X ; • for all x, y ∈ X with ρ ( x, y ) ≤ R , we have k ξ ( x ) − ξ ( y ) k ≤ ε ; • there exists S >
0, such that for each x ∈ X , ξ ( x ) is supported in theclosed ball B ( x ; S ) around x with radius S . The completeness here refers to the metric.
OARSE MEDIANS AND PROPERTY A 6
Geodesicity.
We shall say that a metric space (
X, ρ ) is quasigeodesic , if thereexist constants G , G , such that there exists a ( G , G ) -quasigeodesic betweenany pair of points in X . Note that when X is a quasigeodesic coarse medianspace, all its asymptotic cones are Lipschitz path connected. This is required forapplying the embedding result of Bowditch [Bow14] (and a blanket assumption in[Bow14, Bow13a]). 3. A criterion
We extract a criterion from a proof of Brown and Ozawa [BO08, Theorem 5.3.15]for proving Property A. Its proof is just an excerpt from [BO08]; which is in turninspired by [Kai04].
Proposition 3.1.
Let X be a uniformly finite, discrete metric space. Suppose thatwe have an assignment of a set S ( x, k, l ) ⊂ X to every l ∈ N , k ∈ { , . . . , l } and x ∈ X , such that: (i) For every l ∈ N there exists S l > , such that S ( x, k, l ) ⊂ B ( x, S l ) for all x ∈ X and k ∈ { , . . . , l } . (ii) For every x, y ∈ X , l ≥ ρ ( x, y ) , k ∈ { l + 1 , . . . , l } , we have inclusions S ( x, k − ρ ( x, y ) , l ) ⊂ S ( x, k, l ) ∩ S ( y, k, l ) and S ( x, k, l ) ∪ S ( y, k, l ) ⊂ S ( x, k + ρ ( x, y ) , l ) . (iii) There exists a function p , such that | S ( x, k, l ) | ≤ p ( l ) for every x ∈ X , l ∈ N and k ∈ { , . . . , l } , satisfying lim n →∞ p ( n ) /n = 1 .Then X has property A. To have some mental picture, let us recall that Brown and Ozawa apply thiscriterion to hyperbolic groups Γ, defining the sets as follows: fix a point u ∈ ∂ Γ.Given x, k, l , the set S ( x, k, l ) consists of points that are exactly 3 l steps along ageodesic between a point within the k -ball around x and u . With this definition,the conditions (i) and (ii) follow from the triangle inequality, and (iii) uses thestability of geodesics in hyperbolic spaces (in this case p can be taken to be alinear function). Proof.
Consider the Banach space ℓ ( X ) and for A ⊂ X denote by χ A ∈ ℓ ( X )the normalised characteristic function of A . Given n ∈ N and x ∈ X , define ξ n ( x ) = 1 n n X k = n +1 χ S ( x,k,n ) Note that k ξ n ( x ) k = 1 and supp( ξ n ( x )) ⊂ B ( x, S n ) for all x ∈ X by (i).To establish Property A, we use the formulation from [Wil09, Thm 1.2.4/2.],recalled also in subsection 2.4. We need to show that for a fixed m , we havelim n →∞ sup ρ ( x,y )= m k ξ n ( x ) − ξ n ( y ) k = 0 . OARSE MEDIANS AND PROPERTY A 7
First, observe that for any
A, B ⊂ X , we have k χ A − χ B k = 2 (cid:18) − | A ∩ B | max {| A | , | B |} (cid:19) ≤ (cid:18) − | A ∩ B || A ∪ B | (cid:19) . Take x, y ∈ X with ρ ( x, y ) = m and assume that n ≥ m . Then for any k ∈{ n + 1 , . . . , n } , applying (ii), k χ S ( x,k,n ) − χ S ( y,k,n ) k ≤ (cid:18) − | S ( x, k − m, n ) || S ( x, k + m, n ) | (cid:19) . Consequently k ξ n ( x ) − ξ n ( y ) k ≤ n n X k = n +1 k χ S ( x,k,n ) − χ S ( y,k,n ) k≤ − n n X k = n +1 | S ( x, k − m, n ) || S ( x, k + m, n ) | ! ≤ − n Y k = n +1 | S ( x, k − m, n ) || S ( x, k + m, n ) | ! /n = 2 − Q n + mj = n +1 − m | S ( x, j, n ) | Q n + mj =2 n +1 − m | S ( x, j, n ) | ! /n ≤ (cid:0) − p ( n ) − m/n (cid:1) . We have used the inequality between the arithmetic and geometric mean in themiddle step, magic cancellation of many terms in the penultimate step and thelast step uses (iii) plus a simple estimate of the sizes of sets by 1 from below. By(iii), the last expression converges to 0 as n converges to ∞ . We are done. (cid:3) Remark . In the condition (iii), we ask for a bound in terms of l . However, it isapparent from the proof that a bound in terms of k with analogous property alsosuffices. Remark . It is clear from the proof of the Proposition that we only need to definethe sets S ( x, k, l ) only for an infinite sequence of indices l (and the corresponding k ∈ { , . . . , l } ), not necessarily for all l ∈ N .4. CAT(0) cube complexes
Proposition 3.1 allows us to quickly prove that finite dimensional CAT(0) cubecomplexes have property A. This was first proved by Brodzki et. al. [BCG + Proposition 4.1.
Let X be a finite dimensional CAT(0) cube complex. Then X has property A. OARSE MEDIANS AND PROPERTY A 8
Proof.
Fix a base vertex x ∈ X . Given a vertex x ∈ X , l ∈ N and k ∈ { , . . . , l } ,consider the normal cube path from y to x , where ρ ( y, x ) ≤ k . We define theset S ( x, k, l ) to contain the 3 l -th vertex on such a normal cube path (or x if werun out of space). Note that the conditions (i) and (ii) from Proposition 3.1 areautomatically satisfied, courtesy of the triangle inequality. To be more precise, if z ∈ S ( x, k, l ), then ρ ( x, z ) ≤ ld , where d = dim( X ).To prove the condition (iii) of Proposition 3.1, we shall argue that if z ∈ S ( x, k, l ), then z ∈ [ x, x ]. Or, equivalently: Claim:
Every half-space containing both x and x contains also z .Each hyperplane that we need to consider (i.e. such that one of the associatedhalf-spaces contains both x and x ) either separates x, x from y or it does not. Inthe latter case, the same half-space also clearly contains z , so it remains to dealwith the former case.Denote by C , C , . . . , C m the normal cube path from y to x , and denote by y = v , v , . . . , v m = x the vertices on this cube path. We shall argue thatany hyperplane separating y from x, x is “used” within the first ρ ( x, y ) steps onthe cube path. Suppose that the cube C i does not cross any hyperplane H with { y }| H { x, x } . Hence every hyperplane K crossing C i satisfies { y, x }| K { x , v i +1 } . Ifthere was a hyperplane L with { y }| L { x, x } which was not “used” before C i on thecube path, then necessarily { y, v i +1 }| L { x, x } , hence L crosses all the hyperplanes K crossing C i . This contradicts the maximality of this step on the normal cubepath. Thus there is no such L , and so all the hyperplanes H with { y }| H { x, x } must be crossed within the first ρ ( x, y ) steps (as there is at most ρ ( x, y ) of suchhyperplanes).Since z is the 3 l -th vertex on the cube path and ρ ( x, y ) ≤ k ≤ l , all thehyperplanes H with { y }| H { x, x } must have been crossed before z . Thus any such H actually also satisfies { y }| H { x, x , z } . We have proved our claim.Coming back to showing the condition (iii) of Proposition 3.1, observe that theinterval [ x, x ] embeds isometrically into the cube complex R d [BCG +
09, Theorem1.14], hence it has polynomial growth. This means that as S ( x, k, l ) ⊂ [ x, x ] ∩ B ( x, ld ), its cardinality is bounded by a polynomial in l (of degree d ). Thisfinishes the proof. (cid:3) For the record, we note that dropping the finite dimensionality assumption ren-ders the statement false, namely infinite dimensional CAT(0) cube complexes donot have Property A; this follows from [Now07], as they contain isometric copiesof ( Z / Z ) n for arbitrarily large n .5. Median algebras
Definition 5.1.
Let Φ be a median algebra. Let n ≥ x , . . . , x n , b ∈ Φ.Define µ ( x ; b ) := x OARSE MEDIANS AND PROPERTY A 9 and inductively for 1 ≤ k < n − µ ( x , . . . , x k +1 ; b ) := µ ( µ ( x , . . . , x k ; b ) , x k +1 , b ) . Note that this definition “agrees” with the original median map µ , since µ ( x , x ; b ) = µ ( x , x , b ).Intuitively, µ ( x , . . . , x n ; b ) should be thought of as a projection of b onto thehull { x , . . . , x n } , just as µ ( x , x , b ) is the projection of b onto [ x , x ]. However,we do not prove this in this note (but see Lemma 5.3). Lemma 5.2.
The µ symbol from Definition 5.1 is symmetric in x , . . . , x n .Proof. Recalling that interchanging the points in µ ( · , · , · ) is one of the axioms ofa median algebra, it is clearly sufficient to prove that for n = 3, we can switch x with x . However, applying axioms of median algebras, we have that µ ( x , x , x ; b ) = µ ( µ ( x , x , b ) , x , b )= µ ( µ ( x , b, x ) , µ ( x , b, b ) , x )= µ ( µ ( x , x , b ) , x , b )= µ ( x , x , x ; b ) . We are done. (cid:3)
In fact, it is easy to see that [ x , b ] ∩ [ x , b ] = [ µ ( x , x , b ) , b )] and then byinduction that T nk =1 [ x k , b ] = [ µ ( x , . . . , x n ; b ) , b ]. Lemma 5.3.
Let Φ be a median algebra. Let x , . . . , x n , b ∈ Φ . (i) A wall separates b and x , . . . , x n if and only if it separates b and µ ( x , . . . , x n ; b ) . (ii) If µ ( x , . . . , x n − ; b ) = µ ( x , . . . , x n ; b ) then there exists a wall separating x , . . . , x n − from x n , b . (iii) If, in addition, Φ has rank at most d , then there exists a subset { y , . . . , y k } ⊆{ x , . . . , x n } with k ≤ d , such that µ ( y , . . . , y k ; b ) = µ ( x , . . . , x n ; b ) . (iv) Assume that a ∈ Φ and x , . . . , x n ∈ [ a, b ] . Then { x , . . . , x n } ⊂ [ a, µ ( x , . . . , x n ; b )] .Proof. Since µ ( x, y, b ) ∈ [ x, y ] = J ( { x, y } ), we can easily prove by induction that µ ( x , . . . , x n ; b ) ∈ J n − ( { x , . . . , x n } ) ⊂ hull { x , . . . , x n } . The statement “= ⇒ ” in(i) follows. For the converse, assume for contradiction that there exists a wall W that separates b and µ ( x , . . . , x n ; b ) but does not separate b from (say, usingLemma 5.2) x n . As half-spaces are convex, the whole interval [ b, x n ] is in the samehalf-space than b . But as µ ( · , x n , b ) ∈ [ b, x n ], this contradicts the assumption that b separates µ ( x , . . . , x n ; b ) = µ ( µ ( x , . . . , x n − ; b ) , x n , b ).For (ii), denote c = µ ( x , . . . , x n − ; b ) and note that µ ( x , . . . , x n ; b ) = µ ( c, x n , b ).Hence c = µ ( x , . . . , x n ; b ) implies c [ x n , b ]. As { c } and [ x n , b ] are convex, thisimplies, by [Bow13a, Lemma 6.1], that there is a wall separating c and x n , b . By(i), this wall separates x , . . . , x n − and x n , b . OARSE MEDIANS AND PROPERTY A 10
For (iii), we proceed by contradiction. Assume that there are at least d + 1points in { x , . . . , x n } which cannot be removed from the expression µ ( x , . . . , x n ; b )without changing the result. The previous part of the lemma, together with Lemma5.2, implies that there exist at least d + 1 different walls which all intersect (forinstance, if one wall separates x , . . . , x n − and x n , b , and another one separates x , . . . , x n − , x n and x n − , b , they clearly intersect). This contradicts the rankassumption (see [Bow13a, Proposition 6.2]).The part (iv) follows by induction from the following statement: if x, y ∈ [ a, b ],then x, y ∈ [ a, µ ( x, y, b )] (so in particular [ a, x ] ⊂ [ a, µ ( x, y, b )]). It is of coursesufficient to prove that x ∈ [ a, µ ( x, y, b )], which is done using median axioms asfollows: µ ( a, x, µ ( x, y, b )) = µ ( µ ( a, x, x ) , µ ( a, x, b ) , y ) = µ ( x, x, y ) = x. We are done. (cid:3)
Lemma 5.4.
Let Φ be a median algebra and let a, b, x, y ∈ Φ satisfy x, y ∈ [ a, b ] .Then y ∈ [ a, x ] implies x ∈ [ y, b ] .Proof. Using the median axioms and lemma’s assumptions, we have µ ( y, b, x ) = µ ( µ ( a, x, y ) , b, x ) = µ ( µ ( b, x, a ) , µ ( b, x, x ) , y ) = µ ( x, x, y ) = x. (cid:3) Proposition 5.5.
Let Φ be a topological median algebra of rank at most d . Givenan interval [ a, b ] ⊂ Φ and a compact set C ⊂ [ a, b ] , there exists h , . . . , h d ∈ C ,such that C ⊂ [ a, µ ( h , . . . , h d ; b )] .If Φ is moreover a metric median algebra satisfying the condition (L2), then wehave ρ ( a, µ ( h , . . . , h d ; b )) ≤ d K d max ≤ i ≤ d ρ ( a, h i ) ≤ d K d sup h ∈ C ρ ( a, h ) .Proof. Consider the compact space C d . Given a tuple ξ ∈ C d , write µ ( ξ ; b ) for theshort. Given ξ ∈ C d , define A ξ = (cid:8) η ∈ C d | µ ( η ; b ) ∈ [ µ ( ξ ; b ) , b ] (cid:9) = (cid:8) η ∈ C d | [ a, µ ( ξ ; b )] ⊂ [ a, µ ( η ; b )] (cid:9) . Note that the two conditions are equivalent: if µ ( η ; b ) ∈ [ µ ( ξ ; b ) , b ], then by Lemma5.4 µ ( ξ ; b ) ∈ [ a, µ ( η ; b )], hence [ a, µ ( ξ ; b )] ⊂ [ a, µ ( η ; b )]. Conversely, the last in-clusion implies µ ( ξ ; b ) ∈ [ a, µ ( η ; b )], so again by Lemma 5.4 we have µ ( η ; b ) ∈ [ µ ( ξ ; b ) , b ].Observe that each set A ξ is closed: In fact it is exactly the inverse image of theclosed set [ µ ( ξ ; b ) , b ] ⊂ Φ under the continuous map µ ( · ; b ) : C d → Φ.Finally, the collection { A ξ | ξ ∈ C d } of subsets of C d has the finite intersectionproperty. Given ξ, η ∈ C d , by Lemma 5.3 (iii), there is ω ∈ C d , such that µ ( ω ; b ) = µ ( ξ ∪ η ; b ). However, from the definition of µ and Lemma 5.2, we know that µ ( ξ ∪ η ; b ) belongs to both [ µ ( ξ ; b ) , b ] and [ µ ( η ; b ) , b ]. In other words, ω ∈ A ξ ∩ A η . Intervals are closed in topological median algebras; this just uses continuity of µ . OARSE MEDIANS AND PROPERTY A 11
Now, as C d is compact, there exists ζ ∈ T ξ ∈ C d A ξ . This means that [ a, µ ( ξ ; b )] ⊂ [ a, µ ( ζ ; b )] for all ξ ∈ C d . In particular, by Lemma 5.3 (iv), ξ ⊂ [ a, µ ( ξ ; b )] ⊂ [ a, µ ( ζ ; b )] for all ξ ∈ C d , so [ a, µ ( ζ ; b )] contains all the points of C . Now justenumerate ζ as h , . . . , h d .For the second part of the proposition, we do inductive estimates using (L2).Denote T = max ≤ i ≤ d ρ ( a, h i ). Then, as the first step, ρ ( a, µ ( h ; b )) = ρ ( a, h ) ≤ T . We show inductively that ρ ( a, µ ( h , . . . , h i ; b )) ≤ i − K i − T . Assuming thisinequality for i , denoting µ ( h , . . . , h i ; b ) = g i we estimate ρ ( a, µ ( h , . . . , h i +1 ; b )) = ρ ( a, µ ( g i , h i +1 ; b )) ≤ ρ ( a, g i ) + ρ ( µ ( g i , g i , b ) , µ ( g i , h i +1 , b )) ≤ i − K i − T + Kρ ( g i , h i +1 ) ≤ i − K i − T + K (cid:0) i − K i − T + T (cid:1) = T (cid:0) i − K i + 3 i − K i − + K (cid:1) ≤ i K i T. We are done. (cid:3) Coarse medians
We shall adapt the idea of ‘moving deep into the interval’ from the CAT(0) cubecomplex setting to the more general coarse median spaces.To explain the idea, consider two points a, b and the context–appropriate notionof the interval [ a, b ]. In the CAT(0) cube complex case, we have moved deep intothis interval by stepping sufficiently far along the cube path from a to b . In thecoarse median case we shall be, roughly speaking, looking for ‘the other end’ of theconvex hull of B ( a, l ) ∩ [ a, b ] (cf. the second bullet in Corollary 6.3). Following thesuit of [Bow14], this is done by going to the asymptotic cone (where the results ofSection 5 can be applied).We begin by fixing a fair amount of notation.For the rest of this section, when we say that X is a coarse median space, wemean that X is a coarse median space, with metric denoted ρ , the median functiondenoted µ , with parameters K and H . These will be fixed throughout.When convenient, we shall be using the notation x ∼ s y for ρ ( x, y ) ≤ s .Given τ ≥ a, b ∈ X , we shall denote by [ a, b ] τ the coarse interval between a and b , i.e. [ a, b ] τ := { x ∈ X | µ ( a, b, x ) ∼ τ x } .We will denote by λ ≥ K and H ), such thatfor all x, y, z ∈ X we have µ ( x, y, z ) ∈ [ x, y ] λ ; its existence is proved in [Bow14,Lemma 9.2].Recall that since the median axiom (M3) holds exactly in median algebras, itdoes hold in coarse median spaces up to a constant γ ≥
0, depending only on theparameters
K, H of the coarse median structure (actually γ = 3 K (3 K + 2) H (5) + OARSE MEDIANS AND PROPERTY A 12 (3 K +2) H (0)). By this we mean that µ ( x, y, µ ( z, v, w )) ∼ γ µ ( µ ( x, y, z ) , µ ( x, y, v ) , w )for all x, y, z, v, w ∈ X . We shall be using γ and this fact throughout this section.Fixing some more notation, given r, t ≥
0, denote L ( r ) = ( K + 1) r + Kλ + γ + 2 H (0) ,L ( r ) = ( K + 2) r + H (0) , and L ( r, t ) = 3 d K d rt + r. The point is that L and L are linear functions of r , and L is linear in r with t fixed, and bounded by a linear function of rt (for t ≥ Lemma 6.1.
Let X be a coarse median space, r ≥ , and let a, b ∈ X , x ∈ [ a, b ] λ .Then [ a, x ] r ⊂ [ a, b ] L ( r ) .Proof. Let z ∈ [ a, x ] r . Thus µ ( a, x, z ) ∼ r z and by assumption also µ ( a, b, x ) ∼ λ x .Hence µ ( a, b, z ) ∼ Kr + H (0) µ ( a, b, µ ( a, x, z )) ∼ γ µ ( µ ( a, b, a ) , µ ( a, b, x ) , z ) ∼ Kλ + H (0) µ ( a, x, z ) ∼ r z. Thus ρ ( µ ( a, b, z ) , z ) ≤ ( K + 1) r + Kλ + γ + 2 H (0) = L ( r ), which means bydefinition that z ∈ [ a, b ] L ( r ) . (cid:3) In what follows, r can be thought of ‘a scale’ and t of ‘a distance’. In otherwords, the statements can read as ‘given a distance ( t ) at which we want the spaceto behave, there exists a scale ( r t ), such that on all larger scales ( r ≥ r t ) it behavesas a median space, with an error proportional to r ’. Proposition 6.2.
Let X be a quasigeodesic coarse median space of rank at most d . For every κ > and t > , there exists r t > , such that for all r ≥ r t , a, b ∈ X and A ⊂ B ( a, rt ) ∩ [ a, b ] κ with sep( A ) ≥ r , there exists h ∈ [ a, b ] L ( r ) , such that • ρ ( a, h ) ≤ L ( r, t ) and • A ⊂ [ a, h ] r . Corollary 6.3.
Let X be a quasigeodesic coarse median space of rank at most d .For every κ > and t > , there exists r t > , such that for all r ≥ r t , a, b ∈ X there exists h ∈ [ a, b ] L ( r ) , such that • ρ ( a, h ) ≤ L ( r, t ) and • B ( a, rt ) ∩ [ a, b ] κ ⊂ [ a, h ] L ( r ) .Proof of Corollary 6.3. This readily follows from Proposition 6.2, by observingthat we may choose A to be a maximal r -separated subset of B ( a, rt ) ∩ [ a, b ] κ .Then any point x ∈ B ( a, rt ) ∩ [ a, b ] κ is at most r -far from a point a x ∈ A , hencethe condition A ⊂ [ a, h ] r implies that µ ( a, h, x ) ∼ Kr + H (0) µ ( a, h, a x ) ∼ r a x ∼ r x. OARSE MEDIANS AND PROPERTY A 13
Since we denoted Kr + H (0) + r + r = L ( r ), the above reads x ∈ [ a, h ] L ( r ) . (cid:3) Proof of Proposition 6.2.
We proceed by contradiction: suppose that for some κ and t the statement is not true, i.e. there exists a sequence 0 < r < r < · · · ∈ R and for each n ∈ N there exist a n , b n ∈ X and A n ⊂ [ a n , b n ] κ ∩ B ( a n , r n t ) withsep( A n ) ≥ r n , such that for all h ∈ [ a n , b n ] L ( r n ) with ρ ( a n , h ) ≤ L ( r n , t ) thereexists x ∈ A n with ρ ( x, µ ( a n , h, x )) > r n .It follows from [Bow14, Lemma 9.7] that we can assume that the cardinalities | A n | are uniformly bounded by a constant p depending on K, H, d, κ, t .The next step is to argue that we can arrange that the distances from a n to b n are linear in r n . Claim.
There exist constants δ , δ , κ ≥ K, H, κ, t and p ) and points b ′ n ∈ [ a n , b n ] λ , such that ρ ( a n , b ′ n ) ≤ δ r n + δ and A n ⊂ [ a n , b ′ n ] κ .The Claim follows from [Bow14, Lemma 9.6], which says that in our situationthere are constants ζ , ξ, κ ′ (depending only on K, H, κ, t and p ), and points c n , d n ∈ X , such that A n ⊂ [ c n , d n ] κ ′ and diam( A n ∪ { c n , d n } ) ≤ ζ diam( A n ) + ξ ≤ ζ r n t + ξ .Since A n ⊂ B ( a n , r n t ), by the proof of that Lemma we can assume that c n = a n for every n . Finally, we define b ′ n = µ ( a n , b n , d n ) ∈ [ a n , b n ] λ and check that b ′ n = µ ( a n , b n , d n ) ∼ K (2 ζr n t + ξ )+ H (0) µ ( d n , b n , d n ) = d n ∼ ζr n t + ξ a n , and for every x ∈ A n (so that µ ( a n , b n , x ) ∼ κ x and µ ( a n , d n , x ) ∼ κ ′ x ) µ ( a n , b ′ n , x ) = µ ( a n , µ ( a n , b n , d n ) , x ) ∼ γ µ ( µ ( a n , x, b n ) , µ ( a n , x, d n ) , a n ) ∼ K ( κ + κ ′ )+ H (0) µ ( x, x, a n ) = x. So altogether, we put κ = K ( κ + κ ′ ) + H (0) + γ , δ = 2( K + 1) ζ t and δ =( K + 1) ξ + H (0) and the Claim is proved.We have now set up the situation so that we can conclude the proof by goingto the asymptotic cone.Let ( X ∞ , ρ ∞ , µ ∞ ) be an asymptotic cone of X , with the sequence of scales ( r n ),basepoints ( a n ) and any non-principal ultrafilter on N .The sequences ( a n ) and ( b ′ n ) determine points a, b ∈ X ∞ (with ρ ∞ ( a, b ) ≤ δ ),and the intervals [ a n , b ′ n ] κ converge to the interval [ a, b ] in X ∞ . Also the sets A n converge to a (finite, 1-separated) set A ⊂ [ a, b ] ∩ B ( a, t ).By Proposition 5.5, there exists h ∈ [ a, b ], such that A ⊂ B ( a, t ) ∩ [ a, b ] ⊂ [ a, h ]and ρ ∞ ( a, h ) ≤ d K d t . This implies that we have a sequence of points h n ∈ X , eventually in [ a n , b ′ n ] r n , such that lim ρ ( a n , h n ) /r n ≤ d K d t , thus eventually ρ ( a n , h n ) ≤ d K d r n t + r n = L ( r n , t ).Since b ′ n ∈ [ a n , b n ] λ , and h n ∈ [ a n , b ′ n ] r n , by Lemma 6.1 we have that h n ∈ [ a n , b n ] L ( r n ) . Hence, by our original assumption, there (eventually) exist points x n ∈ A n with ρ ( x n , µ ( a n , h n , x n )) > r n . The sequence of x n yields a point x ∈ A , Since µ ( a, b, h ) = h , we have ρ ( h n , µ ( a n , b ′ n , h n )) /r n →
0, hence the claim.
OARSE MEDIANS AND PROPERTY A 14 such that ρ ∞ ( x, µ ∞ ( a, h, x )) ≥
1. This point witnesses that A [ a, h ], which is acontradiction. (cid:3) Lemma 6.4.
Let X be a coarse median space. There exist constants α, β ≥ (depending only on the parameters of the coarse median structure), such that thefollowing holds: let a, b, h, m ∈ X and r ≥ satisfy m ∈ [ a, h ] L ( r ) , h ∈ [ a, b ] L ( r ) .Then p = µ ( m, b, h ) satisfies ρ ( h, p ) ≤ αr + β .Proof. Note that the assumptions say that m ∼ L ( r ) µ ( a, h, m ) and h ∼ L ( r ) µ ( a, b, h ). We estimate p = µ ( m, b, h ) ∼ KL ( r )+ H (0) µ ( µ ( a, h, m ) , b, h ) ∼ γ µ ( µ ( b, h, h ) , µ ( b, h, a ) , m ) ∼ KL ( r )+ H (0) µ ( h, h, m ) = h. Altogether, ρ ( h, p ) ≤ K ( L ( r ) + L ( r )) + 2 H (0) + γ , which is a linear function of r . (cid:3) Theorem 6.5.
Let X be a uniformly locally finite, quasigeodesic, at most expo-nential growth, coarse median space of finite rank. Then X has property A.Proof. The proof follows the idea of our proof for CAT(0) cube complexes, whichrelies on Proposition 3.1. We shall verify its assumptions. Let α, β > x ∈ X .We now apply Corollary 6.3 for κ = λ and all t ∈ N to obtain a sequence r t ∈ N ,so that the conclusion of the Corollary holds. Furthermore, we can choose the r t inductively to arrange that the sequence N ∋ t l t = tr t − H (0)3 K is increasing.For a moment, fix x ∈ X , t ∈ N and k ∈ { , . . . , l t } . For every y ∈ B ( x, k ),Corollary 6.3 applied for a = y , b = x and r = r t produces for us a point h y ∈ [ y, x ] L ( r t ) . We collect these points into the set S ( x, k, l t ) = { h y ∈ X | y ∈ B ( x, k ) } . Loosely speaking, the set S ( x, k, l t ) contains one point associated to each y ∈ B ( x, k ), which should be thought of as being “ tr t -deep” inside the interval [ y, x ] λ .Defined like this, the condition (ii) of Proposition 3.1 is automatic, (i) followsfrom the first bullet of Corollary 6.3, and finally (iii) requires some checking:Take y ∈ B ( x, k ), with the notation as above. Denote m y = µ ( x, y, x ). Then m y ∈ [ y, x ] λ and ρ ( y, m y ) ≤ Kρ ( x, y ) + H (0) ≤ K · l t + H (0) = tr t . Thus the second bullet of Corollary 6.3 implies that m y ∈ [ y, h y ] L ( r t ) . Since wealso know that h y ∈ [ y, x ] L ( r t ) , Lemma 6.4 now implies that the point p y = µ ( m y , x , h y ) ∈ [ m y , x ] λ satisfies ρ ( h y , p y ) ≤ αr t + β . As m y = µ ( x, y, x ) ∈ [ x, x ] λ , Lemma 6.1 implies p y ∈ [ x, x ] L ( λ ) . OARSE MEDIANS AND PROPERTY A 15
To summarise, for each h y ∈ S ( x, k, l t ) we can associate a point p y ∈ [ x, x ] L ( λ ) satisfying ρ ( h y , p y ) ≤ αr t + β , and consequently also ρ ( x, p y ) ≤ ρ ( x, y ) + ρ ( y, h y ) + ρ ( h y , p y ) ≤ l t + 3 d K d tr t + r t + αr t + β, which is clearly depends linearly on l t . Hence, by [Bow14, Proposition 9.8], thenumber of the possible points p y is bounded by P ( l t ) for some polynomial P (depending only on H , K , d , and uniform local finiteness of X ). Since we assumeat most exponential growth of X , it follows that the cardinality of S ( x, k, l t ) is atmost P ( l t ) c ′ c r t for some constants c, c ′ ≥
1. Finally, as l t → ∞ means by definitionalso t → ∞ , it is easy to see that also r t /l t →
0, thus ( P ( l t ) c ′ c r t ) /l t →
1. Thisfinishes proving the condition (iii) of Proposition 3.1 and we are done. (cid:3)
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Mathematical Sciences, University of Southampton, SO17 1BJ, United Kingdom
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