A four point characterisation for coarse median spaces
AA FOUR POINT CHARACTERISATION FOR COARSEMEDIAN SPACES
GRAHAM A. NIBLO, NICK WRIGHT, AND JIAWEN ZHANGA bstract . Coarse median spaces simultaneously generalise theclasses of hyperbolic spaces and median algebras, and arise nat-urally in the study of the mapping class groups and many othercontexts. Their definition as originally conceived by Bowditch re-quires median approximations for all finite subsets of the space.Here we provide a simplification of the definition in the formof a 4-point condition analogous to Gromov’s 4-point conditiondefining hyperbolicity. We give an intrinsic characterisation ofrank in terms of the coarse median operator and use this to givea direct proof that rank 1 geodesic coarse median spaces are δ -hyperbolic, bypassing Bowditch’s use of asymptotic cones. A keyingredient of the proof is a new definition of intervals in coarsemedian spaces and an analysis of their interaction with geodesics.
1. I ntroduction
Coarse median spaces and groups were introduced by Bowditchin 2013 [5] as a coarse variant of classical median algebras. The no-tion of a coarse median group leads to a unified viewpoint on severalinteresting classes, including Gromov’s hyperbolic groups, mappingclass groups, and CAT(0) cubical groups. Bowditch showed that ge-odesic hyperbolic spaces are exactly geodesic coarse median spacesof rank 1, and mapping class groups are examples of coarse me-dian spaces of finite rank [5]. In 2014 [3, 4], Behrstoke, Hagen andSisto introduced the notion of hierarchically hyperbolic spaces, andshowed that these are coarse median.
Mathematics Subject Classification.
Key words and phrases.
Coarse median space, canonical metric, hyperbolicity,rank.Partially supported by the Sino-British Fellowship Trust by Royal Society. a r X i v : . [ m a t h . M G ] F e b GRAHAM A. NIBLO, NICK WRIGHT, AND JIAWEN ZHANG
Intuitively, a coarse median space ( X , d , µ ) is a metric space ( X , d )equipped with a ternary operator µ (called the coarse median), inwhich every finite subset can be approximated by a finite CAT(0)cube complex, with distortion controlled by the metric. This can beviewed as a wide-ranging extension of Gromov’s observation thatin a δ -hyperbolic space finite subsets can be approximated by trees.See also Zeidler’s Master’s thesis [23].In this paper we simplify the definition of a coarse median space,replacing the requirement to approximate arbitrary finite subsetswith a simplified 4-point condition which may be viewed as a highdimensional analogue of Gromov’s 4-point condition for hyperbol-icity. Our condition asserts that given any four points a , b , c , d thetwo iterated coarse medians µ ( µ ( a , b , c ) , b , d ) and µ ( a , b , µ ( c , b , d )) areuniformly close. As illustrated below this corresponds to an ap-proximation by a CAT(0) cube complex of dimension 3, where thecorresponding iterated medians coincide.F igure
1. The CAT(0) cube complex associated to thefree median algebra on { a , b , c , d } We recall that Gromov gave a 4-point condition characterisinghyperbolicity for geodesic spaces which, in essence, asserts that
FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 3 any four points can be approximated by one of the trees shownin Figure 2. We can visualise each of these trees as degenerateF igure
2. Gromov’s 4-point conditioncases of Figure 1 in which two of the dimensions of the cube arecollapsed, making clear the relationship between Gromov’s 4-pointcondition and ours, and the sense in which coarse median spaces area higher dimensional analogue of δ -hyperbolic spaces. The presenceof the central 3-cube, which may be arbitrarily large, allows for flatgeometry.The coarse 4-point condition is also a coarse analogue of the 4-point characterisation for median algebras, as introduced and stud-ied by Kolibiar and Marcisov ´a in [13]. To establish the equivalencewith Bowditch’s original definition we introduce a model for the freemedian algebra generated by n points, which may be of independentinterest.Any median algebra generated by 4 points can be modelled bythe 3-dimensional CAT(0) cube complex illustrated in Figure 1, soit is a priori di ffi cult to see how to characterise rank using our newdefinition. We overcome this by o ff ering several intrinsic character-isations of rank in terms of the coarse median operator itself andwhich are equivalent to Bowditch’s definition. GRAHAM A. NIBLO, NICK WRIGHT, AND JIAWEN ZHANG
The rank 1 geodesic case is of independent interest, since, as re-marked above, it coincides with the class of geodesic hyperbolicspaces, [5]. Bowditch’s proof that rank 1 geodesic coarse medianspaces are hyperbolic uses an ingenious asymptotic cones argu-ment which conceals in part the strong interaction between quasi-geodesics and coarse median intervals in this case. Introducinga new definition of interval in a coarse median space, we give analternative proof, bypassing the asymptotic cones argument, and in-stead exploiting a result of Papasoglu [17] and Pomroy [18], see also[9]. We also consider the behaviour of quasi-geodesics in higherrank coarse median spaces, giving an example in rank 2 to showthat even geodesics can wander far from the intervals defined bythe coarse median operator. In a subsequent paper, [15] we furtherdevelop the concept of the coarse interval structure associated to acoarse median space and, as an application, we show there that themetric data is determined by the coarse median operator itself.The paper is organised as follows. In Section 2, we recall Bowditch’sdefinition of coarse median spaces and introduce our notion of(coarse) intervals (which di ff er in one small but crucial respect fromthe intervals studied by Bowditch). In Section 3, we establish our4-point condition characterising coarse median spaces. In Section 4,we give several characterisations for rank in terms of the coarse me-dian operator and give a new proof of Bowditch’s result concerningthe hyperbolicity of rank 1 geodesic coarse median spaces. Finallyin Section 5, we construct an example to show that geodesics do nothave to remain close to intervals in a coarse median space of rankgreater than 1. 2. P reliminaries Metrics and geodesics.Definition 2.1.
Let ( X , d ) and ( Y , d (cid:48) ) be metric spaces.(1) ( X , d ) is said to be quasi-geodesic , if there exist constants L , C > x , y ∈ X , there exists a map γ : [0 , d ( x , y )] → X with γ (0) = x , γ ( d ( x , y )) = y , satisfying: for FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 5 any s , t ∈ [0 , d ( x , y )], L − | s − t | − C (cid:54) d ( γ ( s ) , γ ( t )) (cid:54) L | s − t | + C . If we care about the constants we say that ( X , d ) is ( L , C )-quasi-geodesic, and if we do not care about the constant C we saythat ( X , d ) is L-quasi-geodesic . If ( X , d ) is (1 , X is geodesic . When considering integer-valued metrics we make the same definitions restricting theintervals to intervals in Z .(2) A map f : ( X , d ) → ( Y , d (cid:48) ) is bornologous if there exists anincreasing map ρ + : R + → R + such that for all x , y ∈ X , d (cid:48) ( f ( x ) , f ( y )) (cid:54) ρ + ( d ( x , y )).(3) ( X , d ) is said to be uniformly discrete if there exists a constant C > x (cid:44) y ∈ X , d ( x , y ) (cid:62) C .(4) Two points x , y ∈ X are said to be s-close (with respect to themetric d ) if d ( x , y ) (cid:54) s . If x is s -close to y we write x ∼ s y .2.2. CAT(0) Cube Complexes.
Before considering coarse medianspaces, we first recall basic notions and results about CAT(0) cubecomplexes. We will survey the properties we need here, but guidethe interested reader to [7, 10, 11, 14, 20] for more information.A cube complex is a polyhedral complex in which each cell is iso-metric to a unit Euclidean cube and the gluing maps are isometries.The dimension of the complex is the maximum of the dimensionsof the cubes. For a cube complex X , we can associate it with the intrinsic pseudo-metric d int , which is the maximal pseudo-metric on X such that each cube embeds isometrically. When X is connected andhas finite dimension, d int is a complete geodesic metric on X . See [7]for a general discussion on polyhedral complex and the associatedintrinsic metric. A geodesic metric space is CAT(0) if all its geodesictriangles are slimmer than the comparative triangle in the Euclideanspace. For a cube complex ( X , d int ), Gromov gave a combinatorialcharacterisation of the CAT(0) condition [11]: X is CAT(0) if andonly if it is simply connected and the link of each vertex is a flagcomplex (see also [7]). GRAHAM A. NIBLO, NICK WRIGHT, AND JIAWEN ZHANG
We also consider the edge path metric d on the vertex set V ofa CAT(0) cube complex. For x , y ∈ V , the interval is defined to be[ x , y ] = { z ∈ V : d ( x , y ) = d ( x , z ) + d ( x , y ) } , which consists of points onany edge path geodesic between x and y . A CAT(0) cubical complex X can be equipped with a set of hyperplanes [8, 14, 16, 20] such thateach edge is crossed by exactly one hyperplane. Each hyperplanedivides the space into two halfspaces, and the metric d counts thenumber of hyperplanes separating a pair of points. The dimensionof X , if it is finite, is the maximal number of pairwise intersectinghyperplanes. We say that a subset is convex if it is an intersection ofhalf spaces, and we can equivalently define the interval [ x , y ] to bethe intersection of all the halfspaces containing both x and y .Another characterisation of the CAT(0) condition was obtainedby Chepoi [10] (see also [19]): a flag cube complex X is CAT(0) ifand only if for any x , y , z ∈ V , the intersection [ x , y ] ∩ [ y , z ] ∩ [ z , x ]consists of a single point µ ( x , y , z ), which is called the median of x , y , z .Obviously, m ( x , y , z ) ∈ [ x , y ], and[ x , y ] = { m ( x , y , z ) : z ∈ V } = { w ∈ V : m ( x , y , w ) = w } A graph such as X (1) satisfying this condition is called a median graph .Given a CAT(0) cube complex, we always take the canonical me-dian structure ( V , m ) defined by intersection of intervals as above.The pair ( V , m ) is a median algebra [12], as defined in the followingsection.2.3. Median Algebras.
As discussed in [1], there are a number ofequivalent formulations of the axioms for median algebras. We willuse the following formulation from [13], see also [2]:
Definition 2.2.
Let X be a set and m a ternary operation on X . Then m is a median operator and the pair ( X , m ) is a median algebra if:(M1) Localisation: m ( a , a , b ) = a ;(M2) Symmetry: m ( a , a , a ) = m ( a σ (1) , a σ (2) , a σ (3) ), where σ is any per-mutation of { , , } ;(M3) The 4-point condition: m ( m ( a , b , c ) , b , d ) = m ( a , b , m ( c , b , d )). FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 7
Condition (M3) is illustrated by Figure 1, which shows the freemedian algebra generated by the 4 points a , b , c , d . As shown in thefigure, the iterated medians on both sides of the equality in (M3)evaluate to the vertex adjacent to b . Example . An important example is furnished by the “ median n-cube ”, denoted by I n , which is the n -dimensional vector space over Z with the median operator µ n given by majority vote on eachcoordinate. Definition 2.4.
The rank of a median algebra ( X , m ) is the supremumof those n for which there is a subalgebra of ( X , m ) isomorphic to themedian algebra ( I n , µ n ).For the median algebra defined by the vertex set of a CAT(0) cubecomplex, the rank coincides with the dimension of the cube complex.For any points a , b ∈ X we define the interval between a , b to be[ a , b ] : = { m ( a , x , b ) : x ∈ X } . Axioms (M1) ∼ (M3) ensure that [ a , b ] is also equal to the set { c ∈ X : m ( a , c , b ) = c } , since if c = m ( a , b , x ) then: m ( c , a , b ) = m ( m ( x , a , b ) , a , b ) = m ( x , m ( a , b , a ) , b ) = m ( x , a , b ) = c . We think of m ( a , x , b ) as the projection of x onto the interval [ a , b ],and axiom (M3) can be viewed as an associativity axiom: For each b ∈ X the binary operator( a , c ) (cid:55)→ a ∗ b c : = m ( a , b , c )is associative. It is also commutative by (M2) and iterated projectiongives rise to the iterated median introduced in [21]. Definition 2.5 ([21]) . Let ( X , m ) be a median algebra. For x ∈ X ,define m ( x ; b ) : = x , and for k (cid:62) x , . . . , x k + ∈ X , define m ( x , . . . , x k + ; b ) : = m ( m ( x , . . . , x k ; b ) , x k + , b ) . GRAHAM A. NIBLO, NICK WRIGHT, AND JIAWEN ZHANG
Note that this definition “agrees” with the original median operator m , since m ( x , x ; b ) = m ( x , x , b ).In the notation above, the definition reduces to: m ( x , . . . , x k ; b ) = x ∗ b x ∗ b . . . ∗ b x k . A subset Y of A is said to be convex if m ( x , b , y ) ∈ Y for all x , y ∈ Y and b ∈ X , or equivalently, if it is closed under the binary operation ∗ b for all b ∈ X . The set { m ( x , . . . , x n ; b ) | b ∈ X } is the convex hull ofthe points x i and we think of the iterated median m ( x , . . . , x n ; b ) asthe projection of b onto the convex hull.We recall several properties of the iterated median operator provedin the original paper [21]. Lemma 2.6 ([21]) . Let ( X , m ) be a median algebra, and x , . . . , x n , a , b ∈ X.Then:(1) The iterated median operator defined above is symmetric in x , . . . , x n ;(2) n (cid:84) k = [ x k , b ] = [ m ( x , . . . , x n ; b ) , b ] ;(3) If, in addition, X has rank at most d, then there exists a subset { y , . . . , y k } ⊆ { x , . . . , x n } with k (cid:54) d, such thatm ( y , . . . , y k ; b ) = m ( x , . . . , x n ; b ); (4) Assume that x , . . . , x n ∈ [ a , b ] , then { x , . . . , x n } ⊆ [ a , m ( x , . . . , x n ; b )] . We remark that condition (1) here follows immediately from thecommutativity and associativity of the binary operator ∗ b .We will also make use of the following “(n + Lemma 2.7.
Let ( X , m ) be a median algebra and a , b , e , . . . , e n + ∈ X, thenm ( a , e n + , m ( e , . . . , e n ; b )) = m ( m ( a , e n + , e ) , . . . , m ( a , e n + , e n ); b ) . Proof.
We prove this by induction on n . FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 9 n =
1: the equation holds trivially. Assume it holds for all k (cid:54) n .Then m ( a , e n + , m ( e , . . . , e n ; b )) = m (cid:16) a , e n + , m ( m ( e , . . . , e n − ; b ) , e n , b ) (cid:17) = m (cid:16) m ( a , e n + , m ( e , . . . , e n − ; b )) , m ( a , e n + , e n ) , b (cid:17) = m (cid:16) m ( m ( a , e n + , e ) , . . . , m ( a , e n + , e n − ); b ) , m ( a , e n + , e n ) , b (cid:17) = m ( m ( a , e n + , e ) , . . . , m ( a , e n + , e n ); b ) , where we use the inductive assumption in the third equation. (cid:3) We now mention two alternative definitions for median algebras.According to Isbell [12], a ternary operator m defines a median ifand only if it satisfies (M1), (M2) and Isbell’s condition : m ( a , m ( a , b , c ) , m ( b , c , d )) = m ( a , b , c ) . This says that m ( a , b , c ) is in the interval from a to m ( b , c , d ), or ingeometrical terms that the projection of a onto [ b , c ] provided bythe median lies between a and every other point ( m ( b , c , d )) of theinterval [ b , c ].An alternative and algebraically powerful formulation, is that( X , m ) is a median algebra if it satisfies (M1), (M2) and the five pointcondition : m ( m ( a , b , c ) , d , e ) = m ( a , m ( b , d , e ) , m ( c , d , e )) , which is the n = e = b one recovers(M3), while m ( a , m ( d , b , c ) , ( a , b , c )) = m ( m ( a , d , a ) , b , c ) = m ( a , b , c )recovering Isbell’s condition.2.4. Coarse median spaces.
In [5], Bowditch introduced coarse me-dian operators as follows:
Definition 2.8 (Bowditch, [5]) . Given a metric space ( X , d ), a coarsemedian (operator) on X is a ternary operator µ : X → X satisfying thefollowing conditions: (C1). There is an a ffi ne function ρ ( t ) = Kt + H such that for any a , b , c , a (cid:48) , b (cid:48) , c (cid:48) ∈ X , d ( µ ( a , b , c ) , µ ( a (cid:48) , b (cid:48) , c (cid:48) )) (cid:54) ρ ( d ( a , a (cid:48) ) + d ( b , b (cid:48) ) + d ( c , c (cid:48) )) . (C2). There is a function H : N → [0 , + ∞ ), such that for any finitesubset A ⊆ X with 1 (cid:54) | A | (cid:54) p , there exists a finite medianalgebra ( Π , µ Π ) and maps π : A → Π , λ : Π → X such that forany x , y , z ∈ Π , a ∈ A , λµ Π ( x , y , z ) ∼ H ( p ) µ ( λ x , λ y , λ z ) , and λπ a ∼ H ( p ) a . We may assume that Π is generated by π ( A ).We say that two coarse median operators µ , µ on a metric space( X , d ) are uniformly close if there is a uniform bound on the set ofdistances { d ( µ ( a , b , c ) , µ ( a , b , c )) | a , b , c ∈ X } . Remark . The control function ρ in (C1) is required by Bowditchto be a ffi ne, however it seems more natural in the context of coarsegeometry to allow ρ to be an arbitrary function and indeed muchof what we show in this paper works with that variation. Thismay allow wider applications in the future (in [15] we introduceand study such a generalisation in the context of coarse intervalstructures), though we note that when X is a quasi-geodesic spacethe existence of any control function ρ guarantees that ρ may bereplaced by an a ffi ne control function as in (C1). In this paperwe will assume that ρ is a ffi ne throughout, but note that many ofthe statements and arguments can be suitably adapted to the moregeneral context.We refer to the functions ρ, H appearing in conditions (C1), (C2)as parameters of the coarse median, or since ρ has the form ρ ( t ) = Kt + H we will sometimes refer to K , H , H as parameters of thecoarse median. The parameters are not unique, and neither are they FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 11 part of the data of the coarse median; it is merely their existencewhich is required.
Remark . As noted by Bowditch there is a constant κ > • µ ( a , a , b ) ∼ κ a ; • µ ( a , a , a ) ∼ κ µ ( a σ (1) , a σ (2) , a σ (3) ) for any permutation σ ∈ S .It follows that any coarse median operator on ( X , d ) may be replacedby another to which it is uniformly close and which satisfies thefirst two median axioms (M1) and (M2), i.e. we may always assumethat κ =
0. We note here that this is true even if we replace thea ffi ne control function by an arbitrary ρ as discussed above, taking κ = ρ (3 H (3)) + H (3).With this in mind we take the following as our definition of acoarse median space. Definition 2.11. A coarse median space is a triple ( X , d , µ ) where d is ametric on X and µ is a ternary operator on X satisfying conditions(M1), (M2), (C1) and (C2).A map f between coarse median spaces ( X , d X , µ X ) , ( Y , d Y , µ Y ) issaid to be an L-quasi-morphism if for any a , b , c ∈ X , µ Y ( f ( a ) , f ( b ) , f ( c )) ∼ L f ( µ X ( a , b , c )).We note that given a median algebra any metric on this will satisfyaxiom (C2), however the metric must be chosen carefully if we wishit to also satisfy axiom (C1). Of course in the case that the medianalgebra has finite intervals, and hence arises as the vertex set of aCAT(0) cube complex, then both the intrinsic and edge path metricssatisfy this axiom.2.5. The rank of a coarse median space.
Rank is a proxy for dimen-sion in the context of coarse median spaces, directly analogous tothe notion of dimension for a CAT(0) cube complex.
Definition 2.12.
Let n be a natural number. We say X has rank atmost n if there exist parameters ρ, H for which we can always choose the approximating median algebra Π in condition (C2) to have rankat most n .We remark that for a median algebra equipped with a suitablemetric making it a coarse median space, the rank as a median algebragives an upper bound for the rank as a coarse median space, howeverthese need not agree. For example any finite median algebra hasrank 0 as a coarse median space.Zeidler [23] showed that, at the cost of increasing the rank of theapproximating median algebra, one can always assume that the map λπ from condition (C2) is the inclusion map ι A : A (cid:44) → X . We willnow show that this can be achieved without increasing the rank: Lemma 2.13.
Assume ( X , d , µ ) is a coarse median space. Then, at the costof changing the parameter function H, one can always change the triples Π , λ, π provided by axiom (C2), so that λπ = ι A : A (cid:44) → X (the inclusionmap), without changing the rank of Π .Proof. Given a finite subset A ⊆ X with 1 (cid:54) | A | (cid:54) p , let the finitemedian algebra ( Π , µ Π ) and maps π : A → Π , λ : Π → X be as in thedefinition above.We now construct another finite median algebra ( Π (cid:48) , µ (cid:48) Π ). As a set, Π (cid:48) = Π (cid:116) A . Define a map τ : Π (cid:48) → Π by τ x = x if x ∈ Π , and τ a = π a if a ∈ A . Now define a median µ (cid:48) Π on Π (cid:48) by: • µ (cid:48) Π ( a , a , x ) = µ (cid:48) Π ( a , x , a ) = µ (cid:48) Π ( x , a , a ) = a , if a ∈ A and x ∈ Π (cid:48) ; • µ (cid:48) Π ( x , y , z ) = µ Π ( τ x , τ y , τ z ), otherwise.It is a direct calculation to check that µ (cid:48) Π satisfies the axioms of amedian operator. Now define π (cid:48) : A → Π (cid:48) by π (cid:48) a = a ; and λ (cid:48) : Π (cid:48) =Π (cid:116) A → X by λ (cid:48) = λ (cid:116) ι A . For any x , y , z ∈ Π (cid:48) : if two of them areequal and sit in A , say x = y ∈ A , then λ (cid:48) µ (cid:48) Π ( x , y , z ) = λ (cid:48) x = x = µ ( λ (cid:48) x , λ (cid:48) y , λ (cid:48) z );otherwise, we have: λ (cid:48) µ (cid:48) Π ( x , y , z ) = λ (cid:48) µ Π ( τ x , τ y , τ z ) = λµ Π ( τ x , τ y , τ z ) ∼ H ( p ) µ ( λ (cid:48) τ x , λ (cid:48) τ y , λ (cid:48) τ z ) ∼ ρ (3 H ( p )) µ ( λ (cid:48) x , λ (cid:48) y , λ (cid:48) z ) , FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 13 where in the last estimate we use (C1) and the fact that for any x ∈ A , λ (cid:48) τ x = λ (cid:48) π x = λπ x ∼ H ( p ) x = λ (cid:48) x . Now for any a ∈ A , λ (cid:48) π (cid:48) a = λ (cid:48) a = a , and by the construction, it isobvious that rank Π (cid:48) = rank Π .In conclusion we have constructed Π (cid:48) , λ (cid:48) , π (cid:48) such that λ (cid:48) π (cid:48) is theinclusion, Π (cid:48) has the same dimension as Π and λ (cid:48) µ (cid:48) Π ( x , y , z ) ∼ H (cid:48) ( p ) µ ( λ (cid:48) x , λ (cid:48) y , λ (cid:48) z ) , where H (cid:48) : p (cid:55)→ ρ (3 H ( p )) + H ( p ). (cid:3) According to the above lemma there is no loss of generality inassuming that the triples Π , λ, π provided by axiom (C2) satisfy theadditional condition that λπ is the inclusion map. Hereafter we willassume that parameters ρ, H (or K , H , H ) for a coarse median arechosen such that this holds.In the finite rank case we introduce the following terminology. Definition 2.14.
For a coarse median space ( X , d , µ ), we say thatrank X (cid:54) n can be achieved under parameters H , ρ (or K , H , H ) if onecan always choose Π in condition (C2) with rank at most n and λπ = ι A : A (cid:44) → X . By Lemma 2.13 there is no change to the definitionof rank.2.6. Iterated coarse medians.
By analogy with the iterated mediansdefined in Section 2.3, we can define the iterated coarse median operator in a coarse median space.
Definition 2.15.
Let ( X , d , µ ) be a coarse median space and b ∈ X .For x ∈ X define µ ( x ; b ) : = x , and for k (cid:62) x , . . . , x k + ∈ X , define µ ( x , . . . , x k + ; b ) : = µ ( µ ( x , . . . , x k ; b ) , x k + , b ) . Note that this definition “agrees” with the original coarse medianoperator µ in the sense that for any a , b , c in X , µ ( a , b , c ) = µ ( a , b ; c ). We extend the estimates provided by axioms (C1) and (C2) fora coarse median operator to hold more generally for the iteratedcoarse median operators as follows:
Lemma 2.16.
Let ( X , d ) be a metric space with ternary operator µ satisfying(C1) with parameter ρ . Then for any n there exists an increasing (a ffi ne)function ρ n depending on ρ , such that for any a , a , . . . , a n , b , b , . . . , b n ∈ X: d ( µ ( a , . . . , a n ; a ) , µ ( b , . . . , b n ; b )) (cid:54) ρ n (cid:16) n (cid:88) k = d ( a k , b k ) (cid:17) . Proof.
We carry out induction on n . This is trivial for n = ρ ( t ) = t . Now consider the case n > n − d ( µ ( a , . . . , a n ; a ) , µ ( b , . . . , b n ; b )) = d (cid:16) µ ( µ ( a , . . . , a n − ; a ) , a n , a ) , µ ( µ ( b , . . . , b n − ; b ) , b n , b ) (cid:17) (cid:54) ρ (cid:16) d ( µ ( a , . . . , a n − ; a ) , µ ( b , . . . , b n − ; b )) + d ( a n , b n ) + d ( a , b ) (cid:17) (cid:54) ρ (cid:16) ρ n − (cid:16) n − (cid:88) k = d ( a k , b k ) (cid:17) + d ( a n , b n ) + d ( a , b ) (cid:17) (cid:54) ρ (cid:16) ρ n − (cid:16) n (cid:88) k = d ( a k , b k ) (cid:17) + n (cid:88) k = d ( a k , b k ) (cid:17) = ρ n (cid:16) n (cid:88) k = d ( a k , b k ) (cid:17) , where ρ n ( t ) : = ρ ( ρ n − ( t ) + t ). We use (C1) in the third line, and theinductive assumption in the fourth. Note that as ρ n − is increasing ρ n is also increasing. (cid:3) Recall that in Bowditch’s definition of a coarse median space ax-iom (C2) says that for any finite A ⊆ X with | A | (cid:54) p , the approxi-mation map σ : ( Π , µ Π ) → ( X , µ ) is an H ( p )-quasi-morphism, i.e. forany x , y , z ∈ Π , σµ Π ( x , y , z ) ∼ H ( p ) µ ( σ x , σ y , σ z ) . FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 15
Lemma 2.17.
Let ( X , d ) be a metric space with ternary operator µ satisfying(C1) with parameter ρ , ( Π , µ Π ) a median algebra, and σ : Π → X an L-quasi-morphism. Then there exists a constant H n ( L ) (depending on ρ ) suchthat for any x , . . . , x n , b ∈ Π , σ ( µ Π ( x , . . . , x n ; b )) ∼ H n ( L ) µ ( σ ( x ) , . . . , σ ( x n ); σ ( b )) . Proof.
We carry out induction on n . For the case n =
1, set H ( L ) = n =
2, set H ( L ) = L . Now assume n >
2, for any x , . . . , x n , b ∈ Π , we have: σ ( µ Π ( x , . . . , x n − ; b )) ∼ H n − ( L ) µ ( σ ( x ) , . . . , σ ( x n − ); σ ( b )) . Then σ ( µ Π ( x , . . . , x n ; b )) = σ ( µ Π ( µ Π ( x , . . . , x n − ; b ) , x n , b )) ∼ L µ ( σ ( µ Π ( x , . . . , x n − ; b )) , σ ( x n ) , σ ( b )) ∼ ρ ( H n − ( L )) µ ( µ ( σ ( x ) , . . . , σ ( x n − ); σ ( b )) , σ ( x n ) , σ ( b )) = µ ( σ ( x ) , . . . , σ ( x n ); σ ( b )) , where we use the definition of quasi-morphism in the second line,and (C1) as well as the inductive assumption in the third line. Finally,take H n ( L ) = ρ ( H n − ( L )) + L for n > (cid:3) Now we prove a coarse version of Lemma 2.7. Note that the case n = Lemma 2.18.
Let ( X , d , µ ) be a coarse median space with parameters ρ, H, then there exists a constant C n depending on ρ, H such that forany a , b , e , . . . , e n + ∈ X, µ ( a , e n + , µ ( e , . . . , e n ; b )) ∼ C n µ ( µ ( a , e n + , e ) , . . . , µ ( a , e n + , e n ); b ) . Proof.
We prove this by induction on n .The case n = κ = ρ ( H (5)) + ρ (2 H (5)) + H (5) such that: ∀ x , y , z , v , w ∈ X ,(1) µ ( x , y , µ ( z , v , w )) ∼ κ µ ( µ ( x , y , z ) , µ ( x , y , v ) , w ) . So we can take C = κ .Now assume that n > n −
1. Then µ ( a , e n + , µ ( e , . . . , e n ; b )) = µ (cid:16) a , e n + , µ ( µ ( e , . . . , e n − ; b ) , e n , b ) (cid:17) ∼ κ µ (cid:16) µ ( a , e n + , µ ( e , . . . , e n − ; b )) , µ ( a , e n + , e n ) , b (cid:17) ∼ ρ ( C n − ) µ (cid:16) µ ( µ ( a , e n + , e ) , . . . , µ ( a , e n + , e n − ); b ) , µ ( a , e n + , e n ) , b (cid:17) = µ ( µ ( a , e n + , e ) , . . . , µ ( a , e n + , e n ); b ) , where we use the inductive assumption in the third line. Set C n = ρ ( C n − ) + κ and we are done. (cid:3) We note that this result still holds if we replace the a ffi ne controlfunction ρ by an arbitrary increasing control function.We conclude our consideration of iterated coarse medians withthe following lemma. Lemma 2.19.
Let ( X , d , µ ) be a coarse median space with parameters ρ, H, then there exists a constant D n depending on ρ, H such that forany a , b , c , e , . . . , e n ∈ X, µ ( a , b , µ ( e , . . . , e n ; c )) ∼ D n µ ( µ ( a , b , e ) , . . . , µ ( a , b , e n ); µ ( a , b , c )) . Proof.
The proof is by induction; the case n = D =
0. Now assume n > n −
1. Then µ ( a , b , µ ( e , . . . , e n ; c )) = µ (cid:16) a , b , µ ( µ ( e , . . . , e n − ; c ) , e n , c ) (cid:17) ∼ κ µ (cid:16) a , b , µ (cid:16) a , b , µ ( µ ( e , . . . , e n − ; c ) , e n , c ) (cid:17)(cid:17) ∼ ρ ( κ ) µ (cid:16) a , b , µ (cid:16) µ ( a , b , µ ( e , . . . , e n − ; c ) , µ ( a , b , e n ) , c ) (cid:17)(cid:17) ∼ κ µ (cid:16) µ ( a , b , µ ( e , . . . , e n − ; c ) , µ ( a , b , µ ( a , b , e n )) , µ ( a , b , c ) (cid:17)(cid:17) ∼ ρ ( κ ) µ (cid:16) µ ( a , b , µ ( e , . . . , e n − ; c ) , µ ( a , b , e n ) , µ ( a , b , c ) (cid:17)(cid:17) ∼ ρ ( D n − ) µ (cid:16) µ ( µ ( a , b , e ) , . . . , µ ( a , b , e n − ); µ ( a , b , c )) , µ ( a , b , e n ) , µ ( a , b , c ) (cid:17)(cid:17) = µ ( µ ( a , b , e ) , . . . , µ ( a , b , e n ); µ ( a , b , c )) FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 17 where we use the inductive assumption in the sixth inequality. Set D n = ρ ( D n − ) + ρ ( κ ) + κ and we are done. (cid:3) Again this result holds in the context of arbitrary control functions.2.7. (Coarse) intervals.
In CAT(0) cube complexes intervals playan important role, indeed the natural median is determined by theinterval structure and vice versa. Similarly, in coarse median spaces,one needs to consider coarse analogues of intervals. Some of thesewere introduced by Bowditch [6].
Definition 2.20.
Given ( X , µ ) a set equipped with a ternary operator µ , we define the interval between points x and z to be:[ x , z ] = { µ ( x , y , z ) : y ∈ X } . This should be contrasted with Bowditch’s definition, [6], of the λ - coarse interval between points x and z in a coarse median space( X , d , µ ) as: [ x , z ] λ = { y ∈ X : µ ( x , y , z ) ∼ λ y } . Clearly [ x , z ] ⊆ [ x , z ] and, as noted in Section 2.3, if ( X , µ ) is a medianalgebra then [ x , z ] = [ x , z ], however these two notions of intervaldo not always coincide in a coarse median space. An example isprovided in Section 5.In a CAT(0) cube complex the median of three points is always theunique point in the intersection of the three intervals they define.Bowditch [6] showed the same result holds coarsely in a coarsemedian space. We adapt this to our notion of interval as follows (asusual the result works whether or not the control function is a ffi ne). Lemma 2.21.
Let ( X , d , µ ) be a coarse median space with parameters ρ, H.Then for any x , y , z ∈ X, µ ( x , y , z ) ∈ [ x , y ] κ , i.e. [ x , y ] ⊆ [ x , y ] κ , where κ is the constant ρ ( H (5)) + ρ (2 H (5)) + H (5) defined above. In particular, µ ( x , y , z ) ∈ [ x , y ] κ ∩ [ y , z ] κ ∩ [ x , z ] κ . Note that generally, the coarse interval [ a , b ] κ is not closed underthe coarse median µ . However, we can endow it with another coarsemedian µ a , b such that [ a , b ] κ is closed under µ a , b , and µ a , b is uniformlyclosed to µ as follows: Lemma 2.22.
Let ( X , d , µ ) be a coarse median space, and κ be as definedabove. Then there exists a constant C > depending on the chosenparameters ρ, H (and not on ( X , d , µ ) itself), such that for any a , b ∈ X andx , y , z ∈ [ a , b ] κ , µ a , b ( x , y , z ) : = µ ( a , b , µ ( x , y , z )) is C-close to µ ( x , y , z ) . Inconclusion, µ a , b is indeed a coarse median on the coarse interval [ a , b ] κ ,and µ a , b is uniformly close to µ . The same result holds if we use [ a , b ] instead of [ a , b ] κ .Proof. In the above setting, we have µ a , b ( x , y , z ) = µ ( a , b , µ ( x , y , z )) ∼ κ µ ( µ ( a , b , x ) , µ ( a , b , y ) , z ) ∼ ρ (2 κ ) µ ( x , y , z ) , where in the last estimate we use that fact that x , y ∈ [ a , b ] κ .Finally, µ a , b ( x , y , z ) = µ ( a , b , µ ( x , y , z )) sits in [ a , b ] ⊆ [ a , b ] κ . (cid:3)
3. T he point condition defines a coarse median space A fundamental di ffi culty with verifying that a space satisfiesBowditch’s axioms is that one needs to establish approximationsfor subsets of arbitrary cardinality. Here we will provide an alter-native characterisation of coarse median spaces, for which one needonly consider subsets of cardinality up to 4.Applying the axioms (C1), (C2), it is not hard to show that thereis a constant κ = ρ ( H (4)) + H (4) such that in any coarse medianspace ( X , d , µ ) with parameters ρ, H , for any points a , b , c , d we have: µ ( µ ( a , b , c ) , b , d ) ∼ κ µ ( a , b , µ ( c , b , d )) . This provides a coarse analogue of Kolibiar’s 4-point axiom (M3).We will provide the following converse:
Theorem 3.1.
Let ( X , d ) be a metric space, and µ : X → X a ternaryoperation. Then µ is a coarse median on ( X , d ) (i.e., it satisfies conditions(C1) and (C2)) if and only if the following three conditions hold:(C0)’ Coarse localisation and coarse symmetry:
There is a constant κ such that for all points a , a , a in X, µ ( a , a , a ) ∼ κ a , and µ ( a σ (1) , a σ (2) , a σ (3) ) ∼ κ µ ( a , a , a ) for any permutation σ of { , , } ; FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 19 (C1)’ A ffi ne control: There exists an a ffi ne function ρ : [0 , + ∞ ) → [0 , + ∞ ) such that for all a , a (cid:48) , b , c ∈ X,d ( µ ( a , b , c ) , µ ( a (cid:48) , b , c )) (cid:54) ρ ( d ( a , a (cid:48) )); (C2)’ Coarse 4-point condition:
There exists a constant κ > such thatfor any a , b , c , d ∈ X, we have: µ ( µ ( a , b , c ) , b , d ) ∼ κ µ ( a , b , µ ( c , b , d )) . We note that if we are interested in generalising coarse medianspaces to allow arbitrary control functions then the a ffi ne controlaxiom (C1)’ should be replaced by the requirement that the map a (cid:55)→ µ ( a , b , c ) is bornologous uniformly in b , c .Free median algebras will play a crucial role in the proof and nextwe will give a concrete construction of the free median algebra on p points as a quotient of the space of formal ternary expressions on p variables. Under the name of formal median expressions these wereoriginally introduced by Zeidler in [23].Throughout this section, we fix an integer p > α , . . . , α p .3.1. Formal ternary expressions and formal median identities.
Weintroduce a model to define formal ternary expressions on the al-phabet Ω p = { α , . . . , α p } . We introduce two additional symbols: ” (cid:104) ”and ” (cid:105) ”, and set (cid:101) Ω p = Ω p ∪ { (cid:104) , (cid:105) } . Let (cid:101) Ω (cid:63) p be the set of all finitewords in (cid:101) Ω p . Definition 3.2.
Define M p to be the unique smallest subset A in (cid:101) Ω (cid:63) p containing Ω p and satisfying: for any ϕ , ϕ , ϕ ∈ A , we have (cid:104) ϕ ϕ ϕ (cid:105) ∈ A . Elements in M p are called formal ternary expressions invariables α , . . . , α p , or formal ternary expressions with p variables . Notethat M p carries a natural ternary operation:( ϕ , ϕ , ϕ ) (cid:55)→ (cid:104) ϕ ϕ ϕ (cid:105) . Lemma 3.3.
For any ϕ ∈ M p \ Ω p , there exist unique ϕ , ϕ , ϕ ∈ M p such that ϕ = (cid:104) ϕ ϕ ϕ (cid:105) . Proof.
Each word in M p \ Ω p has the form (cid:104) ϕ ϕ ϕ (cid:105) since otherwisewe can delete the words not satisfying this condition to get a smallerset, which is a contradiction to the minimality of M p .Now we claim: for any ϕ, ψ ∈ M p , if ψ is a prefix of ϕ (i.e. thereexists a word w ∈ (cid:101) Ω (cid:63) p such that ϕ = ψ w ), then they are equal. Ifthe claim holds then it follows that each ϕ ∈ M p \ Ω p can be writtenuniquely in the form (cid:104) ϕ ϕ ϕ (cid:105) . Indeed, assume ϕ = (cid:104) ϕ ϕ ϕ (cid:105) = (cid:104) ψ ψ ψ (cid:105) for ϕ i , ψ i ∈ M p . Since ψ is a prefix of ϕ or vice versa,so by the claim, we have ψ = ϕ ; similarly, we have ψ = ϕ and ψ = ϕ .Now we prove the claim by induction on the word length of ϕ .The claim holds trivially for any ϕ of word length 1, i.e. ϕ ∈ Ω p .Assume that ϕ has word length n and that for any word of lengthless than n the claim holds. Suppose that ψ is a prefix of ϕ , andassume ψ = (cid:104) ψ ψ ψ (cid:105) and ϕ = (cid:104) ϕ ϕ ϕ (cid:105) for ϕ i , ψ i ∈ M p . As ψ is aprefix of ϕ , we know ψ is a prefix of ϕ or vice versa. Inductivelywe have ψ = ϕ . Similarly, we have ψ = ϕ and ψ = ϕ , so ψ = ϕ ,as required. (cid:3) Definition 3.4.
For each formal ternary expression ϕ ∈ M p , we canassociate a natural number ξ ( ϕ ) to it inductively as follows, whichis called the complexity of ϕ : • If ϕ ∈ Ω p , i.e. ϕ = α i for some i , we define ξ ( α i ) = • Assume we have defined complexities for all the formalternary expressions with length (as a word in (cid:101) Ω (cid:63) p ) less than n >
1, then for ϕ with length n , by Lemma 3.3, there existsunique ϕ , ϕ , ϕ ∈ M p such that ϕ = (cid:104) ϕ ϕ ϕ (cid:105) . Since each ϕ i has length less than n , ξ ( ϕ i ) has already been defined. Nowwe define the complexity ξ ( ϕ ) : = max { ξ ( ϕ ) , ξ ( ϕ ) , ξ ( ϕ ) } + M p ( n ) ⊆ M p be the set of all formal ternary expressions havingcomplexity less than or equal to n , and S p ( n ) = M p ( n ) \ M p ( n −
1) bethe set of all formal ternary expressions having complexity equal to n (set S p (0) = M p (0)). FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 21
Remark . It’s obvious by definition that M p = ∞ (cid:91) n = M p ( n ) = ∞ (cid:71) n = S p ( n ) , and, by an easy inductive argument, M p ( n ) is finite for each n .Lemma 3.3 says that for any formal ternary expression ϕ (cid:60) M p (0),there exist unique formal ternary expressions ϕ , ϕ , ϕ ∈ M p with ξ ( ϕ i ) < ξ ( ϕ ) and ϕ = (cid:104) ϕ ϕ ϕ (cid:105) . Lemma 3.6.
Let Y be a set with ternary operation µ and M p be the set offormal ternary expressions on the alphabet Ω p . Then any map Ω p → Yextends uniquely to a map of ternary algebras M p → Y. Thus M p is thefree ternary algbera on p variables.Proof. Let y , . . . , y p ∈ Y be the images of the variables { α , . . . , α p } in Ω p . Given ϕ ∈ M p we define an element ϕ Y ( y , . . . , y p ) inductivelyas follows:(1) For ϕ = α i ∈ M p (0), define ϕ Y ( y , . . . , y p ) = y i ;(2) If ϕ = (cid:104) ϕ ϕ ϕ (cid:105) ∈ M p ( n ) , n > ϕ , ϕ , ϕ all lie in M p ( n − ϕ Y ( y , . . . , y p ) = µ (cid:16) ( ϕ ) Y ( y , . . . , y p ) , ( ϕ ) Y ( y , . . . , y p ) , ( ϕ ) Y ( y , . . . , y p ) (cid:17) . By construction the map ϕ (cid:55)→ ϕ Y ( y , . . . , y p ) is a map of ternaryalgebras extending the map from Ω p to Y as required.Conversely, the condition that we have a map of ternary algebrasextending the original map implies that ϕ Y must satisfy conditions1 and 2 above yielding uniqueness. (cid:3) Let Y be a set with ternary operation µ , and y , . . . , y p ∈ Y . Given aformal ternary expression ϕ on p variables, we define its evaluationat ( y , . . . , y p ) to be the image ϕ Y ( y , . . . , y p ) of ϕ provided by Lemma3.6. We call the corresponding map ϕ Y : Y p → Y the realisation of ϕ .Given two formal ternary expressions ϕ, ψ ∈ M p , we say thatthe equation ϕ = ψ is a formal median identity , if it holds in anymedian algebra. To be more precise, for any median algebra ( Y , µ )and y , . . . , y p ∈ Y , we have: ϕ Y ( y , . . . , y p ) = ψ Y ( y , . . . , y p ) . It is easy to see that the axioms for a median operator immediatelylead to the following formal median identities: ϕ = (cid:104) ϕϕψ (cid:105) , ∀ ϕ, ψ ∈ M p ; (cid:104) ϕ ϕ ϕ (cid:105) = (cid:104) ϕ σ (1) ϕ σ (2) ϕ σ (3) (cid:105) , ∀ ϕ , ϕ , ϕ ∈ M p and σ ∈ S ; (cid:104)(cid:104) ϕ ϕ ϕ (cid:105) ϕ ϕ (cid:105) = (cid:104) ϕ ϕ (cid:104) ϕ ϕ ϕ (cid:105)(cid:105) , ∀ ϕ , . . . , ϕ ∈ M p . Two formal ternary expressions are said to be equivalent , written ϕ ∼ med ψ , if ϕ = ψ is a formal median identity. Clearly this isan equivalence relation on M p . Now suppose that ϕ = ψ , ϕ = ψ , ϕ = ψ are formal median identities. Then (cid:104) ϕ ϕ ϕ (cid:105) = (cid:104) ψ ψ ψ (cid:105) is also a formal median identity, so the ternary operator on M p de-scends to a ternary operator on the quotient of M p by the equivalencerelation ∼ med . Moreover this operator makes the quotient itself a me-dian algebra as it now satisfies the median axioms. Given any othermedian algebra ( Y , µ ) and points y , . . . , y p ∈ Y the universal mapof ternary algebras from M p to Y factors through the quotient bydefinition of the formal median identities, so we have shown that: Lemma 3.7.
The quotient M p / ∼ med is the free median algebra on p points. In the case of coarse median spaces the median identities will onlyhold up to bounded error. To quantify the errors we need to expressthe equivalence relation ∼ med explicitly, which we do by describing itin terms of elementary transformations which admit metric control.3.2. Elementary transformations.
We define elementary transforma-tions inductively as follows: • For ϕ ∈ S p (0), the only allowed elementary transformationsare of the form ϕ (cid:55)→ (cid:104) ϕϕψ (cid:105) for some ψ ∈ M p ; • Suppose for elements in M p ( n − ϕ ∈ S p ( n ) we definethe allowed elementary transformations of ϕ as follows: Type I). ϕ (cid:55)→ (cid:104) ϕϕψ (cid:105) for any ψ in M p ; if ϕ = (cid:104) ψψϕ (cid:48) (cid:105) , then ϕ (cid:55)→ ψ ; Type II). If ϕ = (cid:104) ϕ ϕ ϕ (cid:105) , then ϕ (cid:55)→ (cid:104) ϕ σ (1) ϕ σ (2) ϕ σ (3) (cid:105) for some σ ∈ S ; FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 23
Type III). If ϕ = (cid:104)(cid:104) ϕ ϕ ϕ (cid:105) ϕ ϕ (cid:105) , then ϕ (cid:55)→ (cid:104) ϕ ϕ (cid:104) ϕ ϕ ϕ (cid:105)(cid:105) ;or if ϕ = (cid:104) ϕ ϕ (cid:104) ϕ ϕ ϕ (cid:105)(cid:105) , then ϕ (cid:55)→ (cid:104)(cid:104) ϕ ϕ ϕ (cid:105) ϕ ϕ (cid:105) ; Type IV). If ϕ = (cid:104) ϕ ϕ ϕ (cid:105) , and ϕ (cid:48) is obtained from ϕ viaan elementary transformation, then ϕ (cid:55)→ (cid:104) ϕ (cid:48) ϕ ϕ (cid:105) (notethat ϕ ∈ M p ( n − ϕ have already been defined).If ψ is obtained from ϕ by a single elementary transformation wewrite ϕ ∼ ET ψ . It is easy to see that the relation ∼ ET is symmetric, i.e. ϕ ∼ ET ψ if and only if ψ ∼ ET ϕ .3.3. A construction for the free median algebra.
Let ∼ be the tran-sitive closure of ∼ ET on the set M p of formal ternary expressions. Theclosure of ∼ ET is an equivalence relation; we denote the quotient setby M p = M p / ∼ , and use [ ϕ ] to denote the equivalence class of ϕ in M p . Furthermore, we define an induced ternary operator m on M p as follows: m ([ ϕ ] , [ ϕ ] , [ ϕ ]) : = [ (cid:104) ϕ ϕ ϕ (cid:105) ]for ϕ i ∈ M p .We will show that ( M p , m ) is the free median algebra generated by[ α ] , . . . , [ α p ]. Lemma 3.8. m is a well-defined median operator on M p .Proof. We need to check if ϕ ∼ ϕ (cid:48) , ϕ ∼ ϕ (cid:48) and ϕ ∼ ϕ (cid:48) , then (cid:104) ϕ ϕ ϕ (cid:105) ∼ (cid:104) ϕ (cid:48) ϕ (cid:48) ϕ (cid:48) (cid:105) .First suppose that ϕ ∼ ET ϕ (cid:48) . Then (cid:104) ϕ ϕ ϕ (cid:105) (cid:55)→ (cid:104) ϕ (cid:48) ϕ ϕ (cid:105) is atype IV elementary transformation. More generally if ϕ ∼ ϕ (cid:48) thena sequence of type IV transformations will ensure that (cid:104) ϕ ϕ ϕ (cid:105) ∼ (cid:104) ϕ (cid:48) ϕ ϕ (cid:105) . Now in the same way, in addition using type II elementarytransformations, we can show that (cid:104) ϕ (cid:48) ϕ ϕ (cid:105) ∼ (cid:104) ϕ (cid:48) ϕ (cid:48) ϕ (cid:105) ∼ (cid:104) ϕ (cid:48) ϕ (cid:48) ϕ (cid:48) (cid:105) .Hence m is a well defined ternary operator on M p . The fact that itis a median follows immediately by applying the elementary trans-formations to its definition. (cid:3) Proposition 3.9.
The median algebra ( M p , m ) is the free median algebragenerated by { [ α ] , . . . , [ α p ] } . Proof.
One only needs to check the universal property : for any medianalgebra ( Y , ν ) and any map f : Ω p → Y , there exists a unique medianhomomorphism ¯ f : ( M p , m ) → ( Y , ν ) extending f . Since M p is thefree ternary algebra on p points the map f extends uniquely to amap ˜ f : M p → Y . Explicitly ˜ f ( ϕ ) = ϕ Y ( f ( α ) , . . . , f ( α p )).Now we define ¯ f ([ ϕ ]) : = ˜ f ( ϕ ), for any [ ϕ ] ∈ M p . We need to checkthat this is a well-defined median homomorphism. Well-definedness:
It su ffi ces to show if ϕ ∼ ET ϕ for two given for-mal median expressions, then ˜ f ( ϕ ) = ˜ f ( ϕ ). We do it via inductionon ξ ( ϕ ). If ξ ( ϕ ) =
0, we know ϕ = (cid:104) ϕ ϕ ψ (cid:105) for some ψ ∈ M p ,which implies˜ f ( ϕ ) = f ( ϕ ) = ν (cid:16) f ( ϕ ) , f ( ϕ ) , ˜ f ( ψ ) (cid:17) = ν (cid:16) ˜ f ( ϕ ) , ˜ f ( ϕ ) , ˜ f ( ψ ) (cid:17) = ˜ f (cid:16) (cid:104) ϕ ϕ ψ (cid:105) (cid:17) = ˜ f ( ϕ ) . If ϕ = (cid:104) ψ ψ ψ (cid:105) with ξ ( ψ i ) < ξ ( ϕ ), by the definition of elementarytransformations, the proof is divided into four cases: • Type I) to III).
One just needs to observe that these elemen-tary transformations do not change elements in a medianalgebra. For example in the case of a Type III) elementarytransformation, ψ will have the form ψ = (cid:104) ψ ψ ψ (cid:105) forsome ψ , ψ , so that ϕ = (cid:104)(cid:104) ψ ψ ψ (cid:105) ψ ψ (cid:105) , and ϕ will havethe form ϕ = (cid:104) ψ ψ (cid:104) ψ ψ ψ (cid:105)(cid:105) . Then we have:˜ f ( ϕ ) = ν (cid:16) ˜ f ( (cid:104) ψ ψ ψ (cid:105) ) , ˜ f ( ψ ) , ˜ f ( ψ ) (cid:17) = ν (cid:16) ν ( ˜ f ( ψ ) , ˜ f ( ψ ) , ˜ f ( ψ )) , ˜ f ( ψ ) , ˜ f ( ψ ) (cid:17) = ν (cid:16) ˜ f ( ψ ) , ˜ f ( ψ ) , ν ( ˜ f ( ψ ) , ˜ f ( ψ ) , ˜ f ( ψ )) (cid:17) = ν (cid:16) ˜ f ( ψ ) , ˜ f ( ψ ) , ˜ f ( (cid:104) ψ ψ ψ (cid:105) ) (cid:17) = ˜ f (cid:16) (cid:104) ψ ψ (cid:104) ψ ψ ψ (cid:105)(cid:105) (cid:17) = ˜ f ( ϕ ) . Here the third equality follows axiom (M3) for a median alge-bra, while the other equalities all follow from the fact that ˜ f isa map of ternary algebras. The elementary transformationsof Type I) and Type II) can be checked similarly. FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 25 • Type IV). ψ (cid:48) is obtained from ψ via an elementary transforma-tion and ϕ = (cid:104) ψ (cid:48) ψ ψ (cid:105) : by induction one has ˜ f ( ψ (cid:48) ) = ˜ f ( ψ ),which implies˜ f ( ϕ ) = ν (cid:16) ˜ f ( ψ ) , ˜ f ( ψ ) , ˜ f ( ψ ) (cid:17) = ν (cid:16) ˜ f ( ψ (cid:48) ) , ˜ f ( ψ ) , ˜ f ( ψ ) (cid:17) = ˜ f ( ϕ ) . Median homomorphism:
By construction, we have:¯ f (cid:16) m ([ ϕ ] , [ ϕ ] , [ ϕ ]) (cid:17) = ¯ f (cid:16) [ (cid:104) ϕ ϕ ϕ (cid:105) ] (cid:17) = ˜ f (cid:16) (cid:104) ϕ ϕ ϕ (cid:105) (cid:17) = ν (cid:16) ˜ f ( ϕ ) , ˜ f ( ϕ ) , ˜ f ( ϕ ) (cid:17) = ν (cid:16) ¯ f ([ ϕ ]) , ¯ f ([ ϕ ]) , ¯ f ([ ϕ ]) (cid:17) . Uniqueness:
This follows from the definition of M p by an easyinductive argument. (cid:3) As a direct corollary of the above construction, we have the fol-lowing description for formal median identities.
Corollary 3.10.
Given ϕ, ψ ∈ M p , then ϕ = ψ is a formal median identityif and only if ϕ ∼ ψ .Proof. Since M p is a median algebra the equivalence relation ∼ mustcontain ∼ med and the universal map from M p / ∼ med to the quotient M p maps the class of ϕ to the class [ ϕ ]. Since M p is also universal thismap must be an isomorphism and the equivalence relations agree.Thus ϕ = ψ is a formal median identity if and only if ϕ ∼ med ψ if andonly if ϕ ∼ ψ . (cid:3) In [23] Zeidler established that, while a given formal median iden-tity does not have to hold for a coarse median operator, it does holdup to bounded error: for µ a coarse median on a metric space ( X , d )with parameters ρ, H and ϕ = ψ a formal median identity, thereexists a constant R = R ( ρ, H , ϕ, ψ ) such that for all x , . . . , x p ∈ X , ϕ X ( x , . . . , x p ) ∼ R ψ X ( x , . . . , x p ) . In order to prove Theorem 3.1 we need to establish this under our, a priori weaker, axioms ( C (cid:48) ∼ ( C (cid:48) . Proposition 3.11.
Let ( X , d ) be a metric space with a ternary operator µ satisfying (C0)’ ∼ (C2)’ with parameters ρ, κ and κ . Let ϕ = ψ be a formal median identity with p variables. Then there exists a constantR = R ( ρ, κ , κ , ϕ, ψ ) , such that for any x , . . . , x p ∈ X, ϕ X ( x , . . . , x p ) ∼ R ψ X ( x , . . . , x p ) . We remark that if µ is a coarse median on a space ( X , d ) withparameters ρ, H , then (C0)’ ∼ (C2)’ hold with parameters ρ, κ , κ where κ , κ depend only on ρ and H (4), thereby recovering Zeidler’sresult, but with the constant depending only on the parameters ρ, H (4). We also note that if we weaken the a ffi ne control in (C1)’ toa uniform bornology then the result remains true. Proof of 3.11.
Since ρ, κ , κ are fixed throughout the proof, we willomit these parameters from the notation and write R ( ϕ, ψ ) in placeof R ( ρ, κ , κ , ϕ, ψ ).By Corollary 3.10, we may choose a finite sequence of elementarytransformations: ϕ = ψ ∼ ET ψ ∼ ET . . . ∼ ET ψ n = ψ. We will prove the result in the case that ϕ ∼ ET ψ , the general casefollowing from this by adding the constants. By definition, there arefour cases: • Type I) and Type II) : By (C0)’, we have ϕ X ∼ κ ψ X so we choose R ( ϕ, ψ ) = κ ; • Type III) . Assume ϕ = (cid:104)(cid:104) ϕ ϕ ϕ (cid:105) ϕ ϕ (cid:105) and ψ = (cid:104) ϕ ϕ (cid:104) ϕ ϕ ϕ (cid:105)(cid:105) .By (C2)’, we have: ϕ X = µ (cid:16) µ (( ϕ ) X , ( ϕ ) X , ( ϕ ) X ) , ( ϕ ) X , ( ϕ ) X (cid:17) ∼ κ µ (cid:16) ( ϕ ) X , ( ϕ ) X , µ (( ϕ ) X , ( ϕ ) X , ( ϕ ) X ) (cid:17) = ψ X , so we choose R ( ϕ, ψ ) = κ . • Type IV) . Let ϕ, ψ ∈ M p ( n ) and suppose that ϕ is equivalentto ψ by an elementary transformation of Type IV). Thus ϕ = (cid:104) ϕ ϕ ϕ (cid:105) , ψ = (cid:104) ϕ (cid:48) ϕ ϕ (cid:105) with ϕ , ϕ (cid:48) ∈ M p ( n −
1) and ϕ ∼ ET ϕ (cid:48) .If the latter is an elementary transformation of type I), II) orIII) then we know that there is a constant R ( ϕ , ϕ (cid:48) ) such that FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 27 ( ϕ ) X ∼ R ( ϕ ,ϕ (cid:48) ) ( ϕ (cid:48) ) X . Otherwise we have a further type IV)elementary transformation, and by induction on n we mayagain assume that ( ϕ ) X ∼ R ( ϕ ,ϕ (cid:48) ) ( ϕ (cid:48) ) X , the base case n = ϕ X = µ (( ϕ ) X , ( ϕ ) X , ( ϕ ) X ) ∼ ρ ( R ( ϕ ,ϕ (cid:48) )) µ (( ϕ (cid:48) ) X , ( ϕ ) X , ( ϕ ) X ) = ψ X and we set R ( ϕ, ψ ) = ρ ( R ( ϕ , ϕ (cid:48) )). (cid:3) Proof of Theorem 3.1.
We know that (C0)’ ∼ (C2)’ hold in ametric space ( X , d ) with a coarse median µ . Conversely we mustprove that if (C0)’ ∼ (C2)’ hold then (C2) also holds, since (C1)follows easily from (C0)’ and (C1)’.For each p , we choose a splitting M p → M p of the quotient map M p → M p , such that [ α i ] is lifted to α i . In other words, for each p and each element of M p we choose a representative in M p . In amedian algebra, the median stabilisation of finitely many points isitself finite, [22, Lemma 6.20], so M p is finite and thus the splittingsprovide finite subsets Θ p of M p .Now let A be a finite subset of X , with cardinality p , and enumeratethe set A as A = { x , . . . , x p } . Define π : A → M p by x i (cid:55)→ [ α i ]. We con-sider the set of formal median identities of the form (cid:104) ϕ ϕ ϕ (cid:105) = ψ where ϕ , ϕ , ϕ , ψ ∈ Θ p . Since Θ p is finite there are only finitelymany of these. By Proposition 3.11 for each of these formal medianidentities there is a constant R , which does not depend on A , suchthat µ (( ϕ ) X , ( ϕ ) X , ( ϕ ) X ) ∼ R ψ X . Let H p be an upper bound for the(finitely many) constants R arising from these identities. (Note more-over that the constant H p does depend on the parameters ρ, κ , κ but not on the space X itself.)Now we define a map λ : M p → X by [ ϕ ] (cid:55)→ ( ϕ ) X for ϕ ∈ Θ p . Bythe above estimate for ϕ , ϕ , ϕ ∈ Θ p , we have: λ (cid:16) m ([ ϕ ] , [ ϕ ] , [ ϕ ]) (cid:17) = λ (cid:16) [ (cid:104) ϕ ϕ ϕ (cid:105) ] (cid:17) ∼ H p µ (cid:16) ( ϕ ) X , ( ϕ ) X , ( ϕ ) X (cid:17) = µ (cid:16) λ ([ ϕ ]) , λ ([ ϕ ]) , λ ([ ϕ ]) (cid:17) . Clearly, λπ is the inclusion A (cid:44) → X , so we are done. (cid:3)
4. C haracterising rank in a coarse median space
For a coarse median space the rank is determined by the dimen-sions of the approximating median algebras for its finite subsets.However, replacing Bowditch’s approximation axiom (C2) by ouraxiom (C2)’ controlling four point approximations, it is no longersu ffi cient to consider the dimensions of these: the free median alge-bra on four points has dimension 3 (see Figure 1) which clearly doesnot provide a universal bound on the rank.We begin by analysing the case of rank 1 coarse median spaceswhere we reprove Bowditch’s result that for geodesic spaces thisimplies hyperbolicity. We will do this directly, without recourse tothe asymptotic cones that are an essential ingredient in Bowditch’sproof. We will then consider the general case, showing how todefine “coarse cubes” in terms of the coarse median, showing thata coarse median space has rank at most n if and only if it cannotcontain arbitrarily large ( n + − coarse cubes.4.1. Rank 1 coarse median spaces.
According to Bowditch [5], theclass of geodesic coarse median spaces with rank 1 coincides with theclass of geodesic δ -hyperbolic spaces. Given a geodesic hyperbolicspace it is possible to construct a coarse median using the thinnessof geodesic triangles; for the opposite direction Bowditch used themethod of asymptotic cones. Here we give another more direct wayto prove this.We will use the following characterisation of Gromov’s hyper-bolicity for geodesic spaces, developed by Papasoglu and Pomroy,which asserts that for a geodesic metric space stability of quasi-geodesics implies hyperbolicity. Theorem 4.1 ([17, 18, 9]) . Let ( X , d ) be a geodesic metric space. Sup-pose there exists ε > , R > , such that for any a , b ∈ X, all (1 , ε ) -quasi-geodesics from a to b remain in an R-neighbourhood of any geodesicconnecting a , b. Then X is hyperbolic. We will prove that in a rank 1 geodesic coarse median space quasi-geodesics are stable in the above sense. The key idea is to use coarse
FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 29 intervals in place of the geodesics connecting a and b . We will provethe following: Theorem 4.2.
Let ( X , d , µ ) be a geodesic coarse median space with rank . For any ζ, ε > , there exists a constant (cid:101) D = (cid:101) D ( ζ, ε ) , such that for anya , b ∈ X and any ( ζ, ε ) -quasi-geodesic γ : [ s , t ] → X connecting a and b,the Hausdor ff distance between [ a , b ] and Im( γ ) is less than (cid:101) D. Applying Theorem 4.1 we obtain the following corollary.
Corollary 4.3.
Let ( X , d , µ ) be a geodesic coarse median space with rank . Then ( X , d ) is hyperbolic.Proof. According to Theorem 4.2, any (1 , ε ) quasi-geodesic is withinHausdor ff distance (cid:101) D of the interval [ a , b ]. In particular any geodesicfrom a to b is within Hausdor ff distance (cid:101) D of the interval. We nowapply Theorem 4.1 with R = (cid:101) D . (cid:3) We will also give a characterisation of rank 1 coarse median spacesin terms of the geometry of intervals. Note that here we do notrequire the space to be (quasi)-geodesic.
Theorem 4.4.
Let ( X , d , µ ) be a coarse median space. Then X has rank atmost if and only if there exists a constant λ such that for any a , b , x ∈ Xwith x ∈ [ a , b ] , we have: (2) [ a , b ] ⊆ N λ ([ a , x ]) ∪ N λ ([ x , b ]) . We regard the inclusion (2) as an analog of Gromov’s thin trianglescondition for coarse intervals, and begin by proving that, moreover,in a rank 1 space an analogous condition holds for neighbourhoodsof intervals.
Lemma 4.5.
Let ( X , d , µ ) be a coarse median space with rank 1 achievedunder parameters ρ, H. Then for any ζ (cid:62) , there exists a constant ζ (cid:48) = H (4) + ζ such that for any a , b , c ∈ X, we have: N ζ ([ a , b ]) ⊆ N ζ (cid:48) ([ a , c ]) ∪ N ζ (cid:48) ([ c , b ]) . Proof.
For any y ∈ N ζ ([ a , b ]), there exists some x ∈ X , such that µ ( a , b , x ) ∼ ζ y , and we set A = { a , b , c , x } . By Definition 2.11 and Definition 2.14 there exists a finite rank 1 median algebra ( Π , µ Π )(i.e., a tree) and maps π : A → Π , λ : Π → X satisfying the condi-tions in (C2); furthermore λ ◦ π is the inclusion A (cid:44) → X . Denote π ( a ) , π ( b ) , π ( c ) , π ( x ) by a , b , c , x respectively.Set m = µ Π ( a , b , x ), then µ Π ( a , b , m ) = m . Since Π is a tree, we have: µ Π ( a , c , m ) = m or µ Π ( b , c , m ) = m . Without loss of generality, assume the former, so λ ( m ) = λ ( µ Π ( a , c , m )) ∼ H (4) µ ( a , c , λ ( m )) . On the other hand, by the definition of m , we have: λ ( m ) = λ ( µ Π ( a , b , x )) ∼ H (4) µ ( a , b , x ) ∼ ζ y . Combine the above two estimates: y ∼ H (4) + ζ µ ( a , c , λ ( m )) , which implies y ∈ N H (4) + ζ ([ a , c ]). Finally, take ζ (cid:48) = H (4) + ζ , thenthe lemma holds. (cid:3) Recall that in a hyperbolic space, any point on a geodesic from a to b sits logarithmically far, with respect to path-length, from anypath connecting a and b . The analogous statement holds for rank 1spaces, if one replaces geodesics with intervals: Proposition 4.6.
Let ( X , d , µ ) be a geodesic coarse median space with rank1, γ a rectifiable path in X connecting a , b ∈ X and (cid:96) ( γ ) the length of γ ,then there exist constants C , C such that [ a , b ] ⊆ N C log (cid:96) ( γ ) + C (Im γ ) . Proof.
Let K , H , H be parameters of ( X , d , µ ) under which rank X (cid:54) γ has been reparameterised to have unit speed. At the cost of varyingthe constants C , C we can simplify the argument, again withoutloss of generality, and assume (cid:96) ( γ ) = N for some integer N . We willargue by induction on N . FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 31 N = (cid:96) ( γ ) =
1, which implies d ( a , b ) (cid:54)
1. For any c ∈ [ a , b ], thereexists some x ∈ X such that c = µ ( a , b , x ). Now by axiom (C1), wehave: c = µ ( a , b , x ) ∼ K + H µ ( a , a , x ) = a , which implies c ∈ B ( a , K + H ). So [ a , b ] ⊆ N K + H ( γ ) for (cid:96) ( γ ) =
1. Weset s = K + H .Induction step: Assume that we have established that for all pairs a , b ∈ X and rectifiable paths γ (cid:48) connecting them of length 2 N − , thereis a constant s N − such that [ a , b ] ⊆ N s N − (Im γ (cid:48) ). Now consider a path γ of (cid:96) ( γ ) = N joining a to b . We will denote the midpoint of γ by c = γ (2 N − ). By Lemma 4.5, we have:[ a , b ] ⊆ N H (4) ([ a , c ]) ∪ N H (4) ([ c , b ]) . By our inductive hypothesis, we have:[ a , c ] ⊆ N s N − ( γ ([0 , N − ])) and [ c , b ] ⊆ N s N − ( γ ([2 N − , N ])) , which implies [ a , b ] ⊆ N H (4) ([ a , c ]) ∪ N H (4) ([ c , b ]) ⊆ N s N − + H (4) (Im γ ),i.e. we can take s N = s N − + H (4), in other words, s N = K + H + ( N − · H (4). In conclusion, take C = H (4) and C = K + H − H (4),we have [ a , b ] ⊆ N C log (cid:96) ( γ ) + C (Im γ ). (cid:3) As in the hyperbolic case, we need the following lemma to tamequasi-geodesics, replacing an arbitrary quasi-geodesic by a rectifi-able quasi-geodesic close to the original.
Lemma 4.7 ([7]) . Let ( X , d ) be a geodesic metric space. Given any ( ζ, ε ) -quasi-geodesic γ : [ s , t ] → X, one can find a continuous ( ζ, ε (cid:48) ) -quasi-geodesic γ (cid:48) : [ s , t ] → X such that1) γ ( s ) = γ (cid:48) ( s ) , γ ( t ) = γ (cid:48) ( t ) ;2) ε (cid:48) = ζ + ε ) ;3) (cid:96) ( γ (cid:48) | [ s (cid:48) , t (cid:48) ] ) (cid:54) k d ( γ (cid:48) ( s (cid:48) ) , γ (cid:48) ( t (cid:48) )) + k for all s (cid:48) , t (cid:48) ∈ [ s , t ] , where k = ζ ( ζ + ε ) and k = ( ζε (cid:48) + ζ + ε ) ;4) the Hausdor ff distance between Im γ and Im γ (cid:48) is less than ζ + ε . We also need the following two lemmas which hold trivially in themedian case. Recall in a coarse median space with given parameters ρ, H , we defined the constant κ = ρ ( H (5)) + ρ (2 H (5)) + H (5). Lemma 4.8.
Let ( X , d , µ ) be a coarse median space with parameters ρ, H.Then there exists a constant C = ρ ( κ ) + κ such that for all a , b ∈ X,x ∈ [ a , b ] , y ∈ [ a , x ] and z ∈ [ x , b ] , we have x ∈ N C ([ y , z ]) .Proof. By definition there exist w , v , v (cid:48) ∈ X such that x = µ ( a , w , b ), y = µ ( a , v , x ) and z = µ ( x , v (cid:48) , b ). Now µ ( y , w , b ) = µ ( µ ( a , v , x ) , w , b ) ∼ κ µ ( µ ( a , w , b ) , µ ( v , w , b ) , x ) = µ ( x , µ ( v , w , b ) , x ) = x , which implies µ ( y , w , z ) = µ ( y , w , µ ( x , v (cid:48) , b )) ∼ κ µ ( x , µ ( y , w , v (cid:48) ) , µ ( y , w , b )) ∼ ρ ( κ ) µ ( x , µ ( y , w , v (cid:48) ) , x ) = x . In other words, x ∈ N ρ ( κ ) + κ ( { µ ( y , w , z ) } ) ⊂ N ρ ( κ ) + κ ([ y , z ]). Take C = ρ ( κ ) + κ and the result holds. (cid:3) The next lemma generalises Lemma 2.22 to the context of coarseneighbourhoods of intervals N λ ([ a , b ]). Lemma 4.9.
Let ( X , d , µ ) be a coarse median space with parameters ρ, H,let λ be a positive constant and a , b be points in X. Then x ∈ N λ ([ a , b ]) implies that µ ( a , b , x ) ∼ ρ ( λ ) + λ + κ x.Proof. By definition, x ∼ λ x (cid:48) ∈ [ a , b ], which implies there exists z ∈ X such that x (cid:48) = µ ( a , b , z ). So µ ( a , b , x ) ∼ ρ ( λ ) µ ( a , b , µ ( a , b , z )) ∼ κ µ ( a , b , z ) = x (cid:48) ∼ λ x . (cid:3) Proof of Theorem 4.2.
Take a ( ζ, ε )-quasi-geodesic connecting points a and b . By Lemma 4.7, we may, at the cost of moving the quasi-geodesic at most ζ + ε , obtain a ( ζ, ε )-quasi-geodesic γ which iscontinuous and such that there exist constants k and k dependingonly on ζ, ε , such that for any s (cid:48) , t (cid:48) ∈ [ s , t ], (cid:96) ( γ | [ s (cid:48) , t (cid:48) ] ) (cid:54) k d ( γ ( s (cid:48) ) , γ ( t (cid:48) )) + k . Let K , H , H be parameters of ( X , d , µ ) under which rank X (cid:54) a , b ] sits within a uniformlybounded neighbourhood of γ . FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 33
Let D = sup { d ( x , Im γ ) : x ∈ [ a , b ] } , which is finite since it is boundedby sup { d ( x , a ) | x ∈ [ a , b ] } ≤ Kd ( a , b ) + H . Indeed for x = µ ( a , b , c ) ∈ [ a , b ], we have x = µ ( a , b , c ) ∼ Kd ( a , b ) + H µ ( a , a , c ) = a . Let x be a point in [ a , b ] such that d ( x , Im γ ) > D −
1. We have µ ( a , b , x ) ∼ κ x by Lemma 2.21. Now if y ∈ [ a , x ] then y = µ ( x , a , z )for some z ∈ X , so µ ( b , a , y ) = µ ( b , a , µ ( x , a , z )) ∼ κ µ ( µ ( b , a , x ) , a , z ) ∼ K κ + H µ ( x , a , z ) = y . Hence y ∈ N λ ([ a , b ]) where λ = K κ + κ + H . Claim: If d ( x , a ) (cid:62) D + λ , then there exists some y ∈ [ x , a ] suchthat 2 D + λ (cid:54) d ( x , y ) (cid:54) D + λ + K + H . Proof of Claim:
First take a geodesic γ (cid:48) : [0 , d ( x , a )] → X from x to a . We will approximate γ (cid:48) by a discrete geodesic as follows. Let x i = γ (cid:48) ( i ) for i = , . . . , k = (cid:98) d ( a , x ) (cid:99) and let y i denote the projectionof x i into the interval [ x , a ], i.e., y i = µ ( a , x i , x ). Then for each i , d ( y i , y i + ) (cid:54) K + H . If d ( y j , x ) (cid:62) D + λ , for some j then let i be thefirst j for which this occurs and set y = y i . Since d ( y i − , x ) < D + λ and d ( x i − , x i ) = d ( x , y ) = d ( x , y i ) < D + λ + K + H .Otherwise set y = a and note that d ( y k , x ) (cid:54) D + λ so again d ( x , y ) = d ( x , a ) (cid:54) D + λ + K + H . This completes the proof of the claim.As shown above y ∈ N λ ([ a , b ]), so we can find a point y (cid:48) ∈ [ a , b ]such that d ( y (cid:48) , y ) (cid:54) λ . Hence by the claim, we have 2 D (cid:54) d ( y (cid:48) , x ) (cid:54) D + λ + K + H .If on the other hand d ( x , a ) < D + λ , then we take y = y (cid:48) = a andwe have D − < d ( y (cid:48) , x ) (cid:54) D + λ .Repeating the argument with b in place of a , we can find points z ∈ [ x , b ] , z (cid:48) ∈ [ a , b ] with: 2 D + λ (cid:54) d ( x , z ) (cid:54) D + λ + K + H and d ( z , z (cid:48) ) (cid:54) λ if d ( x , b ) (cid:62) D + λ ; z = z (cid:48) = b with D − < d ( z (cid:48) , x ) (cid:54) D + λ otherwise.If y (cid:48) = a then we set y (cid:48)(cid:48) = a , otherwise by the definition of D , thereexists a point y (cid:48)(cid:48) ∈ Im γ such that d ( y (cid:48)(cid:48) , y (cid:48) ) (cid:54) D , and we can choosea geodesic segment α from y (cid:48) to y (cid:48)(cid:48) . Similarly if z (cid:48) = b then we set z (cid:48)(cid:48) = b , otherwise there exists a point z (cid:48)(cid:48) ∈ Im γ such that d ( z (cid:48)(cid:48) , z (cid:48) ) (cid:54) D ,and we choose a geodesic segment β from z (cid:48)(cid:48) to z (cid:48) .Now consider the path ϑ from y (cid:48) to z (cid:48) that transverses α thenfollows γ from y (cid:48)(cid:48) to z (cid:48)(cid:48) , then transverses β from z (cid:48)(cid:48) to z (cid:48) , see Figure3. We have: d ( y (cid:48)(cid:48) , z (cid:48)(cid:48) ) (cid:54) d ( y (cid:48)(cid:48) , y (cid:48) ) + d ( y (cid:48) , x ) + d ( x , z (cid:48) ) + d ( z (cid:48) , z (cid:48)(cid:48) ) (cid:54) D + · (2 D + λ + K + H ) = D + λ + K + H ) , which implies, by Lemma 4.7, that (cid:96) ( ϑ ) (cid:54) D + k · (6 D + λ + K + H )) + k = (6 k + D + k (2 λ + K + H ) + k .F igure
3. Construction of ϑ Independent of our choice of the points y (cid:48)(cid:48) , z (cid:48)(cid:48) , the distance from x to any point on γ is at least D −
1. If y (cid:48) = a then we have set y (cid:48)(cid:48) = y (cid:48) , so the distance from x to the (only) geodesic arc from y (cid:48) to y (cid:48)(cid:48) = y (cid:48) is d ( x , a ) > D −
1. If on the other hand y (cid:48) (cid:44) a , then byhypothesis d ( x , y (cid:48) ) (cid:62) D , while d ( y (cid:48) , y (cid:48)(cid:48) ) (cid:54) D so the distance from x to the chosen geodesic arc from y (cid:48) to y (cid:48)(cid:48) must be at least D . A similarargument shows that the distance from x to the chosen geodesic arcfrom z (cid:48)(cid:48) to z (cid:48) must be at least D as well, so that d ( x , Im ϑ ) > D − D , but x might not sit inside [ y (cid:48) , z (cid:48) ]. However, by Lemma FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 35 C such that x ∈ N C ([ y , z ]),which implies there is a point µ ( y , z , w ) ∈ [ y , z ] which is C -close to x . Now µ ( y , z , w ) ∼ K λ + H µ ( y (cid:48) , z (cid:48) , w ) ∈ [ y (cid:48) , z (cid:48) ] by axiom (C1). Set x (cid:48) = µ ( y (cid:48) , z (cid:48) , w ) and C (cid:48) = C + K λ + H so that x ∼ C (cid:48) x (cid:48) ∈ [ y (cid:48) , z (cid:48) ].Hence by Proposition 4.6, D − < d ( x , Im ϑ ) (cid:54) d ( x (cid:48) , Im ϑ ) + C (cid:48) (cid:54) C log (cid:96) ( ϑ ) + C + C (cid:48) (cid:54) C log ((6 k + D + k (2 λ + K + H ) + k ) + C + C (cid:48) . The right hand side of the inequalities grows logarithmically fastwith respect to D , hence there exists some constant D (cid:48) = D (cid:48) ( ζ, ε )such that D (cid:54) D (cid:48) . In other words, [ a , b ] ⊆ N D (cid:48) (Im γ ).Now we will show that the quasi-geodesic γ sits within a uni-formly bounded neighbourhood of the interval [ a , b ]. Assume thereexists some point x (cid:48) ∈ Im γ such that d ( x (cid:48) , [ a , b ]) > D (cid:48) . Take a max-imal non-empty subinterval ( s , t ) ⊆ [ s , t ] such that γ | ( s , t ) sits out-side N D (cid:48) ([ a , b ]). As in the first part of the proof, pick a discretegeodesic a = x , x , . . . , x n = b from a to b and set y i = µ ( a , b , x i ).For each i , y i ∈ [ a , b ] and the distance d ( y i , Im γ ) (cid:54) D (cid:48) . So either d ( y i , Im γ | [ s , s ] ) (cid:54) D (cid:48) or d ( y i , Im γ | [ t , t ] ) (cid:54) D (cid:48) . Note that d ( y , Im γ | [ s , s ] ) (cid:54) D (cid:48) and d ( y n , Im γ | [ t , t ] ) (cid:54) D (cid:48) . Take the first i satisfying d ( y i , Im γ | [ s , s ] ) (cid:54) D (cid:48) and d ( y i + , Im γ | [ t , t ] ) (cid:54) D (cid:48) , and set w = y i . Then it follows thatthere exists s (cid:48) ∈ [ s , s ] and t (cid:48) ∈ [ t , t ] such that d ( w , γ ( s (cid:48) )) (cid:54) D (cid:48) , d ( w , γ ( t (cid:48) )) (cid:54) D (cid:48) + H + K , which implies d ( γ ( s (cid:48) ) , γ ( t (cid:48) )) (cid:54) D (cid:48) + H + K .Since γ is assumed to be tame we have (cid:96) ( γ | [ s (cid:48) , t (cid:48) ] ) (cid:54) k ( D (cid:48) + H + K ) + k ,which implies (cid:96) ( γ | [ s , t ] ) (cid:54) k (2 D (cid:48) + H + K ) + k . So we have Im γ ⊆N k (2 D (cid:48) + H + K ) + k + D (cid:48) ([ a , b ]). Finally, take (cid:101) D = k (2 D (cid:48) + H + K ) + k + D (cid:48) + ζ + ε , then the Hausdor ff distance between [ a , b ] and the original( ζ, ε )-quasi-geodesic is controlled by (cid:101) D . (cid:3) In our proof of Theorem 4.4 we will make use of the followinggeneralisation of Zeidler’s parallel edge lemma, [23, Lemma 2.4.5].The proof in this more general context is more or less identical, andis therefore omitted.
Lemma 4.10.
Let ( X , d ) be a metric space with a ternary operator µ sat-isfying the weak form of axiom (C1) with (arbitrary) control parameter ρ . Let C be a CAT(0) cube complex, and f : ( C (0) , µ C ) → ( X , µ ) an L-quasi-morphism, then for any parallel edges ( x , y ) , ( x (cid:48) , y (cid:48) ) in C, we haved ( f ( x (cid:48) ) , f ( y (cid:48) )) (cid:54) ρ ( d ( f ( x ) , f ( y ))) + L . (cid:3) Proof of Theorem 4.4.
Necessity is a special case of Lemma 4.5, so wejust need to show su ffi ciency. Let ρ, H be parameters of ( X , d , µ ). Bydefinition and Lemma 2.13 for any p ∈ N and A ⊆ X with | A | = p ,there exists a finite median algebra ( Π , µ Π ), and maps π : A → Π , σ : Π → X satisfying axioms (C1), (C2), (cid:104) π ( A ) (cid:105) = Π and σπ = i A .Denote the associated finite CAT(0) cube complex also by Π . Nowwe want to modify Π , π, σ so that Π has dimension 1.If dim Π =
1, nothing needs to be modified. Assume dim Π (cid:62) a , ¯ b , ¯ c , ¯ d in Π . To be more explicit, assume¯ a is connected to ¯ b and ¯ c in Π . Denote σ ( ¯ a ) , σ (¯ b ) , σ ( ¯ c ) , σ ( ¯ d ) by a , b , c , d .By axiom (C2), we have: b (cid:48) : = µ ( a , b , d ) = µ ( σ ( ¯ a ) , σ (¯ b ) , σ ( ¯ d )) ∼ H ( p ) σ ( µ ( ¯ a , ¯ b , ¯ d )) = σ (¯ b ) = b . Similarly, c (cid:48) : = µ ( a , c , d ) ∼ H ( p ) c . Since b (cid:48) = µ ( a , b , d ) ∈ [ a , d ], byassumption, there exists a constant λ such that [ a , d ] ⊆ N λ ([ a , b (cid:48) ]) ∪N λ ([ b (cid:48) , d ]). Now since c (cid:48) ∈ [ a , d ], without loss of generality, we canassume that c (cid:48) ∈ N λ ([ a , b (cid:48) ]), which implies c (cid:48) ∼ ρ ( λ ) + λ + κ µ ( a , b (cid:48) , c (cid:48) ) byLemma 4.9. So c ∼ H ( p ) µ ( a , c , d ) ∼ ρ ( λ ) + λ + κ µ ( a , µ ( a , b , d ) , µ ( a , c , d )) ∼ κ µ ( a , d , µ ( a , b , c )) . On the other hand, µ ( a , b , c ) = µ ( σ ( ¯ a ) , σ (¯ b ) , σ ( ¯ c )) ∼ H ( p ) σ ( µ ( ¯ a , ¯ b , ¯ c )) = a ,so µ ( a , d , µ ( a , b , c )) ∼ ρ ( H ( p )) µ ( a , d , a ) = a . To sum up, we showed c is [ ρ ( H ( p )) + ρ ( λ ) + H ( p ) + λ + κ )]-closeto a . For brevity, if we take α (cid:48) ( t ) = ρ ( t ) + t + ρ ( λ ) + λ + κ , then c is α (cid:48) ( H ( p ))-close to a .By Lemma 4.10, we obtain another function α ( t ) = ρ ( α (cid:48) ( t )) + t suchthat for any edge ( ¯ x , ¯ y ) in Π parallel to ( ¯ a , ¯ c ), σ ( ¯ x ) is α ( H ( p ))-close to σ ( ¯ y ). Note that the function α ( t ) = ρ ( α (cid:48) ( t )) + t = ρ ( ρ ( t ) + t + ρ ( λ ) + λ + κ ) + t FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 37 depends only on ρ, κ , λ . In particular it does not depend on the set A .Now consider the quotient CAT(0) cube complex Π (cid:48) of Π deter-mined by all the hyperplanes except the hyperplane h crossing ( ¯ a , ¯ c )(see [8, 16]). Let pr : Π → Π (cid:48) be the natural projection, and choose asection s : Π (cid:48) → Π , i.e. pr ◦ s = id Π (cid:48) . For any vertex ¯ v not adjacent to h we have s ◦ pr ( ¯ v ) = ¯ v , while if ¯ v is adjacent to h then either s ◦ pr ( ¯ v ) = ¯ v or there is an edge crossing h joining s ◦ pr ( ¯ v ) , ¯ v . The map s is notuniquely determined, but for s , s (cid:48) two di ff erent sections, accordingto the analysis above the compositions σ ◦ s , σ ◦ s (cid:48) are α ( H ( p ))-close.Now consider the diagram X Π (cid:48) s (cid:42) (cid:42) Π σ (cid:59) (cid:59) pr (cid:106) (cid:106) A i (cid:79) (cid:79) π (cid:99) (cid:99) and take π (cid:48) = pr ◦ π , σ (cid:48) = σ ◦ s . We have σ (cid:48) ◦ π (cid:48) ∼ α ( H ( p )) i A , andfor any x (cid:48) , y (cid:48) , z (cid:48) ∈ Π (cid:48) , we want to estimate the distance between σ (cid:48) ( µ Π (cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) )) and µ ( σ (cid:48) ( x (cid:48) ) , σ (cid:48) ( y (cid:48) ) , σ (cid:48) ( z (cid:48) )). Claim. µ Π (cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) ) = pr ( s ( µ Π (cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) ))) = pr ( µ Π ( s ( x (cid:48) ) , s ( y (cid:48) ) , s ( z (cid:48) ))).Indeed, by the construction of Π (cid:48) , for any hyperplane h (cid:48) (cid:44) h of Π and ¯ u , ¯ v ∈ Π , h (cid:48) separates ¯ u , ¯ v if and only if h (cid:48) , as a hyperplane of Π (cid:48) ,separates pr ( ¯ u ) , pr ( ¯ v ). Now assume that(3) pr ( s ( µ Π (cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) ))) (cid:44) pr ( µ Π ( s ( x (cid:48) ) , s ( y (cid:48) ) , s ( z (cid:48) ))) , then there exists a hyperplane h (cid:48) of Π (cid:48) such that h (cid:48) separates pr ( s ( µ Π (cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) )))and pr ( µ Π ( s ( x (cid:48) ) , s ( y (cid:48) ) , s ( z (cid:48) ))), which implies h (cid:48) , as a hyperplane of Π rather than h , separates s ( µ Π (cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) )) and µ Π ( s ( x (cid:48) ) , s ( y (cid:48) ) , s ( z (cid:48) )). Weselect a vertex ¯ v ∈ Π such that s ◦ pr ( ¯ v ) = ¯ v and let h (cid:48) + denotethe half space corresponding to the hyperplane h (cid:48) containing ¯ v and h (cid:48)− the complementary halfspace. In Π (cid:48) we will abuse notation andalso denote by h (cid:48) + the halfspace containing pr ( ¯ v ). This ambiguityis tolerated since the points will tell us which space we are focus-ing on. It follows that s ( h (cid:48) + ) ⊆ h (cid:48) + and s ( h (cid:48)− ) ⊆ h (cid:48)− . At least two of any three points must lie in the same half space corresponding toa given hyperplane h (cid:48) , so without loss of generality we can assume s ( x (cid:48) ) , s ( y (cid:48) ) ∈ h (cid:48) + , which implies µ Π ( s ( x (cid:48) ) , s ( y (cid:48) ) , s ( z (cid:48) )) ∈ h (cid:48) + ; on the otherhand, according to the choice of orientation, we have x (cid:48) , y (cid:48) ∈ h (cid:48) + ,which implies µ Π (cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) ) ∈ h (cid:48) + , so s ( µ Π (cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) )) ∈ h (cid:48) + . This is acontradiction to our assumption (3), hence the claim is proved.Returning to the proof we have pr ( s ( µ Π (cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) ))) = pr ( µ Π ( s ( x (cid:48) ) , s ( y (cid:48) ) , s ( z (cid:48) ))),so either the two points s ( µ Π (cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) )) , µ Π ( s ( x (cid:48) ) , s ( y (cid:48) ) , s ( z (cid:48) )) are equal,or, by the analysis above, they are joined by an edge crossing h in Π .It follows that σ ( s ( µ Π (cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) ))) ∼ α ( H ( p )) σ ( µ Π ( s ( x (cid:48) ) , s ( y (cid:48) ) , s ( z (cid:48) ))) . Now for any x (cid:48) , y (cid:48) , z (cid:48) , we have: σ (cid:48) ( µ Π (cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) )) = σ ( s ( µ Π (cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) ))) ∼ α ( H ( p )) σ ( µ Π ( s ( x (cid:48) ) , s ( y (cid:48) ) , s ( z (cid:48) ))) ∼ H ( p ) µ ( σ ( s ( x (cid:48) )) , σ ( s ( y (cid:48) )) , σ ( s ( z (cid:48) ))) = µ ( σ (cid:48) ( x (cid:48) ) , σ (cid:48) ( y (cid:48) ) , σ (cid:48) ( z (cid:48) )) , which means if we take H (cid:48) ( p ) = β ( H ( p )) where β ( t ) = α ( t ) + t , then σ (cid:48) ( µ Π (cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) )) ∼ H (cid:48) ( p ) µ ( σ (cid:48) ( x (cid:48) ) , σ (cid:48) ( y (cid:48) ) , σ (cid:48) ( z (cid:48) )) . To sum up, if we have a 2-square in the original approximation( Π , π, σ ), then we can replace it by another approximation ( Π (cid:48) , π (cid:48) , σ (cid:48) ).The controlling parameter H (cid:48) ( p ) = β ( H ( p )) for the new approxima-tion depends only on the original parameters H ( p ) , ρ, κ and theconstant λ . Since a CAT(0) cube complex generated by p verticeshas at most 2 p hyperplanes we can iterate this process at most 2 p times to remove all squares, ending with a new approximating tree Π (cid:48) with controlling parameter H (cid:48) ( p ) = β p ( H ( p ))where the function β defined above only depends on ρ, κ and theconstant λ .This process defines a parameter H (cid:48) for which the approximatingmedian algebras required by axiom (C2) can always be taken to betrees. Hence our space has rank 1 with parameters ρ, H (cid:48) . (cid:3) FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 39
Higher rank spaces.
In the proof above we deduced that thespace X satisfying the thin interval triangles condition (2) cannotpossess arbitrary large “coarse squares”. More explicitly, for a“coarse square” a , b , c , d as in the proof, we showed that at leastone of its parallel edge pairs must have bounded length. This inter-mediate result is crucial in our analysis of higher rank spaces, andit is exactly the rank 1 case of our general characterisation of rank.Intuitively, our characterisation states that a coarse median spacehas rank at most n if and only if it does not contain arbitrarily large( n + Theorem 4.11.
Let ( X , d , µ ) be a coarse median space and n ∈ N . Thenthe following conditions are equivalent.(1) rank X (cid:54) n;(2) For any λ > , there exists a constant C = C ( λ ) such that for anya , b ∈ X, any e , . . . , e n + ∈ [ a , b ] with µ ( e i , a , e j ) ∼ λ a for all i (cid:44) j,there exists i such that e i ∼ C a;(3) For any L > , there exists a constant C = C ( L ) such that for anyL-quasi-morphism σ from the median n-cube I n + to X, there existadjacent points x , y ∈ I n + such that σ ( x ) ∼ C σ ( y ) . In condition (2) of the above theorem, one should imagine a as acorner of an ( n + e , . . . , e n + as endpoints ofedges adjacent to a . Proof of Theorem 4.11. (1) ⇒ (2): Let ρ, H be parameters of ( X , d , µ )under which rank X (cid:54) n can be achieved. Given λ > a , b ∈ X and e , . . . , e n + ∈ [ a , b ] with µ ( e i , a , e j ) ∼ λ a ( i (cid:44) j ), we set A = { a , b , e , . . . , e n + } . By axiom C2, there exists a finite rank n medianalgebra Π , and maps π : A → Π and σ : Π → X satisfying the con-ditions in C2 and σ ◦ π = i A . Denote π ( a ) , π ( b ) , π ( e i ) by ¯ a , ¯ b , ¯ e i . Ac-cording to Lemma 2.22, without loss of generality, we can alwaystake Π = [ ¯ a , ¯ b ] after changing the parameters within some controlled bounds if necessary. To make it clear, consider the following dia-gram: X [ ¯ a , ¯ b ] inclusion (cid:42) (cid:42) Π σ (cid:59) (cid:59) µ Π (¯ a , ¯ b , · ) (cid:108) (cid:108) A . i (cid:79) (cid:79) π (cid:99) (cid:99) Define π (cid:48) : A → [ ¯ a , ¯ b ] by x (cid:55)→ µ Π ( ¯ a , ¯ b , π x ), and σ (cid:48) : [ ¯ a , ¯ b ] → X by¯ c (cid:55)→ σ ( ¯ c ). Note that σ (cid:48) ◦ π (cid:48) ( e i ) = σ ( µ Π ( ¯ a , ¯ b , ¯ e i )) ∼ H ( n + µ ( a , b , e i ) ∼ κ e i ,since e i ∈ [ a , b ]. For convenience, we still write π, σ instead of π (cid:48) , σ (cid:48) .To sum up, after modifying ρ, H if necessary, we can always find afinite rank n median algebra Π and maps π : A → Π and σ : Π → X such that Π = [ ¯ a , ¯ b ], and conditions in (C2) hold (here we cannotrequire σ ◦ π = i A again, but just close to i A ).Since [ ¯ a , ¯ b ] has rank at most n , without loss of generality, we canassume, by Lemma 2.6, that µ Π (¯ e , . . . , ¯ e n + ; ¯ b ) = µ Π (¯ e , . . . , ¯ e n ; ¯ b ) , which implies that n + (cid:92) i = [¯ e i , ¯ b ] = n (cid:92) i = [¯ e i , ¯ b ] . So n (cid:92) i = [¯ e i , ¯ b ] ⊆ [¯ e n + , ¯ b ] , i.e. µ (¯ e , . . . , ¯ e n ; ¯ b ) ∈ [¯ e n + , ¯ b ] . Equivalently, ¯ e n + ∈ [ ¯ a , µ (¯ e , . . . , ¯ e n ; ¯ b )] since ¯ e i ∈ [ ¯ a , ¯ b ]. By Lemma 2.7,we have:¯ e n + = µ Π ( ¯ a , ¯ e n + , µ (¯ e , . . . , ¯ e n ; ¯ b )) = µ Π ( µ Π ( ¯ a , ¯ e n + , ¯ e ) , . . . , µ Π ( ¯ a , ¯ e n + , ¯ e n ); ¯ b ) . Now translate the above equation into X , recall that σ (¯ e i ) ∼ H ( n + e i .By Lemma 2.17, there exists a constant α ( n , ρ, H ): e n + ∼ H ( n + σ (¯ e n + ) = σ ( µ Π ( µ Π ( ¯ a , ¯ e n + , ¯ e ) , . . . , µ Π ( ¯ a , ¯ e n + , ¯ e n ); ¯ b )) ∼ α ( n ,ρ, H ) µ ( σ ( µ Π ( ¯ a , ¯ e n + , ¯ e )) , . . . , σ ( µ Π ( ¯ a , ¯ e n + , ¯ e n )); σ (¯ b )) . By (C1) and (C2), we know σ ( µ Π ( ¯ a , ¯ e n + , ¯ e i )) ∼ H ( n + µ ( σ ( ¯ a ) , σ (¯ e n + ) , σ (¯ e i )) ∼ ρ (3 H ( n + µ ( a , e n + , e i ) ∼ λ a . FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 41
Now by Lemma 2.16, there exists a constant β ( λ, n , ρ, H ) such that µ ( σ ( µ Π ( ¯ a , ¯ e n + , ¯ e )) , . . . , σ ( µ Π ( ¯ a , ¯ e n + , ¯ e n )); σ (¯ b )) ∼ β ( λ, n ,ρ, H ) µ ( a , . . . , a ; b ) = a , which implies there exists some constant C = C ( λ ) such that e n + is C -close to a . (2) ⇒ (3): Let ρ, H be parameters of ( X , d , µ ). Let ¯0 denote the zerovector in the median cube ( I n + , µ n + ), let ¯1 denote the vector with1 in all coordinates, and ¯ e i denote the basis vector with a single 1in the i th coordinate. Given an L -quasi-morphism σ : I n + → X , let a = σ (¯0) , b = σ (¯1) and e i = σ (¯ e i ). Since ¯ e i ∈ [¯0 , ¯1], µ n + (¯0 , ¯ e i , ¯1) = ¯ e i ,which implies e (cid:48) i : = µ ( a , b , e i ) = µ ( σ (¯0) , σ (¯1) , σ (¯ e i )) ∼ L σ ( µ n + (¯0 , ¯ e i , ¯1)) = e i and e (cid:48) i ∈ [ a , b ]. Now µ ( e (cid:48) i , a , e (cid:48) j ) = µ ( µ ( a , b , e i ) , a , µ ( a , b , e j )) ∼ κ µ ( a , b , µ ( e i , e j , a )),and µ ( e i , e j , a ) = µ ( σ (¯ e i ) , σ (¯ e j ) , σ (¯0)) ∼ L σ ( µ n + (¯ e i , ¯ e j , ¯0)) = a . So µ ( e (cid:48) i , a , e (cid:48) j ) ∼ ρ ( L ) + κ µ ( a , b , a ) = a . Now take λ = ρ ( L ) + κ , andby the assumption, there exists a constant C (cid:48) depending on λ andhence implictly on L , such that one of e (cid:48) , . . . , e (cid:48) n + is C (cid:48) -close to a . Thisimplies that one of the points σ ( ¯ e i ) = e i is C -close to σ (¯0) = a for C = C (cid:48) + L . (3) ⇒ (1): Let ρ, H be parameters of ( X , d , µ ). By definition, forany p ∈ N and A ⊆ X with | A | = p , there exists a finite medianalgebra ( Π , µ Π ), and maps π : A → Π , σ : Π → X satisfying axioms(C1), (C2), the conditions in Remark 2.10 and σπ = i A . Denote theassociated finite CAT(0) cube complex also by Π . Now we need tomodify Π , π, σ to ensure that Π has rank n .If rank Π = n , nothing need to be modified. Assume rank Π (cid:62) n +
1, so that, as a CAT(0) cube complex it has dimension at least n + n + I n + , µ n + ). By condition (3) with L = H ( p ), we know there exists a constant C depending on L andhence on p such that, without loss of generality, σ (¯ e n + ) is C -close to σ (¯0). Now construct the quotient CAT(0) cube complex Π (cid:48) from Π by deleting the hyperplane h crossing the edge ( ¯0 , ¯ e n + ). Then therest of the proof follows exactly that of Theorem 4.4. (cid:3) As mentioned in previous sections, the above theorem o ff ers away to define the rank of a coarse median space in the simplifiedsetting. More explicitly, Theorem 3.1 and Theorem 4.11 combine toshow : Theorem 4.12.
Let ( X , d ) be a metric space, and µ : X → X a ternaryoperation. Then ( X , d , µ ) is a coarse median space of rank at most n if andonly if the following conditions hold:(M1). µ ( a , a , b ) = a for any a , b ∈ X(M2). µ ( a σ (1) , a σ (2) , a σ (3) ) = µ ( a , a , a ) , for any a , a , a ∈ X and σ apermutation;(C1)’. There exists an a ffi ne control function ρ : [0 , + ∞ ) → [0 , + ∞ ) suchthat for all a , a (cid:48) , b , c ∈ X,d ( µ ( a , b , c ) , µ ( a (cid:48) , b , c )) (cid:54) ρ ( d ( a , a (cid:48) )); (C2)’. There exists a constant κ > such that for any a , b , c , d ∈ X, wehave µ ( µ ( a , b , c ) , b , d ) ∼ κ µ ( a , b , µ ( c , b , d )); (C3)’. ∀ λ > , ∃ C = C ( λ ) such that for any a , b ∈ X, any e , . . . , e n + ∈ [ a , b ] with µ ( e i , a , e j ) ∼ λ a for all i (cid:44) j, there exists i such thate i ∼ C a.
5. A counterexample
Recall that in a discrete median algebra, or equivalently, a CAT(0)cube complex, the interval between two points is the set of pointswhich lie on geodesics connecting them. This makes a bridge be-tween the algebraic aspect and the geometry of the object. It isnatural to ask to what extent this holds in a coarse median space?As we have already seen in Theorem 4.2, it is “almost” true in rank1: intervals are “about the same” as the union of quasi-geodesics.As we shall see, the interaction of geodesics and intervals in higherrank is considerably less well behaved. We will show:
FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 43
Theorem 5.1.
There is a rank geodesic coarse median space ( X , d , µ ) such that for any C (cid:62) there exist a , b ∈ X, with the property that nogeodesic γ connecting a , b lies within Hausdor ff distance C of the interval [ a , b ] .Proof. Let X = Z , and µ the canonical median operator on X , i.e. µ (( x , y ) , ( x , y ) , ( x , y )) = ( m ( x , x , x ) , m ( y , y , y )) , where ( x , y ) , ( x , y ) , ( x , y ) ∈ Z and m ( · , · , · ) is the classical medianof three real numbers. Equip ( X , µ ) with a metric d as follows. Wewill view X as a graph in the canonical way, and define d to be aweighted edge-path metric on X , induced by assigning a positivenumber to each edge as its length. We define all horizontal edges tohave length 1, i.e. d (( x , y ) , ( x + , y )) = x , y ) ∈ Z ; lengths ofvertical edges are listed as follows. See Figure 4. • If x (cid:54) d (( x , y ) , ( x , y + = • If x = d (( x , y ) , ( x , y + = • If x (cid:62) d (( x , y ) , ( x , y + = { γ n } of edge paths in ( X , d ): for each n ∈ N , define a path starting from a n = ( n + , n + , n + ,
0) andfinally horizontally rightwards to b n = ( n + , γ n are geodesic for all n . We claimthat ( X , d , µ ) is a coarse median space. In fact, consider the identitymap id : ( X , d , µ ) → ( X , d , µ ), where d is the edge-path metricdefined by each edge having length 1. It is obvious that this is a bi-Lipschitz map and a median morphism, so ( X , d , µ ) is indeed a coarsemedian space of rank 2. Now notice that [ a n , b n ] = { n + } × [0 , n + d H (Im γ n , [ a n , b n ]) = n +
1, which implies there is no uniform boundon the Hausdor ff distance between geodesics and intervals.Unfortunately the space we have constructed is not geodesic, forinstance the points (3 , , (3 ,
1) are distance 5 apart, but there is no(integer) path of length less than 7 connecting them. We can modifythe space easily to rectify this problem by subdividing the edges of F igure
4. Lengths of vertical edges in the metric d on Z length n > n − d to includethese points is now geodesic and we continue to denote it by d . Weare left with the issue of how to define coarse medians involving theinserted points. This is dealt with by projecting each point p = ( x , y )to its “floor” p = ( x , (cid:98) y (cid:99) ), the nearest original point to the vertex ator below it on a vertical line. We then define the ternary operator µ (cid:48) by µ (cid:48) ( p , q , r ) = µ ( p , q , r ) when p , q , r are distinct and if two areequal then we define µ (cid:48) ( p , q , r ) to be that point. Since the floor mapmoves points at most a distance of 4 in the d metric, µ (cid:48) is still a FOUR POINT CHARACTERISATION FOR COARSE MEDIAN SPACES 45 coarse median operator with respect to the extended metric d , andintervals between a n and b n remain the same. (cid:3) R eferences [1] Hans-J Bandelt and Jarmila Hedl´ıkov´a. Median algebras. Discrete mathemat-ics , 45(1):1–30, 1983.[2] Hans-Jurgen Bandelt and Victor Chepoi. Metric graph theory and geometry:a survey.
Contemporary Mathematics , 453:49–86, 2008.[3] Jason Behrstock, Mark Hagen, Alessandro Sisto, et al. Hierarchically hyper-bolic spaces, I: Curve complexes for cubical groups.
Geometry & Topology ,21(3):1731–1804, 2017.[4] Jason Behrstock, Mark F Hagen, and Alessandro Sisto. Hierarchically hy-perbolic spaces II: combination theorems and the distance formula. arXivpreprint arXiv:1509.00632 , 2015.[5] B. H. Bowditch. Coarse median spaces and groups.
Pacific Journal of Mathe-matics , 261(1):53–93, 2013.[6] B. H. Bowditch. Embedding median algebras in products of trees.
GeometriaeDedicata , 170(1):157–176, 2014.[7] M. R. Bridson and A. H¨afliger.
Metric spaces of non-positive curvature , volume319 of
Grundlehren der Mathematischen Wissenschaften [Fundamental Principlesof Mathematical Sciences] . Springer-Verlag, Berlin, 1999.[8] I. Chatterji and G. A. Niblo. From wall spaces to CAT(0) cube complexes.
International Journal of Algebra and Computation , 15(05n06):875–885, 2005.[9] Indira Chatterji and Graham A Niblo. A characterization of hyperbolicspaces.
Groups Geometry and Dynamics , 1(3):281, 2006.[10] V. Chepoi. Graphs of some CAT(0) complexes.
Advances in Applied Mathemat-ics , 24(2):125–179, 2000.[11] M. Gromov. Hyperbolic groups. In
Essays in group theory , pages 75–263.Springer, 1987.[12] J. R. Isbell. Median algebra.
Transactions of the American Mathematical Society ,260(2):319–362, 1980.[13] Milan Kolibiar and Tamara Marcisov´a. On a question of J. Hashimoto.
Matem-atick`y ˇcasopis , 24(2):179–185, 1974.[14] G. A. Niblo and L. D. Reeves. The geometry of cube complexes and thecomplexity of their fundamental groups.
Topology , 37(3):621–633, 1998.[15] G.A. Niblo, N.J. Wright, and J. Zhang. The intrinsic geometry of coarsemedian spaces and their intervals. arXiv:1802.02499, 2018.[16] B. Nica. Cubulating spaces with walls.
Algebr. Geom. Topol , 4:297–309, 2004.[17] Panos Papasoglu. Strongly geodesically automatic groups are hyperbolic.
Inventiones mathematicae , 121(1):323–334, 1995.[18] J. Pomroy. A characterisation of hyperbolic spaces. Master’s thesis, Univer-sity of Warwick, 1994.[19] M. Roller. Poc sets, median algebras and group actions. an extended studyof Dunwoody’s construction and Sageev’s theorem.
Southampton PreprintArchive , 1998. [20] M. Sageev. Ends of group pairs and non-positively curved cube complexes.
Proceedings of the London Mathematical Society , 3(3):585–617, 1995.[21] J´an ˇSpakula and Nick Wright. Coarse medians and property A.
Algebraic &Geometric Topology , 17(4):2481–2498, 2017.[22] Marcel L.J. van De Vel.
Theory of convex structures , volume 50. Elsevier, 1993.[23] R. Zeidler. Coarse median structures on groups. Master’s thesis, Universityof Vienna, Vienna, Austria, 2013.S chool of M athematics , U niversity of S outhampton , H ighfield , SO17 1BJ,U nited K ingdom . E-mail address : { g.a.niblo,n.j.wright,jiawen.zhang }}