The n-point Condition and Rough CAT(0)

We show that for n>4 a length space (X,d) satisfies a rough n-point condition if and only it it is rough CAT(0).


Introduction
Gromov hyperbolic spaces and CAT(0) spaces have been intensively studied; see [CDP], [GH], [Va], [BH] and the references therein. Their respective theories display some common features, notably the canonical boundary topologies. Rough CAT(0) spaces, a class of length spaces that properly contains both CAT(0) spaces and those Gromov hyperbolic spaces that are length spaces, were introduced by the first author and Kurt Falk in a pair of papers to unify as much as possible of the theories of CAT(0) and Gromov hyperbolic spaces: the basic "finite distance" theory of rough CAT(0) spaces was developed in [BF1], and the boundary theory was developed in [BF2]. As in the earlier papers, we usually write rCAT(0) in place of rough CAT(0) below. Rough CAT(0) is closely related to the class of bolic spaces of Kasparov and Skandalis [KS1], [KS2] that was introduced in the context of their work on the Baum-Connes and Novikov Conjectures, and is also related to Gromov's class of CAT(-1,ε) spaces [Gr], [DG].
One gap in the theory developed so far is the absence of results indicating that the class of rCAT(0) spaces are closed under reasonably general limit processes such as pointed and unpointed Gromov-Hausdorff limits and ultralimits. The purpose of this paper is to fill that gap.
The fact that the CAT(0) class is closed under such limit processes is a consequence of the following well-known result (for which, see [BH,II.1.11

]):
Theorem A. A complete geodesic metric space X is CAT(0) if and only if it satisfies the 4-point condition.
In [BF1,Theorem 3.18], it was shown that a rough variant of the 4-point condition is quantitatively equivalent to a weak version of rCAT(0), and it follows that the class of weak rCAT(0) spaces is closed under reasonably general limit processes. However it seems difficult to decide whether or not all weak rCAT(0) spaces are necessarily rCAT(0). To establish similar limit closure properties for rCAT(0), we prove the following rough analogue to Theorem A; rough n-point conditions are defined in Section 2. Theorem 1.1. Let (X, d) be a length space. If n ≥ 5 and (X, d) satisfies a C-rough n-point condition for some After some preliminaries in Section 2, we prove a pair of preparatory lemmas in Section 3. We then prove the main theorem and discuss its limit closure consequences in Section 4.

Preliminaries
Whenever we write R 2 in this paper, we always mean the plane with the Euclidean metric attached. Throughout this section, X is a metric space with metric d attached; any extra assumptions on d will be explicitly stated.
We denote h-short segments connecting points x, y ∈ X by [x, y] h . It is convenient to use [x, y] h also for the image of this path, so instead of writing z = γ(t) for some 0 ≤ t ≤ L, we often write z ∈ [x, y] h . Given such a path γ and point z = γ(t), we denote by [x, z] h and [z, y] h respectively the subpaths γ| [0,t] and γ| [t,L] , respectively; note that both of these are h-short segments. A 0-short segment is called a geodesic segment, and we write [x, y] in place of [x, y] 0 .
A metric space (X, d) is a geodesic space if for every x, y ∈ X, there exists at least one geodesic segment [x, y]. More generally, (X, d) is a length space if for every x, y ∈ X and every h > 0, there exists a h-short path [x, y] Given such a h-short triangle T , a comparison triangle will mean a Euclidean triangleT : A geodesic triangle T = T (x, y, z) is just a 0-short triangle. Note that in this case ifT : A geodesic space (X, d) is a CAT(0) space if given any geodesic triangle T = T (x, y, z) with comparison triangleT =T (x,ȳ,z), and any two points u ∈ [x, y] and v ∈ [x, z], we have d(u, v) ≤ |ū −v|, whereū andv are comparison points for u and v.
Definition 2.1. Given C > 0, and a function H : X × X × X → (0, ∞), a length space (X, d) is said to be a C-rCAT(0; H) space if the following C-rough CAT(0) condition is satisfied: • u, v lie on different sides of T ; •ū,v ∈T are comparison points for u, v, respectively; We call (T h (x, y, z), u, v) the metric space data and (T (x,ȳ,z),ū,v) the comparison data.
The explicit H in the rCAT(0) condition has proved to be useful, but one situation where rCAT(0; * ) is needed is when the parameter C is close to 0. In particular, we show in Theorem 4.16 that if (X n ) is a sequence of C n -rCAT(0; * ) spaces with C n → 0, then under rather general conditions the resulting limit space is necessarily CAT(0). A fortiori, we could change the C n -rCAT(0; * ) hypothesis in this result to C n -rCAT(0), but such a variant is of no real interest since a length space satisfying a C-rCAT(0) condition for C < 1/2 has diameter at most C (as a hint, in a space of diameter larger than this, consider a triangle T (x, x, x) containing a side [x, x] that moves away from x and back again). In particular, only a one-point space can be C-rCAT(0) for all C > 0. By contrast, the class of spaces that are C-rCAT(0; * ) for all C > 0 is quite large: it includes, for instance, all CAT(0) spaces (as mentioned above), as well as the deleted Euclidean plane.
We now introduce the concept of C-rough subembeddings (into R 2 ), which we use to define rough n-point conditions. Definition 2.3. Let (X, d) be a metric space, C ≥ 0 and n ≥ 3 be an integer. Suppose x i ∈ X and x i ∈ R 2 for 0 ≤ i ≤ n, with x 0 = x n and x 0 = x n . We say that Definition 2.4. Let n ≥ 3 be an integer. A metric space (X, d) satisfies the C-rough n-point condition, where C ≥ 0, if every n-tuple in X has a C-rough subembedding into R 2 . We say that X satisfies a rough n-point condition if it satisfies a C-rough npoint condition for some C. The n-point condition is the 0-rough n-point condition.
We note that our notion of a rough 5-point condition is somewhat analogous to the mesoscopic curvature notion of Delzant and Gromov [DG] which they call CAT ε (κ), although that paper is concerned with κ < 0, whereas our notion corresponds to κ = 0.
Before proceeding further, let us discuss these conditions. If we vary just one of the parameters C and n in the C-rough n-point condition, it is easy to see that decreasing C or increasing n gives a stronger condition; note that to deduce the Crough (n − 1)-point condition from the C-rough n-point condition, we simply take x n = x n−1 . The 3-point condition is satisfied by all metric spaces.
For geodesic spaces, the 4-point condition is equivalent to CAT(0); see [BH,II.1.11]. For length spaces, a C-rough 4-point condition is quantitatively equivalent to a weaker version of rCAT(0) in which the C-rough CAT(0) condition is assumed for metric space data (T h (x, y, z), u, v) only when v is one of the vertices x, y, z; see [BF1,Theorem 3.18]. However it seems difficult to decide whether or not weak rCAT(0) spaces are necessarily rCAT(0). We do not address that issue in this paper, but we will show that, among length spaces, rCAT(0) is quantitatively equivalent to a rough n-point condition for n > 4. Thus the class of weak rCAT(0) spaces coincides with the class of length spaces satisfying a rough 4-point condition, and the class of rCAT(0) spaces coincides with the class of length spaces satisfying an n-point condition for any value (or all values) of n > 4, but we cannot say whether or not a rough 4-point condition implies a rough n-point condition for n > 4.

Two lemmas
The proof of Theorem 1.1 requires the following two lemmas. The first is a restatement of [BF1, Lemma 3.12].
Lemma 3.1. Let x, y be a pair of points in the Euclidean plane R 2 , with l := |x−y| > 0. Fixing h > 0, and writing L := l + h, let γ : [0, L] → R 2 be a h-short segment from x to y, parameterized by arclength. Then there exists a map λ : [0, L] → [x, y] such that λ(0) = x, λ(L) = y, and In and We assume without loss of generality that s ≤ t. An elementary calculation using the parallelogram law shows that given x, y, z in the Euclidean plane with w ∈ [y, z] and |w − y| = r|z − y| we have Using (3.3) twice, we get In particular d ≤ √ 3ε, as required.

Proof and consequences
Here we prove Theorem 1.1 and discuss some consequences. First we need a definition.
Definition 4.1. Suppose (S, d S ) is a metric space, and that for i = 1, 2, we have a metric space (X i , d i ), a closed subspace S i ⊂ X i , and a surjective isometry f i : S → S i . We then define the gluing of X 1 and X 2 along S 1 , S 2 (denoted by X = X 1 ⊔ S X 2 ) as the quotient of the disjoint union of X 1 and X 2 under the identification of f 1 (s) with f 2 (s) for each s ∈ S. The glued metric d on X is defined by the equations We note the following easily verified facts about (X, d) := X 1 ⊔ S X 2 defined by gluing as above: • d restricted to X i , i = 1, 2, coincides with d i ; • every geodesic segment in X i , i = 1, 2, is also a geodesic segment in X.
We now prove the following slight improvement of Theorem 1.1.
Proof. Assume that (X, d) is a length space. We first prove the forward implication, so we assume that n ≥ 5 and that (X, d) satisfies a C-rough n-point condition for some C ≥ 0. It follows trivially that (X, d) satisfies a C-rough 5-point condition. Let T := T h (x, y, z) be a h-short geodesic triangle in X, where , From the definition of a C-rough subembedding and the fact that T is h-short, it follows that the piecewise linear paths Now letT = T (x,ȳ,z) be a comparison triangle for T and chooseū ∈ [x,ȳ],v ∈ [x,z] satisfying: Since |x −ȳ| = d(x, y) ≤ |x ′ − y ′ |, it follows from (4.6) and (4.8) that soū is a comparison point for u. Similarlyv is a comparison point for v. Finally, using (4.4) and (4.5), we see that and so by Lemma 3.2, we get Thus (X, d) is C ′ -rCAT(0; * ), with C ′ = C + 2 √ 3ε. Taking ε = 1, we see that X is C ′ -rCAT(0), where C ′ = C + 2 √ 3. Letting ε > 0 be sufficiently small, we see that X is C ′′ -rCAT(0; * ).
We next proceed with the reverse implication, so let us assume that (X, d) is C ′ -rCAT(0; * ). We will prove that (X, d) satisfies the C n -rough n-point condition, where C n := (n − 2)C ′ and n ≥ 3.
The proof will involve induction, but using a stronger inductive hypothesis which involves not just a set of n points, but an n-gon with these points as vertices. Additionally, the inductive process requires us to establish simultaneously a CAT(0) version of the result. Note that it suffices to prove the result for sets of distinct points, since the desired conditions for n points with at least one repeated point follows immediately from the condition for n − 1 points.
Given u 1 , u 2 , . . . , u n ∈ X, n ≥ 3, we say that P is a h-short n-gon (with vertices u 1 , u 2 , . . . , u n = u 0 ) if P is the union of h-short paths [u i−1 , u i ] h for i = 1, 2, . . . , n. An n-gon is geodesic if it is 0-short. We say that h is suitably small if h < H(u i , u j , u k ) for all 1 ≤ i, j, k ≤ n.

Suppose
• Q is a geodesic n-gon with distinct vertices (v i ) n i=1 and associated metric d ′ ; • P a h-short n-gon with distinct vertices (u i ) n i=1 and associated metric d; Since a geodesic segment is isometrically equivalent to a segment on R, we can view the restriction of F to a single side of Q as being a path, and hence define the path length len(F ; x, y) to be the length of the associated path segment from F (x) to F (y). We call F : Q → P a constant speed n-gon map if P, Q, F are as above, and if for each 1 ≤ i ≤ n there is a constant K i such that len(F ; x, y) It is easy to see that, given any P, Q as above, a constant speed n-gon map always exists.
Given the following data: • a h-short n-gon P with distinct vertices u 1 , u 2 , . . . , u n ∈ X, where (X, d) is a metric space and h is suitably small; • a constant speed n-gon map F : Q → P , where Q is a geodesic n-gon with distinct vertices v 1 , v 2 , . . . , v n ∈ Y , and (Y, d ′ ) is a CAT(0) space, we define a hypothesis A n (P, h; F, Q, d ′ , C n ): (4.10) The inductive hypothesis for n is that for all P, h as above, there exist data (F, Q, d ′ ) such that A n (P, h; F, Q, d ′ , C n ) holds, and such that (Q, d ′ ) is a convex Euclidean n-gon in R 2 with d ′ being the Euclidean metric. This implies the desired C n -rough n-point embedding: the vertices of Q give the rough subembedding of the vertices of P . We have defined the hypothesis A n (P, h; F, Q, d ′ , C n ) in the more general context of a CAT(0) space Y because we will need this along the way.
The CAT(0) version of our inductive hypothesis for n is that for all geodesic n-gons P as above, there exist data (F, Q, d ′ ) such that A n (P, 0; F, Q, d ′ , 0) holds, and such that (Q, d ′ ) is a convex Euclidean n-gon in R 2 with d ′ being the Euclidean metric. Note also that with h = 0 and C n = 0 we get equality in (4.9), and (4.10) simplifies to It is a routine task to use the C ′ -rCAT(0) condition to verify the inductive hypothesis for n = 3 (and CAT(0) to verify the CAT(0) variant of the inductive hypothesis for n = 3), so assume that it holds for n = k ≥ 3. Let P be a given h-short (k + 1)-gon, where h is sufficiently small. We draw a h-short path from u 1 to u k that splits P into a h-short k-gon P 1 with vertices u 1 , . . . , u k , and a h-short triangle P 2 with vertices u 1 , u k , u k+1 . Let F i : Q i → P i , i = 1, 2 be the maps guaranteed by our inductive hypothesis for n = k and the easy case n = 3, where Q 1 is a convex k-gon with vertices v 1 , v 2 , . . . , v k ∈ R 2 and Q 2 is a triangle with vertices v 1 , v k , v k+1 . By use of isometries of R 2 , we may assume that the sides from v 1 to v k in Q 1 and in Q 2 are the same, and that Q 1 and Q 2 are on opposite sides of this line segment (so the interiors of Q 1 and Q 2 are disjoint).
We now let (Q, d ′ ) be the metric space formed by gluing Q 1 and Q 2 together along S = [v 1 , v k ], so Q ′ = Q 1 ⊔ S Q 2 . Let Q be the (k + 1)-gon with vertices v 1 , v 2 , . . . , v k+1 and define F : P → Q by Note that the fact that each F i is a constant speed map ensures that F is welldefined.
We wish to prove A k+1 (P, h; F, Q, d ′ , C k+1 ). In view of the construction, it suffices to verify (4.10), and for this we may assume that x ∈ Q 1 and y ∈ Q 2 . Let γ be the geodesic in Q connecting x to y. It follows that Using (4.10) for P 1 and P 2 and the definition of the gluing metric d ′ on Q, we thus get Essentially the same argument allows us to deduce A k+1 (P, 0; F, Q, d ′ , 0) from the CAT(0) version of our inductive hypothesis.
If Q happens to be convex, we are done with the proof so assume that Q is not convex. Then the interior angle at either v 1 or v k exceeds π. Assume without loss of generality that the interior angle at v 1 is larger than π. Now the union of the two geodesic segments [v k+1 , v 1 ] and [v 1 , v 2 ] is also a geodesic segment, and so by eliminating v 1 as a vertex, we may consider Q to be a geodesic k-gon with vertices v 2 , v 3 , . . . , v k+1 . We also note that Q is CAT(0) since Q 1 and Q 2 are CAT(0) and the gluing set [v 1 , v k ] is convex; see [BH,II.11.1]. Applying the CAT(0) version of our induction assumption to Q, we get a map G : R → Q, where R is a convex k-gon in R 2 with vertices w 2 , w 3 , . . . , w k+1 satisfying: whenever y ∈ R, w i a vertex adjacent to y , We now view R as a convex (k + 1)-gon by identifying G −1 (v 1 ) as an extra vertex (with interior angle π). Then F • G is the desired mapping (for both the rCAT(0) and CAT(0) variants of our inductive hypothesis). Thus we have established the inductive hypothesis for n = k + 1 and we are done with the proof.
For completeness we state a CAT(0) variant of Theorem 1.1. Proof. Since Theorem A already tells us that the 4-point condition is equivalent to CAT(0), it suffices to prove that CAT(0) implies the n-point condition for each n > 4. But this follows from the CAT(0) version of our inductive hypothesis which was established in the proof for all n ∈ N.
As mentioned in the Introduction, CAT(0) is preserved by various limit operations, including pointed Gromov-Hausdorff limits and ultralimits [BH,II.3.10]. The trick is to use the 4-point condition and the concept of a 4-point limit. A very similar argument, with the 4-point condition replaced by our rough 5-point condition, will give us similar results for rCAT(0) spaces. We begin with a definition of n-point limits.
Definition 4.15. A metric space (X, d) is an n-point limit of a sequence of metric spaces (X m , d m ), m ∈ N, if for every {x i } n i=1 ⊂ X, and ε > 0, there exist infinitely many integers m and points x i (m) ∈ X m , 1 ≤ i ≤ n, such that |d(x i , x j ) − d m (x i (m), x j (m))| < ε for 1 ≤ i, j ≤ n.
We are now ready to state a 5-point limit result. Note that, since any n-point limit of ((X m , d m )) ∞ m=1 (X, d) is also n ′ -point limit of this sequence of spaces for all n ′ ≤ n, the following result also holds if 5 is replaced by any larger integer. The proof of this result, which is very similar to the corresponding result for CAT(0) and 4-point limits given in [BH,II.3.9], is included for completeness.
Theorem 4.16. Suppose the length space (X, d) is a 5-point limit of (X m , d m ), m ∈ N, where (X m , d m ) is C m -rCAT(0; * ) for some constant C m . If C m ≤ C for all m ∈ N, then (X, d) is a C-rCAT(0) space, where C = 3C + 2 √ 3. If C m → 0, and (X, d) is complete, then (X, d) is a CAT(0) space.
Proof. Suppose first that C m ≤ C for all m ∈ N. Let (x i ) 5 i=1 be an arbitrary 5tuple of points in (X, d), and suppose that it is the 5-point limit of the 5-tuples (x i (m)) 5 i=1 in X m . By passing to a subsequence if necessary, we may assume that d(x i (m), x j (m)) → d(x i , x j ) for all 1 ≤ i, j ≤ 5.
By Theorem 4.2, every (X m , d m ) satisfies a C ′ -rough 5-point condition, where C ′ := 3C, so there exists a C ′ -rough subembedding (x 1 (m), x 2 (m), . . . , x 5 (m)) of (x 1 (m), x 2 (m), . . . , x 5 (m)) into R 2 , for each m ∈ N. Since translation is an isometry in R 2 , we may assume that the points x 1 (m) coincide for all m ∈ N. Thus all 5-tuples are contained in a disk of finite radius and by passing to a subsequence if necessary we may assume that x i (m) converges to some point x i as m → ∞, for all 1 ≤ i ≤ m. It follows readily that (x i ) 5 i=1 is a C ′ -rough subembedding of (x i ) 5 i=1 in R 2 . Thus