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Hyperbolic Spaces and Ptolemy Moebius Structures

We characterize the class of Gromov hyperbolic spaces, whose boundary at infinity allow canonical M\"obius structures.


Introduction
There is a deep and well studied relation between the geometry of the classical hyperbolic space and the Möbius geometry of its boundary at infinity. This relation can be generalized in a nice way to CAT(−1) spaces.
Let X be a CAT(−1) space with boundary Z = ∂ ∞ X. For every basepoint o ∈ X one can define the Bourdon metric ρ o (x, y) = e −(x|y)o on Z, where ( | ) o is the Gromov product with respect to o, compare [B1]. For different basepoints o, o ′ ∈ X the metrics ρ o , ρ o ′ are Möbius equivalent and thus define a Möbius structure on Z. By [FS1] this Möbius structure is ptolemaic.
We discuss this curvature condition in more detail later and compare it in section 3.3 with the asymptotically CAT(−1) condition, which is formulated in more familiar comparison terms. It turns out that CAT(−1) are asymptotically PT −1 and that the relation between these spaces and the Möbius geometry of their boundaries can be expressed in the following two results: Theorem 1.2. Let X be asymptotically PT −1 , then X is a boundary continuous Gromov hyperbolic space. For every basepoint o ∈ X, ρ o (x, y) = e −(x|y)o defines a metric on ∂ ∞ X. For different basepoints these metrics are Möbius equivalent and thus define a canonical Möbius structure M on ∂ ∞ X. The Möbius structure M is complete and ptolemaic. Theorem 1.3. Let (Z, M) be a complete and ptolemaic Möbius space. Then there exists an asymptotically PT −1 space X such that ∂ ∞ X with its canonical Möbius structure is Möbius equivalent to (Z, M).
The results can be viewed as a characterization of the class of Gromov hyperbolic spaces, whose boundary allows a canonical Möbius structure.
In the proof we use a hyperbolic cone construction due to Bonk and Schramm [BoS]. which associates to a metric space (Z, d) a cone (Con(Z), ρ). We show in Proposition 4.1 that if (Z, d) is ptolemaic, then the cone is asymptotically PT −1 . This method can also be used obtain a characterization of visual Gromov hyperbolic spaces in the spirit of the [BoS]. Recall that two metric spaces X and Y are roughly similar, if there are constants K, λ > 0 and a map f : A theorem of Bonk and Schramm states that a visual Gromov hyperbolic space with doubling boundary is roughly similar to a convex subset of the real hyperbolic space H n for some integer n.
We have a version without conditions on the boundary: Theorem 1.4. Every visual Gromov hyperbolic space is roughly similar to some asymptotically PT −1 space.
We discuss some open questions in Remark 3.2 The structure of the paper is as follows. In section 2 we recall the basic facts about metric Möbius geometry and boundary continuous Gromov hyperbolic spaces. In section 3 we introduce the PT κ property, discuss asymptotically PT −1 spaces and prove Theorem 1.2. In section 4 we introduce hyperbolic cones and prove Theorem 1.3. The proof of Theorem 1.4 is in section 5.
If Ω(d) is not empty, we call the unique ω ∈ Ω(d) simply the point at infinity of (Z, d). We write Z ω for the set Z \ {ω}.
The topology considered on (Z, d) is the topology with the basis consisting of all open distance balls B r (z) around points in z ∈ Z ω and the complements D C of all closed distance balls D = B r (z).
We call an extended metric space (Z, d) complete, if first every Cauchy sequence in Z ω converges and secondly if the infinitely remote point ω exists in case that Z ω is unbounded. For example the real line (R, d), with its standard metric is not complete (as extended metric space), while (R ∪ {∞}, d) is complete.
We say that a quadruple (x, y, z, w) ∈ Z 4 is admissible, if no entry occurs three or four times in the quadruple. We denote with Q ⊂ Z 4 the set of admissible quadruples. We define the cross ratio triple as the map crt : Q → Σ ⊂ RP 2 which maps admissible quadruples to points in the real projective plane defined by here Σ is the subset of points (a : b : c) ∈ RP 2 , where all entries a, b, c are nonnegative or all entries are non positive.
Similar as for the classical cross ratio there are six possible definitions by permuting the entries and we choose the above one.
A map f : Z → Z ′ between two extended metric spaces is called Möbius, if f is injective and for all admissible quadruples (x, y, z, w) of X, crt(f (x), f (y), f (z), f (w)) = crt(x, y, z, w). The Möbius equivalence of metrics of metrics on a given set Z is clearly an equivalence relation. A Möbius structure M on Z is an equivalence class of extended metrics on Z.
A pair (Z, M) of a set Z together with a Möbius structure M on Z is called a Möbius space. A Möbius structure well defines a topology on Z, thus a Möbius space is in particular a topological space. Since completeness is also a Möbius invariant we can speak about complete Möbius structures.
In general two metrics in M can look very different. However if two metrics have the same remote point at infinity, then they are homothetic (see [FS2]): Lemma 2.1. Let M be a Möbius structure on a set X, and let d, d ′ ∈ M, such that ω ∈ X is the remote point of d and of d ′ . Then there exists λ > 0, such that d ′ (x, y) = λd(x, y) for all x, y ∈ X.
An extended metric space (Z, d) is called a Ptolemy space, if for all quadruples of points {x, y, z, w} ∈ Z 4 the Ptolemy inequality holds We can reformulate this condition in terms of the cross ratio triple. Let ∆ ⊂ Σ be the set of points (a : b : c) ∈ Σ, such that the entries a, b, c satisfy the triangle inequality. This is obviously well defined.
Then an extended space is Ptolemy, if crt(x, y, z, w) ∈ ∆ for all allowed quadruples Q.
This description shows that the Ptolemy property is Möbius invariant and thus a property of the Möbius structure M.
The importance of the Ptolemy property comes from the following fact (see e.g. [FS2]): The metric d ω can be obtained by metric involution. If d is a metric on Z then gives the required metric.

Boundary continuous Gromov hyperbolic spaces
We recall some basic facts from the theory of Gromov hyperbolic spaces, compare e.g [BS] A metric space (X, d) is called Gromov hyperbolic if there exists some δ > 0 such that for all quadruples x 1 , x 2 , x 3 , x 4 ∈ X we have For three points x, y, z ∈ X one defines the Gromov product where we write |xy| as a short version of d(x, y).
A sequence (x i ) converges at infinity, if for some (and hence every) The boundary ∂ ∞ X consist of the equivalence classes of these sequences.
For two points ζ, ξ ∈ ∂ ∞ X and a base point o ∈ X one defines where the infimum is taken over all sequences (x i ) ∈ ζ and (y i ) ∈ ξ. In a similar way we also define (x|ξ) o , where o, x ∈ X and ξ ∈ ∂ ∞ X.
We remark that the sequence (x i |y i ) o does not necessarily converge, therefore we need the complicated definition in (1).
A Gromov hyperbolic space is called boundary continuous, if the Gromov product extends continuously to the boundary in the following way: if (x i ), (y i ) are sequences in X which converge to points x, y in X or ∂ ∞ X, then (x i |y i ) o → (x|y) o for all base points o ∈ X. For boundary continuous spaces one can define nicely Busemann functions. If ω ∈ ∂ ∞ X and o ∈ X a base point, then where w i → ω is the Busemannfunction of ω normalized to have the value 0 at the point o ∈ X. We have the formula: We also define form ω ∈ ∂ ∞ X a base point o ∈ X and x, y, z from X or

Asymptotic PT κ spaces
In subsection 3.1 we define general PT κ spaces. Then in section 3.2 we introduce the more general notion of asymptotic PT κ spaces and compare it in section 3.3 with the notion of asymptotically CAT(κ) spaces. Finally in section 3.4 we prove Theorem 1.2.

The P T κ inequality
In the case that κ > 0 we assume in addition that the diameter is bounded by π √ κ . It is well known that the standard space forms M n κ of constant curvature κ satisfy the PT κ inequality. For the euclidean space this is the classical ptolemaic inequality and for the other spaces it is proved in [H]. By comparison we obtain the result also for CAT(κ)-spaces.
Remark 3.2. The following questions arises naturally: is a geodesic PT κ space CAT(κ)? A positive answer would imply a nice four point characterization of CAT(κ) spaces. In the case κ = 0 this is not true (see [FLS]), but the counterexamples are not locally compact and there are partial positive results in the locally compact case (see e.g. [BuFW], [MS]). For κ < 0 the question is completely open.
3.2 Asymptotic PT κ inequality for κ < 0 One obtains the asymptotic PT κ property (for κ < 0) by weakening equation the PT κ inequality and allowing some error term. Instead of equation (4) we require that for some universal δ ≥ 0 we have It is more convenient to formulate this condition using only exponential functions. It is easy to check that these conditions are equivalent.
Definition 3.3. A metric space is called asymptotic P T κ for some κ < 0, if there exists some δ ≥ 0 such that for all quadruples Remark 3.4. The asymptotic PT κ condition is a strong curvature condition.
Proof. From the asymptotic P T κ inequality, we obtain that Here ρ = max i,j ρ i,j for i, j = 1, 2, 3, 4. Since we know that for 0 ≤ x ≤ 1 It satisfies the asymptotic P T κ ′ inequality.
By scaling an asymptotic PT κ space with the factor 1 √ −κ we obtain an asymptotic PT −1 space. Therefore we will discuss in the sequel mainly PT −1 spaces.

Asymptotically CAT(κ) spaces
We relate the asymptotic PT(κ) property to some condition which is formulated in familiar comparison terms.
Definition 3.6. Let κ < 0. A metric space (X, d) is called asymptotically CAT(κ), if there exists some δ ≥ 0 such that for every quadruple of points 2 ) + δ. Remark 3.7. This definition makes also sense for κ = 0, then the last inequality is just d(x 2 , x 4 ) ≤ |x 2 x 4 | + 2δ. Then the condition is the rough CAT(0) condition of [BuF]. In general, if one replaces the last condition ( also for κ < 0) simply by the condition d(x 2 , x 4 ) ≤ |x 2 x 4 | + δ, then one obtains , what is called rough CAT(κ) in [BuF]. For κ < 0 this is equivalent to Gromov hyperbolicity and brings no new information Lemma 3.8. If (X, d) is asymptotically CAT(κ), then it is also asymptotically PT κ .
Proof. Let X be asymptotically CAT(κ) with constant δ. Let x 1 , x 2 , x 3 , x 4 ∈ X be given and let x 1 , x 2 , x 3 , x 4 ∈ M 2 κ be comparison points according to the asymptotic CAT(κ) property. Let ρ i,j = d(x i , x j ), ρ i,j = |x i x j | and ρ = max ρ i,j . Then Thus X is asymptotically PT κ with constant δ √ −κ .

Properties of asymptotically PT −1 spaces
Proposition 3.9. An asymptotic P T −1 metric space is a Gromov hyperbolic space.
Thus X is a Gromov hyperbolic space.
Lemma 3.10. Let X be an asymptotic Then lim Proof. From the asymptotic P T −1 inequality, we obtain Note that by triangle inequalities and hence E i → 0 by our assumptions. Taking the limit, we obtain As an immediate consequence we get Corollary 3.11. An asymptotic PT −1 space is boundary continuous.
Theorem 3.12. Let X be an asymptotic P T −1 metric space and o ∈ X, then Proof. First, we show that ρ o is a metric on ∂ ∞ X. For given three points From the asymptotic P T −1 inequality, we have . Again we easily check that E i → 0 and we obtain in the limit the triangle inequality for ρ o .
We use the similar argument to show that ρ o satisfies the ptolemaic inequality i.e. Choose sequences (x i ) ∈ x, (y i ) ∈ y, (z i ) ∈ z, (w i ) ∈ w. Since we have Again we see that E i → 0 and we obtain in the limit the desired ptolemaic inequality.
Remark 3.13. The above result implies in particular that the asymptotic upper curvature bound (see [BF]) of an asymptotic PT κ space is bounded above by κ.

Hyperbolic cones over Möbius spaces
In this chapter we prove Theorem 1.3. Therefore we give (bases on [BoS]) a construction, how to associate to a ptolemaic Möbius space (Z, M) a hyperbolic space X (which turns out to be asymptotically PT −1 ), such that (Z, M) is the canonical Möbius structure of ∂ ∞ X.
Let (Z, M) be a complete ptolemaic Möbius space. We choose a point ω ∈ Z and an extended metric d ∈ M from the Möbius structure, such that {ω} = Ω(d) is the point at infinity. Such a metric exists by Theorem 2.2 and this metric is unique (up to homothety) by Lemma 2.1.
We take now the metric space (Z ω , d), where Z ω = Z \ {ω} and apply the cone construction of [BoS] to it. The space Con(Z ω ) has properties analogous to the hyperbolic convex hull of a set in the boundary of a real hyperbolic space. Set It turns out that ρ satisfies the triangle inequality and is thus a metric, see [BoS]. We write |zz ′ | = d(z, z ′ ) for z, z ′ ∈ Z ω . Proposition 4.1. (Con(Z ω ), ρ) is asymptotically P T −1 .
Proof. Given arbitrary four points x i = (z i , h i ) ∈ Con((Z ω , d)), z i ∈ (Z ω , d), i = 1, 2, 3, 4, we have Since (Z, M) is a complete ptolemaic Möbius space, (Z ω , d) is a complete metric space which satisfies the P T 0 inequality, hence we obtain Replacing |z i z j | by (6), we have the following inequality This can be written as Using again (6) we obtain which shows that the last term in (7) is nonnegative and can be omitted. We use below that Therefore X is asymptotic P T −1 .
To finish the proof of Theorem 1.3, we have to show that ∂ ∞ X can be canonically identified with Z.
We chose a base point z 0 ∈ Z ω and then o := (z 0 , 1) as base point of X. We define for simplicity |z| := |zz 0 |. For x = (z, h) and Lemma 4.2. A sequence x i = (z i , h i ) in X converges at infinity, if and only if one of the following holds 1.
Proof. We show first the if implication: Assume 1. that (z i ) is a Cauchy sequence and h i → 0. Then equation One easily sees For the only if part assume that we have given a sequence x i = (z i , h i ) with lim i,j→∞ (x i |x j ) o = ∞.
We first show that there cannot exist two subsequences (x i k ) and (x i l ) of (x i ), such that |z i k | + h i k → ∞ for k → ∞ and |z i l | + h i l ≤ M for all l. If to the contrary such sequences would exist, then we easily obtain using triangle inequalities that and hence lim sup(x i k |x i l ) o is finite, a contradiction.
Thus either |z i | + h i → ∞ and we are in case 2 or |z i | + h i is bounded. The boundedness and (x i |x j ) o →, ∞ implies log(|z i z j | + h i ∨ h j ) → ∞ and hence (z i ) is a Cauchy sequence and h i → 0. Lemma 4.3. One can identify Z with ∂ ∞ X in a canonical way.
Proof. We define a map χ : Z → ∂ ∞ X by z → [(z, 1 i )] for z ∈ Z ω and ω → [(z 0 , i)]; here [ ] denotes the equivalence class of the corresponding sequences. Formula (8) shows that this map is injective. Let now ξ ∈ ∂ ∞ X be given and be represented by a sequence x i = (z i , h i ). If |z i |+h i → ∞ then Proposition 5.1. Assume that X is a visual Gromov hyperbolic space such that e −(·|·)o is bilipschitz to a ptolemaic metric d on ∂ ∞ X, the X is rough isometric to an asymptotically PT −1 space.
Proof. Consider the truncated Cone Con T (∂ ∞ X, d), which is defined as Con T (∂ ∞ X) = ∂ ∞ X × (0, D], where D = diam(∂ ∞ X, d) again with the metric defined by (5). This is the cone considered in [BoS], where it is shown that X is rough isometric to Con T (∂ ∞ X, d). Since by Proposition 4.1 the cone is PT −1 , the result follows. Now we can finish the proof of Theorem 1.4. We start with some visual Gromov hyperbolic space (X, d X ) with some base point o ∈ X. There exists some ε > 0, such that the function e −ε(·|·)o is bilipschitz to a metric ρ(·, ·) on ∂ ∞ X (see e.g. [BS,Theorem 2.2.7]). By a result of Lytchak (see [FS1,Proposition 8]) ρ 1 2 is a ptolemaic. Clearly e − ε 2 (·|·)o is bilipschitz to the metric ρ 1 2 (·, ·). Thus the visual Gromov hyperbolic space (X, ε 2 d X ) satisfies the assumptions of Proposition 5.1 and is rough isometric to an asymptotically PT −1 space. Hence (X, d X ) is rough similar to this space.