Characterising trees and hyperbolic spaces by their boundaries

We use the language of proper CAT(-1) spaces to study thick, locally compact trees, the real, complex and quaternionic hyperbolic spaces and the hyperbolic plane over the octonions. These are rank 1 Euclidean buildings, respectively rank 1 symmetric spaces of non-compact type. We give a uniform proof that these spaces may be reconstructed using the cross ratio on their visual boundary, bringing together the work of Tits and Bourdon.


Introduction
Euclidean buildings and symmetric spaces of non-compact type are two important families of metric spaces of non-positive curvature, with a number of common properties.For example, their maximal flat subspaces are all of the same dimension (this is the rank) and their boundary points admit the structure of a spherical building.We study the rank 1 case.The n-dimensional hyperbolic spaces K H n over K = R, C, H (n ≥ 2) and O (n = 2) are defined in Section 2.3.These are the rank 1 symmetric spaces of non-compact type.A (thick, locally compact, discrete) rank 1 Euclidean building is the metric realisation of a connected acyclic graph whose vertices have finite degree at least 3, we therefore refer to these spaces as trees, see Section 2.6.Tits gives a characterisation of trees T in terms of their boundary points ∂T and canonical valuation ω T : (∂T ) 4 → R in [11].He then uses this to prove that all Euclidean buildings can be characterised by their boundary.Having already classified spherical buildings of dimension ≥ 2 this allowed him to give a classification for Euclidean buildings of rank ≥ 3. A detailed account of Tits' work is given in [12].
Bourdon defines and studies the cross ratio ω on the boundary of a proper CAT(-1) space in [5].This simultaneously generalises the projective cross ratio on the boundary of the real hyperbolic plane and the canonical valuation defined by Tits for trees.Bourdon proves that a hyperbolic space H is isometrically embedded in a proper CAT(-1) space X if and only if there is a cross-ratio preserving homeomorphism from ∂H onto ∂X.In particular, this characterises when two hyperbolic spaces are isometric, in terms of their boundary points and the cross ratio.
Our main result is a uniform reconstruction of trees and hyperbolic spaces, using their boundary points and the cross ratio.In Tits' work, he shows that a tree may The author was supported by DFG SPP 2026 'Geometry at Infinity'.MSC2020 Subject Classification: 05C05, 51M10.
In Section 2 we outline the necessary preliminaries for this paper, in particular we define hyperbolic spaces, trees and proper CAT(-1) spaces, recalling some of their properties.We also define the Gromov product and Bourdon metric on the visual boundary of a proper CAT(-1) space.
In Section 3 we recall the definition of the cross ratio and outline what is known about its relationship with the geometry of proper CAT(-1) spaces.In particular, we define χ αβ and prove that this models [α, β].We also recall the Ptolemy inequality, proven by Foertsch and Schroeder [9] which gives a condition in terms of the cross ratio for when an ideal quadrilateral in a proper CAT(-1) space is isometric to an ideal quadrilateral in the real hyperbolic plane.
In Section 4 we prove a number of uniform propositions for trees and hyperbolic spaces, showing that one may use the cross ratio to understand how and when any two geodesics intersect.We then use this to define the equivalence relation on the disjoint union of all χ αβ and prove Theorem 1.1.If there exists a geodesic segment from x to y for all x, y ∈ X, then we say that X is a geodesic metric space.If I = [0, ∞), then we call c a geodesic ray and if I = R, then we call c a geodesic line.

Preliminaries
The visual boundary of X is defined to be the set of equivalence classes of geodesic rays where c ∼ c ′ if and only if there exists K > 0 such that d(c(t), c ′ (t)) < K for all t ≥ 0. We will denote boundary points using greek letters and if a geodesic ray c is in the equivalence class α, then we will write c(∞) = α.For a more detailed introduction to geodesic metric spaces and the visual boundary, see [6].

Symmetric Spaces of Non-Compact Type.
A symmetric space is a connected Riemannian manifold M such that for all o ∈ M , there exists an isometry φ o : M → M , fixing o and reversing geodesics through o.It follows easily from the definition, that a symmetric space is geodesically complete and its isometry group acts transitively.If M is a simply connected symmetric space with non-positive sectional curvature and no Euclidean de Rham factor, then it is said to be of non-compact type.The rank of a symmetric space of non-compact type is equal to the dimension of its maximal flat subspaces.i.e. subspaces which are isometric to Euclidean space, ordered by inclusion.The rank 1 symmetric spaces of non-compact type are the hyperbolic spaces defined in the next section.For more details on symmetric spaces of non-compact type, see [8].
2.3.The Hyperbolic Spaces.Let n ≥ 2 and let K denote either the field of real numbers R, complex numbers C or the skew field of quaternions H. Then one may define the following Hermitian form on K n+1 : where x = (x 1 , . . ., x n+1 ) and y = (y 1 , . . ., y n+1 ).We define an equivalence relation on K n+1 \ {0} by x ∼ K y if and only if x = λy for some λ ∈ K, the set of equivalence classes under this relation gives the n-dimensional projective space K P n .The points of the n-dimensional hyperbolic space K H n are then defined to be the set of points [x] ∈ K P n with ⟨x | x⟩ < 0. The distance d H ([x], [y]) between two points of K H n is defined to be the unique positive real number satisfying  , [y] with the property that for some i ̸ = j, x i and y j are real (cf.[2] and [10,Section 19]).The above model is based on [3].
0 , it is clear that there is a canonical isometric embedding of the real hyperbolic plane in the other hyperbolic spaces.In this section we look at other isometric embeddings of R H 2 → K H n , where K = R, C, H (n ≥ 2) or K = O (n = 2), using the ideas in [6,Theorem II.10.16].Let [x] ∈ K H n , since the isometry group acts transitively we may assume without loss of generality that x = 0, . . ., 0, 1 and we can model the tangent space at [x] by x ⊥ , where where u is a unit tangent vector i.e. u ∈ x ⊥ with ⟨u | u⟩ = 1.For all u ∈ x ⊥ \ {0}, ⟨u | u⟩ > 0 (positive definiteness) and the real part of ⟨u | v⟩ is equal to the Riemannian metric for all u, v ∈ x ⊥ .Let u, v ∈ x ⊥ be distinct unit tangent vectors and suppose that ⟨u | v⟩ ∈ R, then the real span of u, v, x is a 3-dimensional vector subspace V ⊂ K × . . .× K × R with the property that for all x, y ∈ V , x ∼ K y if and only if x ∼ R y.By positive definiteness we may find ũ, ṽ ∈ V ∩ x ⊥ such that ⟨ũ | ũ⟩ = 1, ⟨ṽ | ṽ⟩ = 1 and ⟨ũ | ṽ⟩ = 0.It follows that there is a linear isometry (i.e. a bijective 2.5.Geodesic Triangles and The Gromov Product.Let (X, d) be a metric space.For o, x, y ∈ X define the Gromov product of x and y based at o as . By the triangle inequality, the Gromov product (x | y) o is a non-negative real number for all x, y, o ∈ X. Suppose that for x, y, z ∈ X, there exist geodesic segments In a geodesic metric space, the Gromov product can be understood in terms of points in a triangle, see Figure 1.All hyperbolic triangles ∆(x, y, z) where φ denotes the golden ratio.Inspired by this property, a geodesic metric space X is called δ-hyperbolic for some δ ≥ 0, if all geodesic triangles ∆(x, y, z) ⊂ X have the property that if p ∈ [x, y], q ∈ [x, z] such that d(x, p) = d(x, q) ≤ (y | z) x , then d(p, q) ≤ δ, see [7] for more details on δ-hyperbolic spaces.Let (X, d) be a geodesic metric space and consider a geodesic triangle ∆(x, y, z).There exist three unique points p, q, r ∈ ∆(x, y, z), such that d(x, p) = d(x, r), d(y, p) = d(y, q) and d(z, q) = d(z, r).The left picture of this figure depicts an arbitrary geodesic triangle with these three special points marked and their respective distances to the vertices highlighted.These distances are equal to the Gromov products (y Euclidean triangle, these three special points are exactly the three intersection points of ∆(x, y, z) with its incircle, this is pictured in the middle.The right picture depicts a tripod.Here, the three special points meet at the centre.

2.6.
Trees.A building is a simplicial complex ∆ which can be expressed as a union of Coxeter complexes (called apartments) such that any two simplices A, B ∈ ∆ are contained in an apartment and for any two apartments containing A, B ∈ ∆, there exists an isomorphism between them, fixing their intersection, see [1] for a detailed account of buildings.We always require that our buildings are thick, locally finite simplicial complexes i.e. every codimension 1 simplex in ∆ is the face of n maximal simplices where 3 ≤ n < ∞.A building is called Euclidean if its apartments are Euclidean Coxeter complexes.Every Euclidean building may be realised as a complete metric space of non-positive curvature [1, Section 11.2.].The rank of a Euclidean building is defined to be the dimension of its maximal flat subspaces (these are exactly the apartments).A rank 1 Euclidean building is the metric realisation of a connected acyclic graph whose vertices have finite degree at least 3.We will refer to these spaces simply as trees.Note that all trees are 0-hyperbolic since all their geodesic triangles are tripods.

The Cross Ratio on the Boundary of a Proper CAT(-1) Space
In this section X denotes a proper CAT(-1) space and ∂X its visual boundary.Suppose throughout that |∂X| ≥ 3.
If X is a tree, then the point p(α, β; γ) is the centre of the tripod ∆(α, β, γ).To see this, note that for p The existence and uniqueness of p(α, β; γ) can easily be verified using the relationship between the Gromov product and the horospherical distance outlined in Section 2.9.
It also follows from this relationship that the cross ratio, defined below, does not depend on the choice of basepoint o ∈ X.
Lemma 3.5.Let α, β, γ, δ ∈ ∂X be pairwise distinct.Given two different parametrisations of [α, β], one can use the cross ratio to determine whether they are describing the same point: which shows that d ω is a metric and φ αβ is an isometry.□ Foertsch and Schroeder study CAT(-1) spaces and Gromov hyperbolic spaces in [9], proving the 'Ptolemy inequality' for complete CAT(-1) spaces, in terms of the Bourdon metric on the Gromov boundary.We present this theorem for proper CAT(-1) spaces, in terms of the cross ratio on the visual boundary.
It follows from the above theroem, that we can use the cross ratio in order to characterise subspaces of the hyperbolic spaces which are isometric to the real hyperbolic plane.We also have a characterisation of all intersecting geodesics in the real hyperbolic plane in terms of the cross ratio.In the next section, we will build on this, giving a characterisation of all intersecting geodesics in all trees and hyperbolic spaces.In order to do this, we make use of a property of intersecting geodesics in the hyperbolic spaces, outlined by Bourdon in [5] in his proof of Theorem 0.1.

Uniform Reconstruction of X
In this section X denotes either a hyperbolic space or a tree (see Sections 2.3 and 2.6).We use the cross ratio in order to understand how and when two geodesics in X intersect.Finally, we use this to construct a metric space Ω(X) which is isometric to X.
Lemma 4.1.Let X be a tree.Given distinct α, β, γ, δ ∈ ∂X, there are only four possible cases for the values of ω(α, β; γ, δ), ω(α, γ; δ, β) and ω(α, δ; γ, β): Proof.X is a 0-hyperbolic space and therefore by [7, Lemmas 2.1.4.and 2.2.2.], for any o ∈ X and any distinct α, β, γ, δ ∈ ∂X, the two smallest entries of the following triple are equal: Equivalently, the two largest entries of are equal.There are therefore four cases: (1) each corresponding to the four cases listed in the statement.□ The following lemma explains why the four cases outlined in the above lemma correspond to the four cases outlined in Figure 3.
Lemma 4.2.Let X be a tree and consider pairwise distinct α, β, γ, δ ∈ ∂X and all six geodesics between them.There are only four possible cases for how these geodesics intersect and these correspond to the four cases outlined in Figure 3.
□ It follows from Theorem 3.7 that if α, β, γ, δ ∈ ∂X are any four distinct boundary points of a hyperbolic space (not necessarily real) and the above condition on the cross ratios holds, then [α, β] and [γ, δ] intersect.However, the condition will not hold for The following proposition shows that any pair of intersecting geodesics in a tree or hyperbolic space satisfy a condition in terms of cross ratios.Proposition 4.4.Let X be either a tree or a hyperbolic space and let α, β, α Proof.Suppose X is a hyperbolic space and [α, β] and [α ′ , β ′ ] intersect in a point o, then α, β, α ′ , β ′ must be distinct.As in Section 2.4, we may suppose that o = [x] where x = (0, . . ., 0, 1) and let u, v ∈ x ⊥ be the distinct unit tangent vectors corresponding to [α, β] and [α  Proof.Since X is a symmetric space, we know that φ o is an isometry of X which fixes o and reverses the direction of all geodesics through o. for all p ∈ X.
□ Proposition 4.4 tells us exactly when two geodesics in a tree or hyperbolic space intersect, in terms of the cross ratio on the boundary.Combining this with the above proposition, we also understand how two geodesics in a tree or hyperbolic space intersect, i.e. we may give their intersection points using the boundary points and cross ratio.We will now use this to define a model space for X.
Clearly, the right hand side of both of these equivalances are equivalent to each other by the definition of Ω(X) and Proposition 4.6.It follows that φ is a bijection.
Given any two points of X we may find a geodesic [α, β] containing them.We may therefore express any two points of X as [α, β] γ (s) and [α, β] γ (t) for some α, β, γ ∈ ∂X and some s, t ∈ R. Since φ is bijective, we may therefore express any two elements of Ω(X) as (α, β, γ, s), (α, β, γ, t) so d ω is well-defined.Since [α, β] γ is a geodesic we know that which shows that d ω is a metric on Ω(X) and φ is an isometry.□

2. 1 .
Geodesics and The Visual Boundary.Let (X, d) be a metric space.Let I ⊆ R be a closed interval.A curve c : I → X is called a geodesic if d(c(s), c(t)) = |s − t| for all s, t ∈ I.If I = [a, b] then we call c a geodesic segment from x = c(a) to y = c(b) and denote its image [x, y].

1 Figure 1 .
Figure1.Let (X, d) be a geodesic metric space and consider a geodesic triangle ∆(x, y, z).There exist three unique points p, q, r ∈ ∆(x, y, z), such that d(x, p) = d(x, r), d(y, p) = d(y, q) and d(z, q) = d(z, r).The left picture of this figure depicts an arbitrary geodesic triangle with these three special points marked and their respective distances to the vertices highlighted.These distances are equal to the Gromov products (y | z) x , (x | z) y and (x | y) z .If ∆(x, y, z) is a Euclidean triangle, these three special points are exactly the three intersection points of ∆(x, y, z) with its incircle, this is pictured in the middle.The right picture depicts a tripod.Here, the three special points meet at the centre.

Figure 3 .
Figure 3. Let X be a tree, then for distinct α, β, γ, δ ∈ ∂X, there are only four possibilities for how the six geodesics between them intersect.

d
X ([α, β] γ (s), [α, β] γ (t)) = |s − t| and therefore d ω ((α, β, γ, s), (α, β, γ, t)) = d X (φ(α, β, γ, s), φ(α, β, γ, t)) (1)re x, y ∈ K n+1 are any representatives of [x], [y] ∈ K H n .The above model is explained in detail in[6, Chapter II.10].Let O denote the division algebra of octonions and define ⟨x | y⟩ ∈ O for x, y ∈ O 3 as in(1), note that since O is not associative, this is not an Hermitian form.Denote by O 3 0 the set of triples x = (x 1 , x 2 , x 3 ) ∈ O 3 such that the subalgebra of O generated by x 1 , x 2 , x 3 is associative.We define an equivalence relation on O 3 0 \ {0} by x ∼ O y if and only if x = λy and λ, y 1 , y 2 , y 3 are contained in an associative subalgebra of O, in this case ⟨x | x⟩ = λλ⟨y | y⟩ ∈ R. The set of equivalence classes under ∼ O is the projective plane O P 2 .The hyperbolic plane O H 2 is defined to be the set of points [x] ∈ O P 2 with ⟨x | x⟩ < 0. The distance d H ([x], [y]) between two points of O H [6,]as are all trees [6, II.1.15].A metric space is called proper if every closed and bounded subset is compact.Recall that trees and hyperbolic spaces are locally compact and complete, it follows from the Hopf-Rinow theorem[6, Proposition I.3.7] that all trees and hyperbolic spaces are proper metric spaces.2.8.Geodesics in CAT(-1) Spaces.Let X be a CAT(-1) space.For all x, y ∈ X, there exists a geodesic segment from x to y, which is unique up to parametrisation[6, Proposition II.1.4].If X is complete, then for all α ∈ ∂X and all o ∈ X, there exists a unique geodesic ray c such that c(0) = o, c(∞) = α [6, Proposition II.8.2].We denote the image of this geodesic ray [o, α].If X is proper, then for any two distinct boundary points α, β ∈ ∂X, there exists a geodesic line c such that c(−∞) = α and c(∞) = β.We denote the image of this geodesic by [α, β].It follows from the flat strip theorem [6, Theorem II.2.13] that this geodesic is unique up to parametrisation.
2.9.The Gromov Product and Bourdon Metric on ∂X.Let X be a proper CAT(-1) space, then we define the Gromov product of α, β ∈ ∂X based at o ∈ X as(α | β) o := lim t→∞ (x(t) | y(t)) o where x ∈ α, y ∈ β [7, Prop.3.4.2].Define the horospherical distance between x, y ∈ X relative to α ∈ ∂X as Both Bourdon and Tits used boundary points in order to model the geodesics of their spaces.Below, we define χ αβ for α, β ∈ ∂X and prove that this models the geodesic [α, β] ⊂ X.