Incidence axioms for the boundary at infinity of complex hyperbolic spaces

We characterize the boundary at infinity of a complex hyperbolic space as a compact Ptolemy space that satisfies four incidence axioms.


Introduction
We characterize the boundary at infinity of a complex hyperbolic space C H k , k ≥ 1, as a compact Ptolemy space that satisfies four incidence axioms.
Recall that a Möbius structure on a set X, or a Möbius space over X, is a class of Möbius equivalent metrics on X, where two metrics are equivalent if they have the same cross-ratios. A Ptolemy space is a Möbius space with the property that the metric inversion operation preserves the Möbius structure, that is, the function d ω (x, y) = d(x, y) d(x, ω) · (y, ω) (1) satisfies the triangle inequality for every metric d of the Möbius structure and every ω ∈ X. A basic example of a Ptolemy space is the boundary at infinity of a rank one symmetric space M of noncompact type taken with the canonical Möbius structure, see sect. 2.2. In the case M = H k+1 is a real hyperbolic space, the boundary at infinity ∂ ∞ M taken with the canonical Möbius structure is Möbius equivalent to an extended Euclidean space R k = R k ∪ {∞}.

Incidence axioms
Basic objects of our axiom system are R-circles, C-circles and harmonic 4-tuples in a Möbius space X. Any R-circle is Möbius equivalent to the extended real line R and any C-circle is the square root of R. In other words, an R-circle can be identified with ∂ ∞ H 2 , and a C-circle with ∂ ∞ ( 1 2 H 2 ), where the hyperbolic plane 1 2 H 2 has constant curvature −4. An (ordered) 4-tuple (x, z, y, u) ⊂ X of pairwise distinct points is harmonic if d(x, z)d(y, u) = d(x, u)d(y, z) for some and hence for any metric d of the Möbius structure. 1 We use the notation Harm A for the set of harmonic 4-tuples in A ⊂ X. In the case A = X we abbreviate to Harm. In the case X = ∂ ∞ M for M = H 2 or M = 1 2 H 2 a 4-tuple (x, z, y, u) is harmonic if and only if the geodesic lines xy, zu ⊂ M intersect each other orthogonally.
We consider the following axioms.
(E) Existence axioms (E C ) Through every two distinct points in X there is a unique C-circle.
(E R ) For every C-circle F ⊂ X, ω ∈ F and u ∈ X \ F there is a unique R-circle σ ⊂ X through ω, u that intersects F ω = F \ {ω}.
(O) Orthogonality axioms: For every R-circle σ and every C-circle F with common distinct points o, ω the following holds (O C ) given u, v ∈ σ such that (o, u, ω, v) ∈ Harm σ , the 4-tuple (w, u, o, v) is harmonic for every w ∈ F , (O R ) given x, y ∈ F such that (o, x, ω, y) ∈ Harm F , the 4-tuple (w, x, ω, y) is harmonic for every w ∈ σ.

Main result
Our main result is the following Theorem 1.1. Let X be a compact Ptolemy space that satisfies axioms (E) and (O). Then X is Möbius equivalent to the boundary at infinity of a complex hyperbolic space C H k , k ≥ 1, taken with the canonical Möbius structure.
In [BS2] we have obtained a Möbius characterization of the boundary at infinity of any rank one symmetric space assuming existence of sufficiently many Möbius automorphisms. In contrast to [BS2], there is no assumption in Theorem 1.1 on automorphisms of X, and one of the key problems with Theorem 1.1 is to establish existence of at least one nontrivial automorphism.
In this respect, Theorem 1.1 is an analog for complex hyperbolic spaces of a result obtained in [FS1] for real hyperbolic spaces: every compact Ptolemy space such that through any three points there exists an R-circle is Möbius equivalent to R k = ∂ ∞ H k+1 .
The model space Y = ∂ ∞ M , M = C H k , serves for motivation and illustration of various notions and constructions used in the proof of Theorem 1.1. These are reflections with respect to C-circles, pure homotheties, orthogonal complements to C-circles, suspensions over orthogonal complements, Möbius joins. In sect. 2, we describe the canonical Möbius structure on Y , show that this structure satisfies axioms (E), (O), and study in Proposition 2.2 holonomy of the normal bundle to a complex hyperbolic plane in C H k . This Proposition plays an important role in sect. 10, see Lemma 10.16. Now, we briefly describe the logic of the proof. The first important step is Proposition 4.1 which leads to existence for every C-circle F ⊂ X of an involution ϕ F : X → X whose fixed point set is F , and such that every R-circle σ ⊂ X intersecting F at two points is ϕ F -invariant, ϕ F (σ) = σ. The involution ϕ F is called the reflection with respect to F .
In the second step, crucial for the paper, we obtain a distance formula, see Proposition 5.2, which is our basic tool. Using the distance formula, we establish existence of nontrivial Möbius automorphisms X → X beginning with vertical shifts, see sect. 7.1, and then proving that reflections with respect to C-circles and pure homotheties are Möbius, sect. 7.4 and 7.5.
Next, in sect. 8 we introduce the notion of an orthogonal complement A = (F, η) ⊥ to a C-circle F ⊂ X at a Möbius involution η : F → F without fixed points. In the case of the model space Y , A is the boundary at infinity of the orthogonal complement to the complex hyperbolic plane E ⊂ M with ∂ ∞ E = F at a ∈ E, η = ∂ ∞ s a , where s a : E → E is the central symmetry with respect to a.
In the third step, we show that for any mutually orthogonal C-circles F , F ′ ⊂ X the intersection A ∩ A ′ of their orthogonal complements, when nonempty, satisfies axioms (E), (O), see sect. 8.4. This gives a possibility to proceed by induction on dimension.
To do that, we introduce in sect. 9 the notion of a Möbius join F * F ′ ⊂ X of a C-circle F and its canonical orthogonal subspace F ′ ⊂ (F, η) ⊥ . It turns out that X = F * A for any A = (F, η) ⊥ . All what remains to prove is that whenever F ′ is Möbius equivalent to ∂ ∞ C H k−1 , the Möbius join F * F ′ is equivalent to ∂ ∞ C H k . The distance formula works perfectly in the case dim F ′ = 1, i.e., F ′ ⊂ (F, η) ⊥ is a C-circle, however, if dim F ′ > 1, it does not work directly. Thus we proceed in two steps. First, we prove that F * F ′ = ∂ ∞ C H 2 for any mutually orthogonal C-circles F , F ′ ⊂ X, in particular, this proves Theorem 1.1 in the case dim X = 3. Then using this fact as a power tool, we establish that the base B ω of the canonical foliation X ω → B ω , ω ∈ X, see sect. 6, is Euclidean. Using this, we finally prove the general case for dim F ′ > 1. We use the standard notation T M for the tangent bundle of M and U M for the subbundle of the unit vectors. For every unit vector u ∈ U a M , a ∈ M , the eigenspaces E u (λ) of the curvature operator R(·, u)u : u ⊥ → u ⊥ , where u ⊥ ⊂ T o M is the subspace orthogonal to u, are parallel along the geodesic γ(t) = exp a (tu), t ∈ R, and the respective eigenvalues λ = −1, −4 are constant along γ. The dimensions of the eigenspaces are dim E u (−1) = 2(k − 1), dim E u (−4) = 1, u ⊥ = E u (−1) ⊕ E u (−4).
Any u ∈ U a M and a unit vector v ∈ E u (−4) span a 2-dimensional subspace L = L(u, v) ⊂ T a M for which exp a L ⊂ M is a totally geodesic subspace isometric to 1 2 H 2 called a complex hyperbolic plane, while for every unit v ∈ E u (−1) the totally geodesic subspace exp a L(u, v) ⊂ M is isometric to H 2 and called a real hyperbolic plane.
At every point a ∈ M there is an isometry s a : M → M with unique fixed point a, s a (a) = a, such that its differential ds a : T a M → T a M is the antipodal map, ds a = − id. The isometry s a is called the central symmetry at a. For different a, b ∈ M the composition s a • s b : M → M preserves the geodesic γ ⊂ M through a, b and acts on γ as a shift whose differential is a parallel translation along γ. Any such isometry is called a transvection.
Furthermore, there is a complex structure J : T M → T M , where J a : T a M → T a M is an isometry with J 2 a = − id. Then for every u ∈ U a M , the vectors u, v = J a (u) form an orthonormal basis of the tangent space to a complex hyperbolic plane E ∋ a.
Remark 2.1. Given a complex hyperbolic plane E ⊂ M and a point a ∈ E, the orthogonal complement E ⊥ ⊂ M to E at a is a totally geodesic subspace isometric to C H k−1 .

Holonomy of the normal bundle to a complex plane
Let E ⊂ M be a complex hyperbolic plane. Here, we describe the holonomy of the normal bundle E ⊥ along E. The result of this section plays an important role in sect. 10. For every a ∈ E we have a 1-dimensional fibration F a of the unit sphere U a E ⊥ tangent to the orthogonal complement E ⊥ a to E. Fibers of F a are circles T a L ∩ U a E ⊥ , where L ⊂ E ⊥ a are complex hyperbolic planes through a. By a holonomy of E ⊥ we mean a parallel translation along E with respect to the Levi-Civita connection of M .
Proposition 2.2. For every a ∈ E the holonomy of E ⊥ along any (piecewise smooth) loop in E with vertex a preserves every fiber of F a , though E ⊥ is not flat.
Proof. Let x, y = J a (x) ∈ U a E be an orthonormal basis of the tangent space T a E. For an arbitrary z ∈ U a E ⊥ we put u = J a (z). We actually only show that R(x, y)z = 2u, where R is the curvature tensor of M . This implies the Proposition.
For k = 1, where M = C H k , there is nothing to prove, and for k = 2 the first assertion is trivial. Thus assume that k ≥ 3. We take v ∈ U a E ⊥ that is orthogonal to z, u.
We use the following expression of R via k(x, y) := R(x, y)y, x for arbitrary x, y, z, w ∈ T a M , see [GKM], where x, y is the scalar product on T a M , Coming back to our choice of x, y, z, u, v, we obtain k(x, z) = k(y, z) = −1, k(w, y) = k(w, x) = −1 for w = u, v.
For the unit vectors Since the vector R(x, y)z ∈ T a E ⊥ is orthogonal to z, we finally obtain R(x, y)z = 2u.

The canonical Möbius structure on ∂ ∞ C H k
We let Y = ∂ ∞ M be the geodesic boundary at infinity of a complex hyperbolic space M . For every a ∈ M the function d a (ξ, ξ ′ ) = e −(ξ|ξ ′ )a for ξ, ξ ′ ∈ Y is a (bounded) metric on Y , where (ξ|ξ ′ ) a is the Gromov product based at a. For every ω ∈ Y and every Busemann function b : M → R centered at ω the function d b (ω, ω) := 0 and d b (ξ, ξ ′ ) = e −(ξ|ξ ′ ) b , except for the case ξ = ξ ′ = ω, is an extended (unbounded) metric on Y with infinitely remote point ω, where (ξ|ξ ′ ) b is the Gromov product with respect to b, see [BS3,sect.3.4.2]. Since M is a CAT(−1)-space, the metrics d a , d b satisfy the Ptolemy inequality and furthermore all these metrics are pairwise Möbius equivalent, see [FS2].
We let M be the canonical Möbius structure on Y generated by the metrics of type d a , a ∈ M , i.e. any metric d ∈ M is Möbius equivalent to some metric d a , a ∈ M . Then Y endowed with M is a compact Ptolemy space. Every extended metric d ∈ M is of type d = d b for some Busemann function b : M → R, while a bounded metric d ∈ M does not necessary coincide with λd a , for some a ∈ M and λ > 0, see [FS2]. We emphasize that metrics of M are neither Carnot-Carathéodory metrics nor length metrics.

Axioms (E) and (O) for ∂ ∞ C H k
Proposition 2.3. The boundary at infinity Y = ∂ ∞ M of any complex hyperbolic space M = C H k , k ≥ 1, taken with the canonical Möbius structure, satisfies axioms (E), (O).
Proof. Every C-circle F ⊂ Y is the boundary at infinity of a uniquely determined complex hyperbolic plane E ⊂ M , and every R-circle σ ⊂ Y is the boundary at infinity of a uniquely determined real hyperbolic plane R ⊂ M . Axioms (E), (O) are trivially satisfied for C H 1 = 1 2 H 2 . Thus we assume that k ≥ 2.
Axiom (E C ): given distinct x, y ∈ Y there is a unique geodesic γ ⊂ M with the ends x, y at infinity. This γ lies in the unique complex hyperbolic There is a unique geodesic γ ⊂ E through a with ω as one of the ends at infinity. Then [au) and γ span a real hyperbolic plane R ⊂ M , for which For every w ∈ F the geodesic rays [aw) ⊂ E, [au) ⊂ R span a sector wau in a real hyperbolic plane R ′ , which is isometric to the sectors wav ⊂ R ′ , oau, oav ⊂ R. Thus d a (o, u) = d a (o, v) = d a (w, u) = d a (w, v) with respect to the metric d a (p, q) = e −(p|q)a on Y . It follows that (w, u, o, v) ∈ Harm.
(O R ): given x, y ∈ F such that (o, x, ω, y) ∈ Harm F , we can assume that x = y. Then the geodesic xy ⊂ E is orthogonal to γ at a = xy ∩ γ For w ∈ σ, let b ∈ E be the orthogonal projection of w on E. Then b ∈ oω, and the geodesic rays [bw) ⊂ R, [bx) ⊂ E span a sector wbx in a real hyperbolic plane R ′ , which is isometric to the sector wby in another real hyperbolic plane R ′′ , while the sectors xbω, ybω ⊂ E are isometric. Thus 3 Spheres between two points

Briefly about Möbius geometry
Here we briefly recall basic notions of Möbius geometry. For more detail see [BS2], [FS1]. A quadruple Q = (x, y, z, u) of points in a set X is said to be admissible if no entry occurs three or four times in Q. Two metrics d, d ′ on X are Möbius equivalent if for any admissible quadruple Q = (x, y, z, u) ⊂ X the respective cross-ratio triples coincide, crt We consider extended metrics on X for which existence of an infinitely remote point ω ∈ X is allowed, that is, d(x, ω) = ∞ for all x ∈ X, x = ω. We always assume that such a point is unique if exists, and that d(ω, ω) = 0. We use notation X ω := X \ ω and the standard conventions for the calculation with ω = ∞.
A Möbius structure on a set X, or a Möbius space over X, is a class M = M(X) of metrics on X which are pairwise Möbius equivalent. A map f : X → X ′ between two Möbius spaces is called Möbius, if f is injective and for all admissible quadruples Q ⊂ X where the cross-ratio triples are taken with respect to some (and hence any) metric of the Möbius structure of X and of X ′ . If a Möbius map f : X → X ′ is bijective, then f −1 is Möbius, f is homeomorphism, and the Möbius spaces X, X ′ are said to be Möbius equivalent.
If two metrics of a Möbius structure have the same infinitely remote point, then they are homothetic, see [FS1]. We always assume that for every ω ∈ X the set X ω is endowed with a metric of the structure having ω as infinitely remote point, and use notation |xy| ω for the distance between x, y ∈ X ω . Sometimes we abbreviate to |xy| = |xy| ω .
A Möbius space X is Ptolemy, if it satisfies the Ptolemy inequality |xz| · |yu| ≤ |xy| · |zu| + |xu| · |yz| for any admissible 4-tuple (x, y, z, u) ⊂ X and for every metric of the Möbius structure. This is equivalent to the definition of a Ptolemy space given in sect.

Harmonic 4-tuples and spheres between two points
An admissible 4-tuple (x, z, y, u) ⊂ X is harmonic if crt(x, z, y, u) = (1 : * : 1) for some and hence any metric d of the Möbius structure. Note that if x = y or z = u for an admissible Q = (x, z, y, u), then Q is harmonic. We say that x, x ′ ∈ X lie on a sphere between distinct ω, ω ′ ∈ X if the 4-tuple (ω, x, ω ′ , x ′ ) is harmonic. For a fixed ω, ω ′ this defines an equivalence relation on X \ {ω, ω ′ }, and any equivalence class S ⊂ X \ {ω, ω ′ } is called a sphere between ω, ω ′ . Möbius maps preserve spheres between two points: if f : X → X is Möbius, then f (S) is a sphere between f (ω), f (ω ′ ). In the metric space X ω the sphere S is a metric sphere centered at ω ′ , for some r > 0. The points ω, ω ′ are poles of S.
The orthogonality axioms (O) tell us that if (o, u, ω, v) is harmonic on an R-circle, then the C-circle through o, ω lies in a sphere between u, v, and vice versa, if (x, o, y, ω) is harmonic on a C-circle, then any R-circle through x, y lies in a sphere between o, ω.
If we take some point on a sphere as infinitely remote, then the sphere becomes the bisector between its poles, Lemma 3.1. Let S ⊂ X is a sphere between distinct u, v ∈ X. Then for every ω ∈ S the set S ω is the bisector in X ω between u, v, that is, Proof. For every x ∈ X ω we have crt(x, u, ω, v) = (|xu| ω : * : |xv| ω ). Thus x ∈ S ω if and only if |xu| ω = |xv| ω .
Corollary 3.2. Let σ, F be R-circle, C-circle respectively with distinct common points o, ω. Then (i) for any (u, o, v, ω) ∈ Harm σ the C-line F ⊂ X ω through o lies in the bisector between u, v, that is, |wu| ω = |wv| ω for every w ∈ F , (ii) for any (x, o, y, ω) ∈ Harm F any R-line σ ⊂ X ω through o lies in the bisector between x, y, that is, |wx| ω = |wy| ω for every w ∈ σ.
Proof. We have {o, ω} = F ∩ σ. Thus by axiom (O C ) the C-line F lies in the bisector between u, v, and by axiom (O R ) the R-line σ lies in the bisector between x, y.
Lemma 3.3. Any C-circle F and any R-circle σ have at most two points in common.
Proof. Assume x, y, ω ∈ σ ∩ γ. Then for the C-circle F through x, ω we have y ∈ F by Lemma 3.3. By (E R ), there is at most one R-circle σ with x, y, ω ∈ σ.
Corollary 3.5. Given a C-circle F ⊂ X and ω ∈ F , there is a retraction µ F,ω : X → F (continuous on X ω ), µ F,ω (u) = u for u ∈ F and Proof. By Lemma 3.3, the intersection σ ∩ F ω consists of a unique point, Assume to the contrary that |op i | → ∞ for some basepoint o ∈ F in the metric of X ω . Let q i ∈ F such that (o, p i , q i , ω) ∈ Harm F . Then |oq i | → ∞. But this would imply that also |ou i | → ∞ because p i and u i lie on the same sphere between o and q i .
4 Involutions associated with complex circles 4.1 Reflections with respect to complex circles Let F ⊂ X be a C-circle. Then by Corollary 3.5, every u ∈ X \ F defines an involution η u : F → F without fixed points by Proposition 4.1. Given a C-circle F ⊂ X and u ∈ X \ F , there exists a unique v ∈ X \ F , v = u, such that (x, u, η u (x), v) ∈ Harm σ for every x ∈ F and for the R-circle σ = σ x through x, u, η u (x).
This results defines for a C-circle F an involution ϕ F : X → X with fixed point set F in the following way: for u ∈ X \ F let v = ϕ F (u) be the unique point defined by Proposition 4.1. For u ∈ F define ϕ F (u) = u. The involution ϕ F is called the reflection with respect to F . In the case that u ∈ X \F we say that u, v are conjugate poles of F . Note that η u = η v for conjugate poles u, v = ϕ F (u) of F . Thus in a simplified way one can visualize F as equator of a 2-sphere, u, v as the poles, the map η u = η v as antipodal map on the equator such that for all x on the equator the points (x, u, η u (x), v) lie in harmonic position on a circle as in the following picture, where For the proof of Proposition 4.1 we need Lemmas 4.2-4.5.
Thus by induction, the function d u is constant along the sequence β i (o).

A Möbius involution of a complex circle
via a metric of the Möbius structure, is continuous and takes values f ( Then ω with f (ω) = 1 does the job.
Lemma 4.7. Assume a sphere S ⊂ X between ω, w ′ intersects the C-circle We denote by ψ : F → F the metric reflection with respect to o. Then ψ is an isometry with ψ(o) = o, in particular, ψ is Möbius. Note that F is in the bisector in X ω between u and v, that is |wu| = |wv| for every w ∈ F . Furthermore, σ is in the bisector between any w ∈ F and ψ(w).
It follows that x, u, x ′ , v lie on the metric sphere of radius r centered at w ′ . Thus by Lemma 4.7 there is a circle It follows Corollary 4.9. For every C-circle F and every u ∈ X \ F the involution . Hence η u preserves the cross-ratio triple of any admissible 4-tuple in F .

A distance formula
Using Proposition 4.8, we derive here a distance formula (see Proposition 5.2) which plays a very important role in the paper.
Lemma 5.1. Let u, v ∈ X be conjugate poles of a C-circle F . Then for every x, z ∈ F we have Proof. We consider a background metric | | v with infinitely remote point v and we define | | ω and | | z as the metric inversions, i.e.

The canonical foliation of X ω
We fix ω ∈ X and denote by B ω the set of all the C-circles in X through ω. By axiom (E C ), for every x ∈ X ω the is a unique C-circle F ∈ B ω with x ∈ F . This defines a map π ω : X ω → B ω , π ω (x) = F , called the canonical projection, the set B ω is called the base of π ω , and the foliation of X ω by the fibers of π ω is said to be canonical. In this section we use notation |xy| for the distance between x, y ∈ X ω in the metric space X ω .

Busemann functions on
The Busemann function of σ is 1-Lipschitz and by Lemma 6.1 it is constant on F as well as on F ′ . It follows that |yy ′ | ≥ a for every y ∈ F , y ′ ∈ F ′ . In particular, |y ′ µ F,ω (y ′ )| ≥ a for all y ′ ∈ F ′ . By symmetry we get equality.

Canonical metric on the base
The projection π ω : X ω → B ω is 1-Lipschitz and isometric if restricted to any R-line in X ω . This implies that every two points in B ω lie on a geodesic line.
Lemma 6.3. The base B ω with the canonical metric is uniquely geodesic, i.e. between any two points there is a unique geodesic. In particular, B ω is contractible.
. This line is unique by Lemma 3.4. The claim follows.
Proposition 6.6. Given two R-lines l, l ′ ⊂ X ω , the Busemann functions of l are affine on l ′ .
Let m ∈ l ′ be the midpoint between x, y ∈ l ′ , |xm| = |my| = 1 2 |xy|. We have to show that b(m) = 1 2 (b(x) + b(y)). Busemann functions in any Ptolemy space are convex, see [FS1,Proposition 4 By Lemma 6.5, for every sufficiently large i there is an R-circle l i ⊂ X ω through x and ω i such that the sequence l i converges pointwise to l ′ . Thus there are points y i ∈ l i with y i → y. The points x, y i divide l i into two segments. Choose a point m i in the segment that does not contain ω i such that |xm i | = |m i y i |. One easily sees that m i → m.
The points x, m i , y i , ω i lie on the R-circle l i in this order. Thus Lemma 6.7. Busemann functions are constant on the fibers of π ω : Using Lemma 6.1 we can assume that l ∩ F = ∅. We take y ∈ l with b(y) = c and consider the C-line F ′ ⊂ X ω through y. Then F ′ ∩ F = ∅. By Lemma 6.1, the function b takes the constant value c on and there is a uniquely determined R-line σ ⊂ X ω through x, z. By Proposition 6.6 the function b takes the constant value c along σ.
Thus the values of b along σ ′ are uniformly bounded because b is Lipschitz and the distance of any u ∈ σ ′ to σ equals |x ′ x| = |z ′ z| by Lemma 6.4. Since b|σ ′ is affine, we conclude that 6.4 Properties of the base Proposition 6.8. The base B ω is isometric to a normed vector space of a finite dimension with a strictly convex norm.
Proof. By Lemma 6.3, B ω is a geodesic metric space such that through any two distinct points there is a unique geodesic line. We show that affine functions on B ω separate points. Any Busemann function b : X ω → R is affine on R-lines by Proposition 6.6. By Lemma 6.7, b is constant on the fibers of π ω , thus it determines a function b : there is a geodesic line l = π ω (l). Let b be a Busemann function on X ω associated with l. Then b takes different values on the fibers of π ω over x, x ′ respectively. Thus the affine function b separates the points x, x ′ , b(x) = b(x ′ ). Then by [HL], B ω is isometric to a convex subset of a normed vector space with a strictly convex norm. Since B ω is geodesically complete, i.e., through any two points there is a geodesic line, this subset is a subspace, and therefore B ω is isometric to a normed vector space E. The Ptolemy space X is compact, thus B ω is locally compact, and the dimension of E is finite.
Corollary 6.9. The space X is homeomorphic to sphere S k+1 , k ≥ 0.
Proof. By Proposition 6.8, B ω is homeomorphic to an Euclidean space R k , and since π ω : X ω → B ω is a fibration over B ω with fibers homeomorphic to R, the space X ω is homeomorphic to R k+1 . Thus X is homeomorphic to S k .
Proposition 6.10. The Ptolemy space X has the following property: Any 4-tuple Q ⊂ X of pairwise distinct points lies on an R-circle σ ⊂ X provided three of the points of Q lie on σ and the Ptolemy equality holds for the crossratio triple crt(Q).
Choosing y as infinitely remote, we have |xz| y = |xu| y + |uz| y and σ is an Rline in X y . Recall that the canonical projection π y : X y → B y is 1-Lipschitz and isometric on every R-line. Thus |x z| = |xz| y = |xu| y + |uz| y ≥ |xu| + |uz|, where x = π y (x). The triangle inequality in B y implies |x z| = |x u| + |u z|. By Proposition 6.8, B y is isometric to a normed vector space with a strictly convex norm. Thus we conclude that u lies between x and z on a line in B y . This means that the C-line F ⊂ X y through u hits σ, and for o = σ ∩ F we have |xz| y = |xo| y +|oz| y . By Proposition 5.2, |xo| y < |xu| y and |oz| y < |uz| y unless u = o. We conclude that u = o ∈ σ.

Vertical shifts
We fix ω ∈ X and change notation using the letter b for elements of the base B ω and F b for the respective fiber of π ω . For any two b, b ′ ∈ B ω we have by Lemma 6.4 the isometry µ bb ′ : Proof. If a sequence of geodesic segments in a metric space pointwise converges, then the limit is also a geodesic segment. Together with uniqueness of R-lines in X ω and compactness of X, this implies the claim.
We fix an orientation of F b and define the orientation of F b ′ via the isometry µ bb ′ . This gives a simultaneously determined orientation O on all the fibers of π ω .
Lemma 7.2. The orientation O is well defined and independent of the choice of b ∈ B ω .
Proof. By Lemma 6.3, the base B ω is contractible. Using Lemma 7.1, we see that the orientation of F b ′′ induced by µ bb ′′ coincides with that induced by Hence, the claim.
We assume that a simultaneous orientation O of C-lines in X ω is fixed, and we call it the fiber orientation. Now we are able to produce nontrivial Möbius automorphisms of X. Using the fiber orientation O we define for every s ∈ R the map γ = γ s : X ω → X ω which acts on every fiber F on π ω as the shift by |s| in the direction determined in the obvious way by the sign of s and O. The map γ is called a vertical shift.
Proof. This immediately follows from definition of a vertical shift, Lemma 6.2 and Proposition 5.2.
Lemma 7.4. For every C-circle F ⊂ X, every Möbius involution γ : F → F without fixed points extends to a Möbius map γ : X → X.
2 H 2 is the hyperbolic plane of constant curvature −4. For a fixed ω ∈ F any vertical shift α : X ω → X ω restricted to F is induced by a parabolic rotation α : Y → Y , which is an isometry without fixed points in Y having the unique fixed point ω ∈ ∂ ∞ Y . One easily sees that parabolic rotations generate the isometry group of Y preserving orientation.
The involution γ is induced by a central symmetry of Y . Thus γ can be represented by a composition of vertical shifts with appropriate fixed points on F . Hence γ extends to a Möbius map γ : X → X.
Assuming that ω ∈ X is fixed we denote B = B ω . For an isometry α : X ω → X ω preserving the fiber orientation O, we use notation rot α for the rotational part of the induced isometry α : B → B. An isometry γ : X ω → X ω is called a shift, if γ preserves O and rot γ = id.

Lifting isometry
We fix ω ∈ X and use notations B = B ω , π = π ω : X ω → B. Let P ⊂ B be a pointed oriented parallelogram, i.e., we assume that an orientation and a vertex o of P are fixed. We have a map τ P : F → F , where F ⊂ X ω is the fiber of π over o, F = π −1 (o). Namely, given x ∈ F , by axiom (E R ) there is a unique R-line in X ω through x that projects down by π to the first (according to the orientation) side of P containing o. In that way, we lift the sides of P to X ω in the cyclic order according to the orientation and starting with o which is initially lifted to x. Then τ P (x) ∈ F is the resulting lift of the parallelogram sides.
Lemma 7.7. The map τ P : F → F is an isometry that preserves orientation and, therefore, it acts on F as a vertical shift.
Proof. The map τ P is obtained as a composition of four C-line isometries of type µ bb ′ , see sect. 7.1. Any isometry µ bb ′ preserves orientation, see Lemma 7.2. Thus τ P : F → F is an isometry preserving orientation.
There is a unique vertical shift γ : X ω → X ω with γ|F = τ P . Furthermore, every shift η : X ω → X ω commutes with γ, thus the extension γ of τ P coincides with that of τ P ′ for any P ′ obtained from P by a shift of the base B. We use the same notation for the extension τ P : X ω → X ω and call it a lifting isometry.
Since the group of vertical shifts is commutative, we have τ P • τ P ′ = τ P ′ • τ P for any (pointed oriented) parallelograms and even for any closed oriented polygons P , P ′ ⊂ B.
Let Q ⊂ B be a closed, oriented polygon. Adding a segment qq ′ ⊂ B between points q, q ′ ∈ Q we obtain closed, oriented polygons P , P ′ such that Q ∪ qq ′ = P ∪ P ′ , the orientations of P , P ′ coincide with that of Q along Q, and the segment qq ′ = P ∩ P ′ receives from P , P ′ opposite orientations. In this case we use notation Q = P ∪ P ′ .
Lemma 7.8. In the notation above we have τ Q = τ P ′ • τ P .
Proof. We fix q ∈ Q ∩ P ∩ P ′ as the base point. Moving from q along Q in the direction prescribed by the orientation of Q, we also move along one of P , P ′ according to the induced orientation. We assume without loss of generality that this is the polygon P . In that way, we first lift P to X ω starting with some point o ∈ F , where F is the fiber of the projection π : X ω → B over q, such that the side q ′ q ⊂ P is the last one while lifting P . Now, we lift P ′ to X ω starting with o ′ = τ P (o) ∈ F moving first along the side qq ′ ⊂ P ′ . Then clearly the resulting lift of P ′ gives τ Q For a vertical shift γ : X ω → X ω we denote |γ| = |xγ(x)| ω the displacement of γ. This is independent of x ∈ X ω . If γ, γ ′ are vertical shifts in the same direction, then |γ • γ ′ | 2 = |γ| 2 + |γ ′ | 2 .
Lemma 7.9. Given a representation P = T ∪T ′ of an oriented parallelogram P ⊂ B by oriented triangles T , T ′ with induced from P orientations, whose common side qq ′ is a diagonal of P , we have τ T = τ T ′ .
Proof. For every n ∈ N we consider the subdivision P = ∪ a∈A P a of P into |A| = n 2 congruent parallelograms P a with sides parallel to those of P . We assume that the orientation of each P a is induced by that of P . The parallelograms P a , P a ′ are obtained from each other by a shift of the base B, thus τ a = τ a ′ for each a, a ′ ∈ A, where τ a = τ Pa . By Lemma 7.8 we have a∈A τ a = τ P , hence |τ P | 2 = n 2 |τ a | 2 for every a ∈ A. Therefore |τ a | 2 = |τ P | 2 /n 2 → 0 as n → ∞. We subdivide the set A into three disjoint subsets A = C ∪ C ′ ∪ D, where a ∈ D if and only if the interior of P a intersects the diagonal qq ′ of P , and a ∈ C if P a ⊂ T , a ∈ C ′ if P a ⊂ T ′ . Then |C| = |C ′ | = n(n−1) 2 , |D| = n. We conclude that a∈C τ a = a∈C ′ τ a and a∈D τ a 2 = n|τ P | 2 /n 2 → 0 as n → ∞. By Lemma 7.8, τ P = τ T • τ T ′ . It follows that The following Lemma will be used in sect. 10.7, see the proof of Proposition 10.22.
Lemma 7.10. Let T = vyz ⊂ B be an oriented triangle. Then for the triangle P = xyz with x ∈ [vz] we have Proof. Arguing as in Lemma 7.8 for the oriented parallelogram Q = vyzw (for which vz is a diagonal) we find that τ P = τ P ′ for P = xyz, P ′ = vyx in the case x is the midpoint of [vz]. Therefore |τ P | 2 = 1 2 |τ T | 2 = |xz| |vz| |τ T | 2 in this case. Next we obtain by induction the required formula for the case |xz|/|vz| is a dyadic number. Then the general case follows by continuity.
Lemma 7.11. Let P ⊂ B be an oriented parallelogram, λP ⊂ B the parallelogram obtained from P by a homothety h : Proof. We can assume that the parallelograms P , λP have o as a common vertex, and that λ is rational. For a general λ one needs to use approximation. We choose n ∈ N such that λn ∈ N and subdivide P into n 2 congruent parallelograms, P = ∪ a∈A P a , |A| = n 2 , as in Lemma 7.9. This also gives a subdivision of λP into λ 2 n 2 parallelograms congruent to ones of the first subdivision, λP = ∪ a∈A ′ P a , |A ′ | = λ 2 n 2 . Then τ P = a∈A τ a , |τ a | 2 = |τ P | 2 /n 2 and |τ λP | 2 = λ 2 n 2 |τ a | 2 = λ 2 |τ P | 2 .

Reflections with respect to C-circles
In sect. 4 we have defined for every C-circle F ⊂ X the reflection ϕ F : X → X whose fixed point set is F and v = ϕ F (u) is conjugate to u pole of F for every u ∈ X \ F . Proposition 7.12. For every C-circle F ⊂ X the reflection ϕ F : X → X is Möbius.
Proof. We fix ω ∈ F and consider F as a C-line in X ω . By definition, ϕ F : X ω → X ω preserves every R-line σ ⊂ X ω intersecting F and acts on σ as the reflection with respect to σ ∩ F , in particular, ϕ F |σ is isometric. It follows from Lemma 6.4 that ϕ F is isometric on every C-line in X ω , thus ϕ F induces the central symmetry ϕ = ϕ F : where "bar" means the projection by π ω .
The triangles o x ′ z and o y ′ z ′ in B are symmetric to each other by ϕ and have the same orientation. It follows from Lemma 7.9 that any lift of the closed polygon P = x ′ y ′ z ′ z x ′ ⊂ B closes up in X ω , that is, τ P = id. Hence µ Fy,ω (z) = z ′ and therefore |xz| = |yz ′ |. We conclude that |yy ′ | = |xx ′ |.

Pure homotheties
For an R-line σ ⊂ X ω we define the semi-C-plane R = R σ ⊂ X ω as R = π −1 ω (π ω (σ)). Note that R has two foliations: one by C-lines and another by R-lines.
Given a C-line F ⊂ X ω and an R-line σ ⊂ X ω with o = F ∩ σ, let R ⊂ X ω be the semi-C-plane spanned by F and σ. Every point x ∈ R is uniquely determined by its projections the vertical to F , x F = µ F,ω (x), and the horizontal to σ, For o ∈ X ω and λ > 0 we define a map h = h o,λ : X ω → X ω as follows. We put h(o) = o and require that h preserves the C-line F through o and every R-line σ through o acting on F and σ as the homotheties with coefficient λ. Finally, h preserves every semi-C-plane R containing F and acts on R by h(x F , x σ ) = (h(x F ), h(x σ )), where R is spanned by F and the R-line σ through o.
Proof. It follows from definition and Proposition 5.2 that the restriction h|R is the required homothety for every semi-C-plane R ⊂ X ω containing the C-line F through o. Thus the induced map h : B ω → B ω is the homothety with coefficient λ.
Let F x be the C-line through x, σ the R-line through o that intersect F x . Similarly, let F y be the C-line through y, γ the R-line through o that intersect F y . We denote z = µ Fx,ω (y) ∈ F x the projection of y to F x . Then by Proposition 5.2 we have |xy| 4 = |xz| 4 + |zy| 4 .
For the closed polygon P = o y z o ⊂ B ω , where "bar" means the projection by π ω , we have τ P (y F ) = z F , where y = (y F , y γ ), z = (z F , z σ ) are vertical and horizontal coordinates in respective semi-C-planes.
By definition, the homothety h : X ω → X ω , h(o) = o, preserves every R-line σ ⊂ X ω through o. Every homothety with this property is said to be pure.

Definition and properties
Let F ⊂ X be a C-circle. Every u ∈ X \ F determines an involution η u : F → F without fixed points, which is Möbius by Corollary 4.9. In other words, we have a map F : X \ F → J, u → η u , where J = J F is the set of Möbius involutions of F without fixed points. We study fibers of this map, F −1 (η) =: (F, η) ⊥ . The set (F, η) ⊥ = {u ∈ X \ F : η u = η} is called the orthogonal complement to F at η.
Given distinct x, y ∈ F let S x,y ⊂ X be the set covered by all R-circles in X through x, y. By Lemma 4.7, this set can be described as the sphere in X between o, ω ∈ F such that (x, o, y, ω) ∈ Harm F . Then for every u ∈ S x,y \ {x, y} we have η u (x) = y. Thus Lemma 8.1. Given a C-circle F ⊂ X and a Möbius involution η : F → F without fixed points, the orthogonal complement A = (F, η) ⊥ to F at η can be represented as Proof. We have A ⊂ S x,y ∩ S o,ω by Eq.
(2). On the other hand, for every u ∈ S x,y ∩ S o,ω we have η u = η along the 4-tuple (x, o, y, ω). Thus η u = η because any Möbius η : F → F is uniquely determined by values at three distinct points. Hence u ∈ A.
where S, S ′ ⊂ X are spheres between x, y and o, ω respectively that contain u, v. Since (x, o, y, ω) ∈ Harm F , by axiom O R we have γ ⊂ S and σ ⊂ S ′ , where S, S ′ are spheres between x, y through o, ω and between o, ω through x, y respectively. Hence S = S, S ′ = S ′ . We conclude that S = S o,ω , S ′ = S x,y . Then S ∩ S ′ = (F, η) ⊥ by Lemma 8.1, hence F ′ ⊂ (F, η) ⊥ . Lemma 8.3. Given a C-circle F ⊂ X and a Möbius involution η : F → F without fixed points, the orthogonal complement A = (F, η) ⊥ is foliated by C-circles F ′ through pairs u, v ∈ A of conjugate poles of F , and ϕ F : A → A preserves every fiber of this fibration.
Proof. By Lemma 8.2, A is covered by C-circles F ′ . We have ϕ F (F ′ ) = F ′ because ϕ F permutes conjugate poles of F . Thus by axiom E C distinct Ccircles of the covering are disjoint, that is, the C-circles F ′ form a fibration of A.
The fibration of (F, η) ⊥ by C-circles described in Lemma 8.3 is called canonical. We do not claim that every C-circle in (F, η) ⊥ is a fiber of the canonical fibration, this is actually not true in general.
Lemma 8.4. Given a Möbius involution η : F → F without fixed points of a C-circle F , for every fiber F ′ of the canonical fibration of A = (F, η) ⊥ the reflection ϕ ′ = ϕ F ′ : X → X preserves A and its canonical fibration.

Mutually orthogonal C-circles
We say that distinct C-circles F , F ′ ⊂ X are mutually orthogonal to each other, F ⊥ F ′ , if ϕ F (F ′ ) = F ′ and ϕ F ′ (F ) = F . Note that then F , F ′ are disjoint because by Lemma 3.3 a C-circle and an R-circle have in common at most two points.
We also note that if F ⊥ F ′ , then ϕ F acts on F ′ as a Möbius involution without fixed points.
Proof. Since F is the fixed point set for ϕ, we have ϕ • ϕ ′ = ϕ ′ • ϕ along F . Thus to prove the equality ϕ • ϕ ′ (x) = ϕ ′ • ϕ(x) for an arbitrary x ∈ X, we can assume that x ∈ F .

Intersection of orthogonal complements
Proof. We let dim X = k + 1 with k ≥ 0. Then dim A = dim A ′ = k − 1 for A = (F, η) ⊥ , A ′ = (F ′ , η ′ ) ⊥ . By Lemma 8.3, A is foliated by C-circles, thus k is even. If k = 0, then X = ∂ ∞ M for M = C H 1 . We can assume that k ≥ 2. Note that the codimension of A equals two and that F is not contractible in X \ A. Thus the assumption A ∩ A ′ = ∅ implies by transversality argument that 2(k − 1) < k + 1. Hence k = 2 and dim X = 3.

An induction argument
Lemma 8.12. Assume a subset A ⊂ X foliated by C-circles is the fixed point set of a Möbius ψ : X → X. Then A satisfies axioms (E), (O).
Proof. We only need to check the existence axioms (E). Given distinct a, a ′ ∈ A, there is a unique C-circle F ⊂ X through a, a ′ . Since ψ(a) = a, ψ(a ′ ) = a ′ , we have ψ(F ) = F . Then in the space X a , the Möbius ψ : X a → X a acts as a homothety preserving a ′ . On the other hand, X a is foliated by C-lines, one of which, F ′ , lies by the assumption in Fix ψ. Hence, ψ : X a → X a is an isometry pointwise preserving F ′ and a ′ . This excludes a possibility that ψ acts on F a as the reflection at a ′ . Therefore, F ⊂ Fix ψ = A, which is axiom (E C ).
Given a C-circle F ⊂ A, ω ∈ F and u ∈ A \ F , there is a unique R-circle σ ⊂ X through ω, u that hits F ω . Thus at least three distinct points of σ lies in Fix ψ. We conclude that ψ pointwise preserves σ, i.e. σ ⊂ A, which is axiom (E R ).
Then the intersection A ∩ A ′ satisfies axioms (E) and (O).
Proof. Apply Lemma 8.12 to A ∩ A ′ which is by Proposition 8.11 the fixed point set of the Möbius ψ = ϕ ′ • ϕ foliated by C-circles. 9 Möbius join 9.1 Canonical subspaces orthogonal to a C-circle Let (F, η) be a C-circle in X with a Möbius involution η : F → F without fixed points. Let F ′ ⊂ (F, η) ⊥ be a nonempty subspace that satisfies axioms (E), (O), and is invariant under the reflection ϕ F : X → X, ϕ F (F ′ ) = F ′ . In this case we say that F ′ is a canonical subspace orthogonal to F at η, COS for brevity. Note that F ′ carries a canonical fibration by C-circles induced by ϕ F , where the fiber through x ∈ F ′ is the uniquely determined C-circle through x, ϕ F (x). We use notation F = F F ′ for this fibration, and H ∈ F means that H is a fiber of F.
Proof. By definition of (F, η) ⊥ , there is a uniquely determined R-circle σ ⊂ X through u, x that hits F at v = η(u). Then there is a unique y = ϕ F (x) ∈ σ such that (u, x, v, y) ∈ Harm σ . Again, y ∈ F ′ by definition of F ′ .
We define the (Möbius) join F * F ′ as the union of R-circles σ ⊂ X such that (u, x, v, y Assume o ′ ∈ {o, ω}. Without loss of generality, o ′ = o. Then ω ′ = ω and σ ∩ σ ′ = {o, ω} since otherwise σ = σ ′ by Lemma 3.4. Thus we can assume that o ′ = o, ω. We take v ′ = ω and u ′ ∈ F such that (u ′ , o ′ , v ′ , ω ′ ) ∈ Harm F . In the space X ω , σ is an R-line through o, and F is a C-line through o, while S ′ is the metric sphere centered at the midpoint u ′ ∈ F between o ′ , ω ′ , |u ′ o ′ | = |u ′ ω ′ | =: r. The distance function d u ′ : σ → R, d u ′ (x) = |u ′ x|, is convex and by Corollary 3.2(i) it is symmetric with respect to o. Thus d u ′ takes the value r at most two times, that is, σ intersects S ′ at most two times (actually exactly two times because the pair (o, ω) separates the pair (o ′ , ω ′ ) on F by properties of η).
On the other hand, σ intersects F ′ twice and Proof. We show that every point u ∈ X lies on a standard R-circle in F * F ′ . Given ω ∈ F , we put o = η(ω) ∈ F and use notation |xy| = |xy| ω for the distance between x, y in X ω . Then F is a C-line in X ω , and every standard Let w ∈ F be the midpoint between x, y, |xw| = |wy| = r. By Lemma 4.7 the sphere S r (w) ⊂ X ω centered at w of radius r is covered by R-circles through x, y. Thus it suffices to show that for every u ∈ X \ F there is x ∈ F such that u ∈ S r (w), where w ∈ F is the midpoint between x, y = η(x), r = |xw| = |wy|.
This shows that F * F ′ = X.

Möbius join equivalence
Theorem 10.1. Let (F, η) be a C-circle in X with a fixed point free Möbius involution η : F → F , F ′ ⊂ X a canonical subspace orthogonal to F at η. Assume that F ′ is Möbius equivalent to ∂ ∞ C H k , k ≥ 1, taken with the canonical Möbius structure. Then the join F * F ′ is Möbius equivalent to ∂ ∞ C H k+1 .
We start the proof with Lemma 10.2. Let F , G be Möbius spaces which are equivalent to Y = ∂ ∞ C H k , k ≥ 1, taken with the canonical Möbius structure. Given Möbius involutions η : F → F , η ′ : G → G without fixed points, there is a Möbius equivalence g : F → G that is equivariant with respect to η, η ′ , Proof. If we identify F with Y , then there is a ∈ M = C H k such that the central symmetry s a : Lemma 10.3. Let E ⊂ M = C H k+1 be a complex hyperbolic plane and a ∈ E. Let E ′ ⊂ M be the orthogonal complement to E at a. Denote by Proof. We use the fact that every Möbius Y → Y is induced by an isometry M → M . Recall that G is the fixed point set of the reflection ϕ G : Y → Y . Then E is the fixed point set for the isometry ζ : M → M with ∂ ∞ ζ = ϕ G , and ζ acts on E ′ as s a does, ζ|E ′ = s a |E ′ . Hence ϕ G |G ′ = ψ|G ′ .

Constructing a map between Möbius joins
We fix a complex hyperbolic plane E ⊂ M = C H k+1 and a ∈ E. The orthogonal complement Recall that the boundary at infinity Y = ∂ ∞ M taken with the canonical Möbius structure satisfies axioms (E), (O) (see Proposition 2.3). Thus all the notions involved in the definition of the Möbius join G * G ′ are well defined for Y .
The central symmetry s a : M → M induces the Möbius involution ψ = ∂ ∞ s a : Y → Y without fixed points, for which G, G ′ are invariant, ψ(G) = G, ψ(G ′ ) = G ′ . Furthermore, G ′ ⊂ Y is a COS to G at ψ and dim Y = dim G ′ + 2. Then by Proposition 9.3, Y = G * G ′ . By Lemma 10.2, there is a Möbius equivalence g : G → F , which is equivariant with respect to ψ and η, g • ψ|G = η • g. Note that ϕ F : X → X acts on F ′ as a Möbius involution without fixed points. By the assumption, F ′ ⊂ X is Möbius equivalent to G ′ ⊂ Y . By Lemma 10.2 again, there is a Möbius equivalence g ′ : G ′ → F ′ which is equivariant with respect to ψ and Then ψ ′ uniquely extends to a Möbius σ → σ, for which we use the same notation ψ ′ . By Lemma 9.2, different standard R-circles in F * F ′ may have common points only in F ∪ F ′ . Thus we have a well defined involution ψ ′ : F * F ′ → F * F ′ without fixed points, which is Möbius along F , F ′ and every standard R-circle in F * F ′ .
For every standard R-circle σ ⊂ G * G ′ , we have the map f : σ∩(G∪G ′ ) → F * F ′ ⊂ X, which is equivariant with respect to ψ and ψ ′ . The map f uniquely extends to a Möbius f : σ → F * F ′ . By Lemma 9.2 different standard R-circles in G * G ′ , F * F ′ may have common points only in G ∪ G ′ , F ∪F ′ respectively, thus this gives a well defined bijection f : G * G ′ → F * F ′ which is Möbius along G, G ′ and any standard R-circle in Y . Moreover, f is equivariant with respect to ψ, ψ ′ .
It suffices to check that f : For brevity, we use notation f ω := f : Y ω → (F * F ′ ) ω ′ regarding f as a map between respective metric spaces.
During the proof we will also consider the maps f u : Y u → X u ′ , where u ′ = f (u). We view f u as a map between metric spaces, where on Y u the metric is defined as the metric inversion (1) of | | ω and on X u ′ the metric inversion of | | ω ′ .
Note that then f |G u is isometric, if u ∈ G.

Isometricity along standard objects
Lemma 10.4. The map f ω is isometric on every standard R-line in Y ω through o.
Proof. Recall that Y is a compact Ptolemy space with axioms (E), (O). Thus G ′ lies in a sphere between o, ω, that is, in a metric sphere in Y ω centered at o, while F ′ lies in a metric sphere in X ω ′ centered at o ′ , see sect. 8. The respective radii are called the radii of G ′ , F ′ in Y ω , X ω ′ respectively.
There is u ∈ G such that (u, o, v, ω) ∈ Harm G for v = ψ(u). Any standard R-circle σ ⊂ Y through u, v lies in a metric sphere in Y ω centered at o. Then σ ′ = f (σ) lies in a metric sphere in X ω ′ centered at o ′ = f (o). Since f |G preserves distances, the metric spheres in Y ω , X ω ′ centered at o, o ′ containing σ, σ ′ respectively have equal radii. Hence the radius of G ′ in Y ω is the same as the radius of , we observe that f |σ is an isometry, that is, f ω preserves distances along any standard R-line through o.
Lemma 10.5. Assume for u ∈ G the map f u is isometric along R-lines (not necessarily standard) through some v ∈ G u and preserves the distance between some x, y on those lines, Proof. We can assume that u = ω since otherwise there is nothing to prove.
Corollary 10.6. The map f ω is isometric along every standard R-circle σ ⊂ Y .
Proof. We let {u, v} = σ∩G. Then σ is an R-line in Y u , hence by Lemma 10.4, f u is isometric along σ. By Lemma 10.5, f ω is isometric along σ.
Lemma 10.7. For every u ∈ G ′ the map f u , is isometric on G.
Lemma 10.8. The map f ω is isometric on G ′ .

R-and COS-foliations of a suspension
Let H ⊂ Y be a COS to a C-circle K ⊂ Y . For every u ∈ K, x ∈ H there is a uniquely determined R-circle σ ⊂ Y through u, x that meets K at another point v. We denote by S u H = S u,v H the set covered by R-circles in Y through u, v which meet H. Note that σ ∩ H = {x, y}, y = ϕ K (x), for every such an R-circle σ. Topologically, S u H is a suspension over H. The points u, v are called the poles of S u H. The set S u H \ {u, v} is foliated by R-circles through u, v. Slightly abusing terminology, this foliation is called the R-foliation of S u H. In the metric of Y u , the circles of the R-foliation are R-lines through v, and K is a C-line.
Let h = h λ : Y u → Y u be a pure homothety (see sect. 7.5) with coefficient λ > 0 centered at v, h(v) = v. The homothety h is induced by an isometry γ : M → M , which is a transvection along the geodesic uv ⊂ M , h = ∂ ∞ γ. Then h preserves every R-line σ ⊂ Y u through v, acting on σ as the homothety with coefficient λ, which preserves orientations of σ. The image H λ = h(H) ⊂ Y u is the boundary at infinity of E λ = γ( E), where E ⊂ M is the subspace with ∂ ∞ E = H, and it is also a COS to K. In that way, the COS's H λ form a foliation of S u H \ {u, v}, which is called the COS-foliation of S u H. In the case H is a C-circle, we say about C-foliation of S u H.
Lemma 10.9. Assume f u is isometric along every R-line σ of the Rfoliation of S u H and along H. Then f u is isometric along every COS H λ of the C-foliation.
Proof. Note that f u |σ is equivariant with respect to h, h ′ , where h ′ = h ′ λ : X u ′ → X u ′ is the pure homothety (see sect. 7.5) with coefficient λ with the fixed point v ′ = f (v). It follows that f |H λ : Lemma 10.10. Let COS H to K be a C-circle. Under the assumption of Lemma 10.9, the map f u is isometric along the suspension S u H.
Proof. By the assumption, f u is isometric along every R-line in S u H through v. Thus taking x, y ∈ S u H we can assume that neither x nor y coincides with u or v. Then there are uniquely determined an R-line σ ⊂ S u H of the R-foliation through x and a C-circle H λ ⊂ S u H of the C-foliation through y. The intersection σ ∩ H λ consists of two points, say, z, w. Taking the metric inversion with respect to w we obtain |xy| w = |xy| u |xw| u · |yw| u and similar expressions for |xz| w , |yz| w . In the space X we have respectively By the assumption and Lemma 10.9, f u is isometric along σ and H λ .
Lemma 10.11. For every u ∈ G, every v ∈ G ′ , the map f u is isometric along the suspension Proof. Every circle σ of the R-foliation of S v G is standard in Y , and σ intersects G at two points p, q = ψ(p). Then by Lemma 10.4 applied to ω = p, f p is isometric along σ. Thus f v is isometric along σ because v ∈ σ. By Lemma 10.7, f v is isometric along G. Using Lemmas 10.9 and 10.10, we obtain that f v is isometric along S v G, in particular, f is Möbius along (v, x, z, u), where "prime" means the image under f . On the other hand, crt(v, x, z, u) = (|vx| u : |vz| u : |xz| u ), and |v ′ x ′ | u ′ = |vx| u by Corollary 10.6 because v, x lie on a standard R-circle. Therefore |x ′ z ′ | u ′ = |xz| u .

Isometricity along fibers of the R-foliation
Lemma 10.12. The polynomial g(s) = s 4 + c(s + b) 4 − d with positive b, c such that cb 4 − d < 0 has a unique positive root.
Proposition 10.13. Let S u H ⊂ Y be a suspension over a C-circle H with the poles u, v = ϕ H (u), and let K be the C-circle through u, v. Assume f u is isometric along K and along every C-circle H λ of the C-foliation of S u H, and the following assumptions hold where "prime" means the image under f . Then f u is isometric along every R-line σ ⊂ S u H of the R-foliation.
We let H t ⊂ S u H be the C-circle of the C-foliation with σ(t) ∈ H t , and note that x ∈ H t ′ , z, p ∈ H t .
By assumption (ii), there is y ∈ H t such that |x ′ y ′ | u ′ = |xy| u .
Remark 10.14. Since f u is isometric along H t , the triangle xyz has two sides |xy| u , |yz| u preserved by f u . Our aim is to show that the third distance |xz| u is also preserved by f u . In the space Y p the distance |xz| p is uniquely determined by |xy| p , |yz| p , |xz| 4 p + |yz| 4 p = |xy| 4 p by Proposition 5.2. We can pass to Y u by a metric inversion. The change of metrics involves the distances |yp| u , |zp| u which are preserved by f u , and the distance |xp| u = |xz| u +|zp| u . In that way, we obtain an equation of 4th degree for |xz| u to which we apply Lemma 10.12.
Recall that G ′ ⊂ Y is a COS to G at ψ, and G ′ carries the canonical fibration F G ′ by C-circles, where the C-circle H ⊂ G ′ of the fibration through For distinct u, v ∈ G the sphere S u,v ⊂ Y formed by all R-circles in Y through u, v is foliated (except u, v) by COS's to G. One can visualize this picture by taking the geodesic γ = uv ⊂ E and regarding the orthogonal projection pr E : In the case v = ψ(u), i.e., the geodesic uv passes through a, the R-circles through u, v are standard, and the sphere S u,v is called standard. Every standard sphere is a suspension over G ′ . If S u,v is not standard, then for every b ∈ uv ⊂ E the fiber G b is the intersection G b = S u,v ∩ S w for the uniquely determined standard sphere S w = S w,ψ(w) , w ∈ G: one takes the geodesic γ b ⊂ E through a, b, then S w = ∂ ∞ pr −1 E (γ b ), and w, ψ(w) ∈ G are the ends at infinity of γ b .
Every COS G b to G, b ∈ uv, carries the canonical fibration Note that the C-circles G, H are mutually orthogonal, G⊥H, Proof. Let uv ⊂ E be the geodesic with end points at infinity u, v. By the assumption, a ∈ uv. We take b ∈ uv, consider the geodesic γ b ⊂ E through a, b, put {x, y} = ∂ ∞ γ b ⊂ G, and let S x H = S x,y H be the Möbius suspension over H with poles x, y. The suspension S x H intersects the COS For an arbitrary c ∈ uv let γ c ⊂ E be the geodesic through a, c, {w, z} = ∂ ∞ γ c ⊂ G, and let S w H be the suspension over H with poles w, z.
Every transvection along a geodesic γ ⊂ M induces a Möbius automorphism of Y , which is a pure homothety of Y u for any end u ∈ ∂ ∞ γ. For brevity, we speak about the action of transvections on Y .
The fiber H c = S u H ∩G c of the C-foliation of S u H over c is obtained from H by the transvection along uv, which moves b into c, and H is obtained from H by the transvection along γ b , which moves a into b. Thus H c is obtained from H by the composition of these transvections.
On the other hand, by Proposition 2.2, the holonomy of the normal bundle E ⊥ along any loop in E preserves any complex hyperbolic plane in E ⊥ . It follows that H c is obtained from H by the transvection along γ c that moves a into c. Therefore, H c = S u H ∩ S w H.
Lemma 10.17. For every distinct u, v ∈ G the map f ω is isometric along any suspension S u H = S u,v H, v = ϕ H (u), over a C-circle H that is orthogonal to G.
Proof. As above we use a "prime" notation for images under f , and the notation We first note that for every u ∈ G, the map f ω is isometric along any standard suspension S u H = S u,v H, where H ⊂ G ′ is a C-circle of the canonical fibration and v = ψ(u) = ϕ H (u). Indeed, the R-foliation of S u H consists of standard R-circles, and by Lemma 10.4, f u is isometric along every of them. By Lemma 10.8, f u is isometric on H. Thus by Lemma 10.10, f u is isometric on S u H. Then by Lemma 10.5, f ω is isometric on S u H. Now, we show that f ω is isometric along every suspension S u H = S u,v H, u ∈ G, where v = ϕ H (u) = ψ(u). We check that the assumptions of Proposition 10.13 are satisfied. In this case, K = G, and f u is isometric along G because f ω is assumed to be isometric along G and u ∈ G. By the first part of the argument, f u is isometric on every C-circle of the Cfoliation of any standard suspension S w H, w ∈ G. Thus by Lemma 10.16, f u is isometric on every C-circle of the C-foliation of S u H. Property (i) follows from Lemma 10.15.
To check property (ii) of Proposition 10.13, we take x ∈ S u H and a Ccircle H λ of the C-foliation of S u H. There is a C-circle H λ ′ of the C-foliation with x ∈ H λ ′ . By Lemma 10.16, H λ ′ = S u H ∩ S w ′ H for some w ′ ∈ G, and there is an R-circle σ ′ ⊂ S w ′ H of the R-foliation with x ∈ σ ′ . The suspension S w ′ H is standard, and the intersection σ ′ ∩ H consists of two points. We take one of them, q ∈ σ ′ ∩ H. Again, H λ = S u H ∩ S w H for some w ∈ G, and there is a (standard) R-circle σ ⊂ S w H of the R-foliation with q ∈ σ. The intersection σ ∩ H λ consists of two points, and we take one of them, y ∈ σ ∩ H λ . Note that σ, σ ′ are standard R-circles with q ∈ σ ∩ σ ′ . Thus the suspension S q G over G with poles q, ϕ G (q) contains x, y. By Lemma 10.11, f u is isometric on S q G, hence |x ′ y ′ | u ′ = |xy| u . That is, the assumptions of Proposition 10.13 are satisfied. By that Proposition f u is isometric along every R-circle of the R-foliation of S u H. Applying Lemma 10.10, we see that f u is isometric along S u H. Then by Lemma 10.5, f ω is isometric along S u H.
Proposition 10.18. For every C-circle H ∈ F G ′ the map f ω is isometric along the Möbius join G * H ⊂ Y .
Proof. Given x, y ∈ G * H, we show that there is a suspension S u H = S u,v H, u, v = ϕ H (u) ∈ G with x, y ∈ S u H.
There are standard suspensions S p H, S q H ⊂ G * H, p, q ∈ G, over H with x ∈ S p H, y ∈ S q H. We can assume that x = pr E (x), y = pr E (y) ∈ E are distinct since otherwise we can take S p H = S q H as the required suspension. Then there is a unique geodesic γ ⊂ E through x, y. Let u, v ∈ G be the ends at infinity of γ. We can assume that v = ψ(u), since otherwise again S p H = S q H. Then by Lemma 10.16, there is a uniquely determined suspension S u H such that every fiber H c of the C-foliation of S u H is represented as H c = S u H ∩ S w H for some w ∈ G. Then x ∈ S u H ∩ S p H, y ∈ S u H ∩ S q H, in particular, x, y ∈ S u H.
Remark 10.19. Proposition 10.18 implies Theorem 10.1 and Theorem 1.1 in the case dim X = 3 because Y = G * H by Proposition 9.3 in that case. Moreover, it implies Theorem 10.1 in the case F , F ′ are mutually orthogonal C-circles in X.

Properties of the base revisited
Here we make a digression to prove an important fact, Proposition 10.21. Let M = C H 2 , Y = ∂ ∞ M . We fix o ∈ M and consider a complex reflection h : M → M with respect to a complex hyperbolic plane E ⊂ M through o. It acts on Y as a reflection with respect to the C-circle F = ∂ ∞ E ⊂ Y . The tangent space V = T o M is a 2-dimension complex vector space with a hermitian form generated by the Riemannian metric, and g = dh : V → V is a unitary involution whose fixed point set is the complex line T o E ⊂ V . We formalize this situation as follows.
Let V = C 2 be the 2-dimensional complex vector space with the standard hermitian form. A unitary involution g : V → V is said to be reflection if dim C Im(1−g) = 1. Note that det g = −1. Thus any product g ·g ′ of unitary involutive reflections lies in SU (2). We need the following simple Lemma 10.20. Let F ⊂ U (2) be the set of unitary involutive reflections V → V . Then the subgroup G ⊂ SU (2) generated by products g · g ′ with g, g ′ ∈ F is transitive on the unit sphere S ⊂ V , and for g ∈ F there is a dense subset F ⊂ F such that g ′ · g has a finite order for every g ′ ∈ F.
Proof. We identify F with the set of complex lines in V and fix g ∈ F. Then we have a decomposition V = C ⊕ C such that C ⊕ {0} = Fix g is the fixed point set of g, and g acts on V as g(a, b) = (a, −b). Note that g preserves any real 2-subspace L ⊂ V spanned by (a, 0), (0, b) with a, b ∈ C, |a| = |b| = 1, acting on L as the reflection with respect to L ∩ Fix g.
Each pair of opposite points in L ∩ S is represented as L ∩ g ′ ∩ S for some g ′ ∈ F, g ′ · g acts on L by a rotation, and the subgroup in G preserving L acts transitively on L ∩ S. If (g ′ · g) k is identical on L, then it is identical, in Z. Then ϕ K (K ′ ) = K ′ , and since ϕ K acts on K identically, we see that ϕ K : X → X preserves Z, ϕ K (Z) = Z, because Z can be represented as the Möbius join Z = K * K ′ . Using conjugation via a Möbius isomorphism Y → Z and the fact that Y = ∂ ∞ C H 2 , we can consider ϕ K as a unitary involutive reflection of V = C 2 . Applying Lemma 10.20, we find a dense subset F Z ⊂ F Z such that the composition g = ϕ K • ϕ H , when restricted to Z, has a finite order, (g|Z) k = id Z , for every C-circle K ∈ F Z and some k ∈ N depending on K. Note that g|F : F → F being a composition of two Möbius involutions ϕ K |F , ϕ H |F without fixed points has an infinite order or is identical. In the first case there are two fixed points u, v ∈ F for g|F , and g acts on X u as a nontrivial homothety. This contradicts the fact that (g|Z) k = id Z . Therefore, g|F = id F and thus ϕ K |F = ϕ H |F = η. Consequently, K ⊂ A. By continuity, this holds for every K ∈ F Z , hence Z ⊂ A.
For every K, K ′ ∈ F Z the product g = ϕ K • ϕ K ′ : X → X is identical on F , hence g : X ω → X ω is an isometry. By Lemma 10.20 isometries of this type act on Z transitively. Varying H ∈ F, H ′ ∈ (H, ϕ F |H) ⊥ ∩ A, we see that the Möbius joins Z = H * H ′ cover A. Thus the group G acts on A transitively. It follows that the group of isometries of the base B preserving o acts on the unit sphere A ⊂ B centered at o transitively. Using the standard argument with Lövner ellipsoid, we obtain that the unit sphere A is an ellipsoid and thus the metric of B is Euclidean.

Suspension over a COS
We turn back to the proof of Theorem 10.1 and to notations of sect. 10.1.
Given distinct u, v ∈ Y , let K ⊂ Y be the C-circle through u, v. The sphere S u,v ⊂ Y formed by all R-circles in Y through u, v carries R-and COS-foliations, see sect. 10.3. Note that every fiber H of the COS-foliation of S u,v satisfies axioms (E) and (O) because H is the boundary at infinity of an orthogonal complement in M = C H k+1 to the complex hyperbolic plane E ⊂ M with ∂ ∞ E = K. Now we are able to prove the following generalization of Lemma 10.10.
Proposition 10.22. Given distinct u, v ∈ Y assume f u is isometric along every R-circle of the R-foliation of S = S u,v and every fiber of the COSfoliation of S. Then f u is isometric along S.
Proof. We take x, y ∈ S and show that |x ′ y ′ | u ′ = |xy| u , where we use "prime" notation for images under f . Every fiber H of the COS-foliation of S lies in a sphere between u, v, that is, there is r = r(H) > 0 such that |vh| u = r for every h ∈ H. In the space Y u every R-circle σ of the R-foliation of S is an R-line through v. Since f u is isometric along σ, we can assume that neither x nor y coincides with u or v. Then there are uniquely determined an R-line σ ⊂ S of the Using Lemma 7.10 we obtain for the oriented triangles P = x y z ⊂ B w , P ′ = x ′ y ′ z ′ ⊂ B w ′ . We assume that a fiber orientation of X w → B w is fixed and that the fiber orientation of X w ′ → B w ′ is induced by f w . Then f ( x) = f • τ P ( z) = τ P ′ • f ( z) and we conclude that the projections µ K,w , µ K ′ ,w ′ commute with f at x, f ( x) = x ′ for x ′ = µ K ′ ,w ′ (x ′ ). Hence |x ′ y ′ | 4 w ′ = |x ′ y ′ | 4 + |y ′ x ′ | 4 w ′ = |x y| 4 + |y x| 4 w = |xy| 4 w by Proposition 5.2. Applying the metric inversion with respect to u and using that |u ′ x ′ | w ′ = |ux| w , |u ′ y ′ | w ′ = |uy| w , we finally obtain |x ′ y ′ | u ′ = |xy| u .
10.8 Proof of Theorems 10.1 and 1.1 Proof of Theorem 10.1. We show that the map f ω = f : Y ω → (F * F ′ ) ω ′ is isometric. First, we note that for every u ∈ G the map f u is isometric along the standard sphere S u = S u,v , v = ψ(u). Indeed, S u = S u G ′ is a suspension over G ′ , and f u is isometric along the fibers of the R-foliation of S u G ′ because they are standard R-lines in Y u , and along G ′ by Lemma 10.8. Thus by Lemma 10.9, f u is isometric along every fiber of the COS-foliation of S u G ′ . By Proposition 10.22, f u is isometric along S u = S u G ′ . It follows from Lemma 10.5 that f ω is isometric along S u . Given x, y ∈ Y we can assume that the points x = pr E (x), y = pr E (y) ∈ E are distinct and neither of them coincides with o, since otherwise x, y lie in a standard sphere S u for some u ∈ G, hence |x ′ y ′ | ω ′ = |xy| ω . Then there is a unique geodesic γ ⊂ E through x, y. Let u, v ∈ G be the ends at infinity of γ, v = ψ(u) by our assumption. Then x, y ∈ S u,v = pr −1 E (γ). The map f u is isometric along the fibers of the COS-foliation of the sphere S u,v because every such a fiber is represented as the fiber S u,v ∩ S w of the COS-foliation of a standard sphere S w for some w ∈ G. Furthermore, S u,v = S u,v G is a suspension over a fiber G of the COS-foliation of S u,v , and every fiber σ of the R-foliation of S u,v is a fiber of the R-foliation of S u,v H for a C-circle H of the canonical fibration of G. Thus as in Lemma 10.17 we see that f u is isometric along σ. By Proposition 10.22, f u is isometric along S u,v . By Lemma 10.5 again, f ω is isometric along S u,v . Hence |x ′ y ′ | ω ′ = |xy| ω . This completes the proof of Theorem 10.1.
Proof of Theorem 1.1. It follows from Proposition 10.21 that dim X is odd, dim X = 2k − 1, k ≥ 1. If k = 1, then X = ∂ ∞ C H 1 is a C-circle. In the case k = 2, Theorem 1.1 is already proved, see Remark 10.19. Thus we assume that k ≥ 3 and argue by induction over dimension. We take any mutually orthogonal C-circles F , F ′ ⊂ X. By Proposition 8.7 their respective orthogonal complements A, A ′ have nonempty intersection X ′ = A ∩ A ′ . By Corollary 8.13, X ′ satisfies axioms (E) and (O), hence X ′ = ∂ ∞ C H k−2 by the inductive assumption. By Theorem 10.1 the join X ′′ = F ′ * X ′ is Möbius equivalent to ∂ ∞ C H k−1 . Since F ′ , X ′ ⊂ A, we see as in the proof of Proposition 10.21 that X ′′ ⊂ A, and therefore X ′′ is a COS to F . Applying Theorem 10.1 once again to F * X ′′ , we obtain that X = F * X ′′ is Möbius equivalent to ∂ ∞ C H k .