Cross ratios on boundaries of symmetric spaces and Euclidean buildings

We generalize the natural cross ratio on the ideal boundary of a rank one symmetric spaces, or even $\mathrm{CAT}(-1)$ space, to higher rank symmetric spaces and (non-locally compact) Euclidean buildings - we obtain vector valued cross ratios defined on simplices of the building at infinity. We show several properties of those cross ratios; for example that (under some restrictions) periods of hyperbolic isometries give back the translation vector. In addition, we show that cross ratio preserving maps on the chamber set are induced by isometries and vice versa - motivating that the cross ratios bring the geometry of the symmetric space/Euclidean building to the boundary.


Introduction
Cross ratios on boundaries are a crucial tool in hyperbolic geometry and more general negatively curved spaces. In this paper we show that we can generalize these cross ratios to (the non-positively curved) symmetric spaces of higher rank and thick Euclidean buildings with many of the properties of the cross ratio still valid.
On the boundary ∂ ∞ H 2 of the hyperbolic plane H 2 there is naturally a cross ratio defined by cr ∂∞H 2 (z 1 , z 2 , z 3 , z 4 ) = z 1 −z 2 z 1 −z 4 z 3 −z 4 z 3 −z 2 when considering H 2 in the upper half space model, i.e. ∂ ∞ H 2 = R ∪ {∞}. This cross ratio plays an essential role in hyperbolic geometry. For example it characterizes the isometry group by its boundary action and therefore allows to study the geometry of the space from its boundary -an important tool in hyperbolic geometry.
The absolute value of this cross ratio can be generalized in a way broader context, namely CAT(−1) spaces [Bou95]: Let ∂ ∞ Y be the ideal boundary of a CAT(−1) space Y , x, y ∈ ∂ ∞ Y and o ∈ Y . Then one can define a Gromov product (⋅ ⋅) o ∶ ∂ ∞ Y 2 → [0, ∞] by (x y) o = lim t→∞ t− 1 2 d(γ ox (t), γ oy (t)), where γ ox , γ oy are the unique unit speed geodesics from o to x, y, respectively. Then a multiplicative cross ratio cr ∂∞Y ∶ A ⊂ ∂ ∞ Y 4 → [0, ∞] is defined by cr ∂∞Y (x, y, z, w) ∶= exp(−(x y) o − (z w) o + (x w) o + (z y) o ) for all (x, y, z, w) ∈ ∂ ∞ Y 4 with no entry occurring three or four times. As the notation suggests one can show that the cross ratio is independent of the base point. By construction cr ∂∞Y has several symmetries with respect to (R + , ⋅). The logarithm of the above cross ratio log(cr ∂∞Y ) ∶ A → [−∞, ∞] is called an additive cross ratio, as the symmetries are with respect to (R, +). In analogy to the hyperbolic plane, maps f ∶ ∂ ∞ Y → ∂ ∞ Y that leave cr ∂∞Y under the diagonal action invariant are called Moebius maps. It follows from the definition of the cross ratio together with the basepoint independence that isometries are Moebius mpas when restricted to the boundary.
The cross ratios cr ∂∞Y and Moebius maps have been proven to be very useful in hyperbolic geometry. For example Bourdon [Bou96] has shown that Moebus maps of rank one symmetric spaces extend uniquely to isometric embeddings of the interior, and with this he gave a new proof of Hamenstädt's 'entropy against curvature' theorem [Ham90]. Otal [Ota90] has (implicitly) shown that Moebius bijections on boundaries of universal covers of closed negatively-curved surfaces can be uniquely extended to isometries; which yields that marked length spectrum rigidity holds for those manifolds -a prominent conjecture formulated in [BK85]. We want to point out that in general it is not known that Moebius maps always extend to isometries and it seems to be very difficult to answer -for negatively curved manifolds this would be equivalent to marked length spectrum rigidity; see e.g. [Bis15]. Moreover, there is a close relation between the cross ratio on the boundary of the universal cover of a closed negatively curved manifold and the quasiconformal structure on the boundary, and to dynamical properties of the geodesic flow; see e.g. [Led95].
On the boundary ∂ ∞S of the universal cover of a closed surface S there are many other cross ratios, besides the above constructed one, that parametrize classical objects associated to the surface; such as simple closed curves, measured laminations, points of Teichmüller space [Bon88], Hitchin representations [Lab07] and positively ratioed representations [MZ17] 1 -to name a few.
This prominence and importance of cross ratios in negative curvature motivates us to ask if such objects also exists for non-positively curved spaces and how much information about the geometry they carry.
There is already some work done in this context. In [CM17], see also [CCM18], a coarse cross ratio for arbitrary CAT(0) spaces on some subset of the boundary has been constructed. For CAT(0) cube complexes there is a cross ratio on the Roller boundary constructed in [BFIM], using essentially the combinatorial structure of the space. In both cases Moebius (respectively quasi-Moebius) bijections are connected to isometries (respectively quasi-isometries).
In this paper we will construct cross ratios for symmetric spaces and Euclidean buildings, which will generalize the cross ratios of CAT(−1) spaces. There is little need to explain the importance of symmetric spaces in differential geometry and related areas. However, we want to point out that there is currently an active field of research on symmetric spaces going on; namely Anosov representations and subgroups (e.g. [Lab06], [KLP], [GW12] and many more). Anosov representations are representations of hyperbolic groups, for example surface groups, into semi-simple Lie groups which come with a equivariant boundary map satisfying some contracting/expanding properties. They yield a class of very well behaved discrete subgroups which in many situations carry a lot of geometric information (e.g. [CG05]). The natural boundary maps Anosov representations come with can be used to pullback the cross ratios that we construct to get cross ratios on the boundary of the group. This has been done for representation into SL(n, R) in [Lab07] using an ad-hoc definition of a cross ratio and this is in the spirit of [MZ17]; and we hope that the vector valued cross ratio we analyze here allows for further applications in this area.
Euclidean buildings arise in many different areas of mathematics. See [Ji12] for an overview of some applications. Probably most prominently they arise in the study of algebraic groups and geometric group theory; they have also been a crucial tool in the proof of quasi-isometric rigidity of symmetric spaces [KL97] (extending Mostow-Prasad rigidity) -to name a few.
We want to construct (generalized) cross ratios for symmetric spaces or thick (non-locally compact) Euclidean buildings similar as for CAT(−1) spaces. For CAT(0) spaces the Gromov product as above is still well defined. However, the cross ratio might be only defined on very small sets and carry little information; as for example for the Euclidean plane. Denote by M either a symmetric space or a thick Euclidean building. Then we will use the building at infinity ∆ ∞ M of those spaces to extract some subset of the ideal boundary on which the Gromov product and cross ratio will be generically defined and well behaved. More precisely, to any point in the ideal boundary ∂ ∞ M we can associate a type, i.e. we have a map typ ∶ ∂ ∞ M → σ with σ the closed fundamental chamber of the Weyl (actually Coxeter) group. Then one can show that to each type ξ ∈ σ there is a unique type ιξ ∈ σ such that elements in typ −1 (ξ) can be generically joined by a geodesic to elements in typ −1 (ιξ) -for symmetric spaces generically means that there is a dense and open subset in the product. This yields that the Gromov product (⋅ ⋅) o for any o ∈ M restricted to typ −1 (ξ) × typ −1 (ιξ) is generically finite and hence we get a generically defined additive cross ratio on (typ −1 (ξ) × typ −1 (ιξ)) 2 in the same way as for CAT(−1) spaces. While the definition requires a base point, one can show that the cross ratio is independent of the choice. We denote this cross ratio by cr ξ .
Let τ be a face of the simplex σ and let ξ ∈ int(τ ), the interior of τ . Moreover, denote by Flag τ (M ) ⊂ ∆ ∞ M the set of simplices of the building at infinity of type τ -in case of symmetric spaces those are exactly the Furstenberg boundaries. Then one can naturally identify typ −1 (ξ) with Flag τ (M ) and in the same way typ −1 (ιξ) with Flag ιτ (M ). In this way we get a cross ratio cr ξ ∶ A τ ⊂ (Flag τ (M ) × Flag ιτ (M )) 2 → [−∞, ∞] which has by construction similar symmetries as the one on CAT(−1) spaces -for A τ see equation (2.1), for the symmetries see equation (3.1). In general there are less symmetries as in the CAT(−1) situations, since the Gromov product is not symmetric.
It is immediate by construction that we are not getting one cross ratio on the set A τ , but a whole collection parametrized by ξ ∈ int(τ ). We show that we can put together this collection of cross ratios to a single vector valued cross ratio with the same symmetries. In case of a symmetric space X, denote by G = Iso 0 (X), be Lie(G) = g = k + p the Cartan decomposition and a a maximal abelian subspace of p together with the Weyl group W induced by the restricted roots. In case of an Euclidean building E denote by (a, W ) the Coxeter complex over which the building is modeled. In both cases let a + be the positive sector in a and a 1 the unit sphere. Then σ = a + ∩ a 1 . Denote by a τ the unique subspace containing the face τ of σ. Then the vector valued cross ratio cr τ with respect to the face τ of σ takes values in a τ ⊂ a. In particular, if considering the chamber set of the building at infinity of a symmetric space or Euclidean building M , i.e. Flag σ (M ), we get a cross ratio (possibly) taking values in all of a (union ±∞).
It seems natural to consider vector valued cross ratios. On one hand, consider a hyperbolic element g ∈ G in an irreducible symmetric space X not of type A n , D 2n+1 for n ≥ 2 or E 6 and G as above. Let g ± ∈ Flag σ (X) be the attractive and repulsive fixed points. Then the so called period cr σ (g − , g ⋅ x, g + , x) gives exactly the translation vector of g along the unique maximal flat joining g − and g + (for generic, but not all x ∈ Flag σ (X)). On the other hand, there is a nice geometric interpretation of the vector valued cross ratio; in case of the chamber set Flag σ (X) of a symmetric space X this reads as follows: Let x 1 , x 2 , y 1 , y 2 ∈ Flag σ (X) such that x i is opposite of y j for i, j = 1, 2. Let B x i = stab(x i ), i.e. B x i is a Borel subgroup of G, and let N x i be the nilpotent (or horospherical) subgroup of B x i ; set in the same way B y j and N y j . Let n x 1 ∈ N x 1 be the unique element with n x 1 ⋅ y 2 = y 1 (the horospherical subgroup acts simply transitive on the opposite chambers); and define in the same way n x 2 ∈ N x 2 through n x 2 ⋅ y 1 = y 2 , as well as n y j ∈ N y j , by n y 1 ⋅ x 1 = x 2 , n y 2 ⋅ x 2 = x 1 . Moreover, denote by F (x 1 , y 1 ) the unique maximal flat joining x 1 and y 1 , and let o ∈ F (x 1 , y 1 ). Then the vector valued difference of o and n x 1 n y 2 n x 2 n y 1 ⋅o ∈ F (x 1 , y 1 ) equals 2cr σ (x 1 , y 1 , x 2 , y 2 ) -we remark that there is a natural identification of F (x 1 , y 1 ) with a such that o is identified with 0. Similar geometric interpretations hold for Euclidean buildings and all vector valued cross ratios.
Let M 1 , M 2 be either two symmetric spaces or two thick Euclidean buildings. Let σ 1 , σ 2 be the according fundamental chambers of the two spaces and let ξ i ∈ int(σ i ) be two types.
) for all (x, y, z, w) ∈ A σ 1 then f is called a σ 1 -Moebius bijection. Moreover we call a locally compact Euclidean building with discrete translation group a combinatorial Euclidean building. Then we can show the following theorems: Theorem A. Let M 1 , M 2 be either symmetric spaces or thick combinatorial Euclidean buildings and ξ 1 ∈ int(σ 1 ). If M 1 , M 2 are irreducible, then every ξ 1 -Moebius bijection f ∶ Flag σ (M 1 ) → Flag σ (M 2 ) can be extended to an isometry F ∶ M 1 → M 2 . If none of the spaces is a Euclidean cone over a spherical building, then this extension is unique. If M 1 , M 2 are reducible, one can rescale the metric of M 1 on irreducible factors -denote this space byM 1 -such that f can be extended to an isometry F ∶M 1 → M 2 .
Theorem B. Let E 1 , E 2 be thick Euclidean buildings. Then for every σ 1 -Moebius bijection f one can rescale the metric of E 1 on irreducible factors -denote this space byÊ 1 -such that f ∶ Flag σ (E 1 ) → Flag σ (E 2 ) can be extended to an isometry F ∶Ê 1 → E 2 . If none of the irreducible factors is a Euclidean cone over a spherical building, then f can be extended to an isometry F ∶ E 1 → E 2 (without rescaling the metric).
We remark that essentially by definition of the cross ratio every isometry gives rise to a Moebius bijection. Then these theorems show that the cross ratios -at least for the chamber set of the building at infinity -carry a lot of the geometric information of the space, as the characterize isometries by their boundary action. In this spirit we hope that those cross ratios will be a valuable tool in the studies of symmetric spaces and Euclidean buildings.
We want to refer the reader to section 5 to slightly more results in this spirit, e.g. when we get a one-to-one correspondence of Moebius bijections and isometries, and also an analysis in which situations the rescaling of the metric is really necessary.
Concerning the proofs of those theorems: It is essential that a Moebius bijection splits as a product of Moebius bijections of irreducible factors; and that Moebius bijections can be extended to building isomorphisms. For rank one symmetric spaces and trees it is already known that Moebius bijections extend to isometries; for irreducible thick combinatorial Euclidean buildings it is enough that Moebius maps are restrictions of building isomorphisms to the chamber set. For the cases of symmetric spaces and (general) thick Euclidean buildings, we derive additional properties of the building map, using the cross ratio. Those properties will allow us to use theorems (essentially due to Tits) showing that the according maps can be extended to isometries.
Structure: In the preliminaries we try to recall those facts about symmetric spaces and Euclidean buildings (of the huge theory available) that will be relevant for us in the ongoing. We assume the reader to be familiar with those objects, but still briefly mention most facts used. Moreover, we show some basic lemmas we need later on.
In section 3 we define R-valued cross ratios and show basic properties. We illustrate the objects with two examples: the product of two copies of the hyperbolic plane and the symmetric space SL(n, R) SO(n, R).
In section 4 we show that the collections of R-valued cross ratios fit together to a single vector valued cross ratio. We motivate that this is the natural object, by showing that under some assumptions periods of hyperbolic elements give rise to the translation vector of the latter. Moreover, we give a geometric interpretation for the vector valued cross ratios.
In the last section, section 5, we show that Moebius maps, i.e. cross ratio preserving maps, on the chamber set extend to isometries. When considering symmetric spaces and combinatorial Euclidean buildings it is enough to consider a R-valued cross ratio. When considering general Euclidean buildings we need to consider the vector valued cross ratio.
Related Work: In [Kim10] essentially the same R-valued cross ratio as in Definition 3.6 has been constructed. However, few of further properties have been shown. Labourie [Lab07] has given one of the cross ratios in Example 3.13 ad-hoc and used it as tool to understand Hitchin representations. Moreover, Martone and Zhang [MZ17] have constructed cross ratios on boundaries of surface groups, which in particular for SL(n, R)-Hitchin representations coincide with the pullback under the boundary map of some of the cross ratios in Example 3.13.
In the case of a symmetric space when writing affine apartment we mean a maximal flat.
We begin with recalling several facts about symmetric spaces and Euclidean buildings -mainly those relevant for us in the ongoing. Then we prove some basic lemmas we need later on.
Coxeter complex and spherical buildings [Ron09], [AB08]: Let W be a finite Coxeter group and S the standard set of generators consisting of involutions. Then W can be realized as a reflection group along hyperplanes in R r with r = S . The hyperplanes decompose R r and the unit sphere S r−1 into (cones over) simplical cells. The maximal, i.e. rdimensional, closed cells in R r are called Weyl sectors. Lower dimensional cells will be called conical cells. The maximal, i.e. r − 1-dimensional, closed simplical cells in S r−1 are called Weyl chambers. The set S corresponds to exactly the hyperplanes bounding a Weyl sector. This Weyl sector will be called the positive sector, the corresponding chamber in S r−1 will be called positive chamber. We can give each simplex adjacent to the positive chamber or positive sector a different label. Then the action of W on the simplical complex induces a unique labeling for all simplices. A fixed label will be called type.
In this paper we refer to (R r , W ) as the Coxeter complex and to (S r−1 , W ) as the spherical Coxeter complex.
A spherical building is a simplical complex B together with a collection of subcomplexes Apt(B), called apartments, which are isomorphic to a fixed spherical Coxeter complex (S r−1 , W ), such that the following holds: 1. For any two simplices a, b ∈ B there is an apartment A ∈ Apt(B) with a, b ∈ A 2. If A, A ′ are apartments containing the simplices a, b, then there is a type preserving simplical isomorphism A → A ′ fixing a, b.
We say that the building is modeled over the spherical Coxeter complex (S r−1 , W ).
A spherical building is called thick if each non-maximal simplex is contained in at least three chambers. A (spherical) Coxeter complex is called irreducible if the Coxeter group can not be written as a product W = W 1 ×W 2 of two nontrivial Coxeter groups. A spherical building is called irreducible if the spherical Coxeter complex over which it is modeled is irreducible. If a building B is reducible, i.e. modeled over the spherical Coxeter complex W 1 × W 2 , then it can be written as the spherical join of two buildings, i.e. B = B 1 ○ B 2 for two spherical buildings B 1 , B 2 modeled over W 1 , W 2 and ○ being the spherical join [KL97,Sc.3.3].
Given a simplex x ∈ B with B a thick spherical building. We denote by Res(x) ∶= {y ∈ B x ⊊ y} and call this the residue of x. Let A be an apartment containing x, i.e. a Coxeter complex containing x. Let W be the Coxeter group of A and denote by W x the stabilizer of x under W . If x is not a chamber then Res(x) is itself a spherical building modeled over the Coxeter complex to W x [Tit74, 3.12].
Euclidean buildings [KW14], [Par00], [Tit86], [KL97]: LetŴ be an affine Coxeter group, i.e.Ŵ can be realized as a subgroup of the isometry group of R r and can be decomposed as a semi-direct productŴ = W ⋉ T W , where W is a finite reflection group and T W < R r is a co-bounded subgroup of translations. Here we assume r = S , where S is the standard generating set of W . Moreover, let (E, d) be a metric space. A chart is an isometric embedding φ ∶ R r → E, and its image is called affine apartment; the image of a Weyl sectors and conical cells are again called Weyl sectors and conical cells. Two charts φ, ψ are calledŴ -compatible if Y = φ −1 ψ(R r ) is convex in the Euclidean sense and if there is an element w ∈Ŵ such that ψ ○w Y = φ Y . A metric space E together with a collection of charts C, called apartment system, is called a Euclidean building (with Coxeter groupŴ ) if it has the following properties: 1. For all φ ∈ C and w ∈Ŵ , the composition φ ○ w is in C.
2. Any two points p, q ∈ E are contained in some affine apartment.
3. The charts areŴ -compatible. 4. If a, b ⊂ E are Weyl sectors, then there exists an affine apartment A such that the intersections A ∩ a and A ∩ b contain Weyl sectors. 5. If A is an affine apartment and p ∈ A a point, then there is a 1-Lipschitz From this properties it follows that the metric space E is necessarily CAT(0). The dimension of R r is called the rank of E, i.e. rk(E) = r. While the definition depends on a fixed set of affine apartments, there is always a unique maximal set of affine apartments, called the complete apartment system. A set is an affine apartment in the complete apartment system if and only if it is isometric to R r . In the ongoing we will always consider E with its complete apartment system. If the subgroup of translations T W is discrete and E is locally compact we call E a combinatorial Euclidean building.
The ideal boundary and Busemann functions [BH99, Part II, Ch.8]: We recall here several properties valid for CAT(0) spaces; hence for Euclidean buildings and symmetric spaces.
We denote by ∂ ∞ M the ideal boundary, i.e. the equivalence classes of geodesic rays -here equivalence means finite Hausdorff distance. Equipped with the cone topology ∂ ∞ M is naturally a topological space. For every o ∈ M and every x ∈ ∂ ∞ M we denote by γ ox the unique unit-speed geodesic ray joining o to x, i.e. γ ox (0) = o and γ ox in the class of x. For o, p, q ∈ M the Gromov product on M is defined by , the Gromov product on the ideal boundary with respect to o, is given by We remark that the convexity of the distance functions guarantees the existence of the limit in [0, ∞]. Given x ∈ ∂ ∞ M the Busemann function with respect to x, which will be denoted by b An easy argument in Euclidean geometry yields that the level sets of Busemann functions in R n with respect to x in the boundary sphere are affine hyperplanes orthogonal to the direction x. In general Busemann level sets with respect to one coordinate are called horospheres and the collection of horospheres is independent of the choice of the other coordinate.
The isometry group Iso(M ) acts naturally by homeomorphisms on ∂ ∞ M , since they map equivalence classes of geodesic rays to equivalence classes of geodesic rays. Moreover, by construction of the Busemann function, it fol- Symmetric spaces [Ebe96, Ch.2]: Let X be a symmetric space. We will always assume that X is of non-compact type. In particular X is a Hadamard manifold and therefore CAT(0). We denote by G = Iso 0 (X), i.e. the connected component of the identity of the isometry group.
Let g = Lie(G) and g = k + p the Cartan decomposition. If we fix a maximal flat F in X together with a basepoint o ∈ F , we get the identification T o M ≅ p. This identification yields T o F ≅ a where a a maximal abelian subspace of p. The restricted root system of g with respect to a defines hyperplanes in a -namely the zero sets of the restricted roots. The Weyl group W of X is the group generated by the reflections along those hyperplanes with respect to the metric that a inherits from T o F ⊂ T o X. Hence we can associate to X a Coxeter complex (a, W ). Let a 1 be the unit sphere in a, then we also get a spherical Coxeter complex (a 1 , W ). It is well known that up to isometry the Coxeter complex is independent of the choices. We fix a Weyl sector in a which we denote by a + and call positive sector. Then a + 1 will be called the positive chamber. 2 The rank of X is the usual rank and equals rk(X) = dim a. To keep the notation consistent with buildings we will call maximal flats in X affine apartments.
The building at infinity [Ebe96, Ch.3], [KW14], [Par00], [Tit86], [KL97]: Let M now be either a symmetric space or a Euclidean building. To keep notation simple, we will denote by (a, W ) also the Coxeter complex over which a Euclidean building is modeled. Moreover, a 1 is the unit sphere in a and hence (a 1 , W ) a spherical Coxeter complex. We fix a positive Weyl sector a + ⊂ a and the according positive chamber a + 1 = a 1 ∩ a + . Let S denote the generating set of W consisting of reflections along the walls of a + . By definition we have rk(M ) = dim a.
The ideal boundary ∂ ∞ M carries naturally the structure of a spherical building ∆ ∞ M modeled over the spherical Coxeter complex (a 1 , W ). The building ∆ ∞ M will be called the building at infinity.
For a Euclidean building E the building at infinity arises as follows: Let A ⊂ E be an affine apartment. Then A being the image of (a, W ) under a chart implies that A is decomposed into conical cells. Each conical cell defines a simplex in ∂ ∞ E by taking the geodesic rays contained in the cell for all times. One can show that two conical cells define the same set in ∂ ∞ E if and only if they have finite Hausdorff distance. In the latter case we say the conical cells are equivalent. Therefore, taking all conical cells in E modulo the equivalence relation we get a simplical structure on ∂ ∞ E. Which can be shown the be a spherical building over the spherical Coxeter complex (a 1 , W ).
In the same way we get the building at infinity of symmetric spaces X: Every maximal flat F with fixed basepoint can be isometrically identified with a. Then the conical cells of a descend to conical cells F ⊂ X. Again taking all conical cells in X modulo the equivalence relation of finite Hausdorff distance gives ∂ ∞ X a simplicial structure, which yields a spherical building modeled over (a 1 , W ).
Apartments in ∆ ∞ M correspond to the ideal boundaries of affine apartments of M . It is well known that ∆ ∞ X is a thick building. We call E a thick Euclidean building if ∆ ∞ E is thick. If the rank of E is one, i.e. E is a tree we call E thick if ∂ ∞ E contains at least three points, i.e. E ≠ R.
In particular the following important property holds: To every two points p, q ∈ M ∪∂ ∞ M we find an affine apartment A in M such that p, q ∈ A∪∂ ∞ A. We say that A joins p and q.
Given two affine apartments A, A ′ in a Euclidean building E that have a common chamber at infinity, i.e. c ∈ ∆ ∞ E such that c ⊂ ∂ ∞ A and c ⊂ ∂ ∞ A ′ . Then the intersection A ∩ A ′ contains a Weyl sector with boundary c. Such a Weyl sector is called a common subsector of A and A ′ .
there is a chamber c x ∈ ∆ ∞ M with x ∈ c x and an affine apartment A with c x ⊂ ∂ ∞ A. Then this yields a isometry from c x to a + 1 with respect to the Tits metric on c x and the angular metric on a + 1 . In this way we can assign to each element of ∂ ∞ M an unique element of a + 1 . It can be shown that the image is independent of the chamber and the apartment chosen, hence we get a well defined map typ ∶ ∂ ∞ M → a + 1 . The type map is consistent with the types of the spherical building ∆ ∞ M , i.e. two simplices of ∆ ∞ M are of the same type if and only if they are mapped to the same face of a + 1 under typ. Hence we also call the faces of a + 1 types (a + 1 will be a face of itself). When speaking of types we denote σ = a + 1 , i.e. a simplex of ∆ ∞ M is a chamber if and only if it is of type σ. Faces of σ will usually be denoted by τ . The set of simplices in ∆ ∞ M of type τ will be denoted by Flag τ (M ), or just by Flag τ if M is clear out of the context and will be called flag space. If we consider chambers we denote this by Flag σ and call it full flag space.
We (ambiguously) call elements in ξ ∈ σ = a + 1 types. However, out of the context it is clear if an element or a simplex is meant. We denote by int(τ ) the interior of a simplex (and set the interior of a point to be the point itself). Given a simplex in x ∈ Flag τ and ξ ∈ τ , we denote by Any isometry F ∶ M 1 → M 2 between either two symmetric spaces or two thick Euclidean buildings induces a building isomorphism The map F ∞ is in general not type preserving. However, that M 1 , M 2 are isometric implies that they are modeled over the same Coxeter complex and hence have the same fundamental chamber σ. Then we can associate to F a type map F σ ∶ σ → σ such that typ(F ∞ (x)) = F σ (typ(x)) for every x ∈ ∂ ∞ M 1 and F σ is a isometry with respect to the angular metric. Moreover, we have that F (Flag τ (M 1 )) = Flag Fσ(τ ) (M 2 ).
The G-action and flag manifolds [Ebe96, Ch.3], [KLP, Sc.2.4]: Let X be a symmetric space and G = Iso 0 (X). Then the cone topology on ∂ ∞ X induces a topology on ∆ ∞ X such that all flag spaces are compact. Moreover, given x ∈ Flag τ (X) be P x = stab(x). Then we can identify Flag τ (X) ≃ G P x with the identification being G-equivariant and homeomorphic; the group P x is a parabolic subgroup of G (every parabolic subgroup arises as the stabilizer of an element of ∆ ∞ X) and G P x is equipped with the quotient topology of the topological group G. Moreover, the identification Flag τ (X) ≃ G P x gives a smooth structure on Flag τ (X) making it a compact connected manifold. The spaces G P x are called Furstenberg boundaries or flag manifolds, motivating our notion of flag space.
Let K be a maximal compact subgroup of G. Then already K acts transitive on the flag manifolds and given x ∈ Flag τ we can identify Kequivariant and homeomorphically Flag τ (X) Moreover, we remark that the G-action is type preserving, i.e. g σ = id for all g ∈ G.
The opposition involution: A important map for us will be the opposition involution ι ∶ a → a, which is given by ι = −id ○ w 0 with w 0 ∈ W the maximal element of the Coxeter group with respect to the generating set S. If W is an irreducible Weyl group, then ι = id if and only if W is not of type A n with n ≥ 2, D 2n+1 with n ≥ 2 or E 6 [Tit74, 2.39]. Moreover, we remark that we can restrict ι ∶ a + 1 → a + 1 and that ι is an isometry with respect to the angular metric.
Opposite simplices [KLP, Sc.2.2, 2.4]: There is a natural notion of opposition in spherical buildings. This corresponds to the following: Let The action of the spherical Coxeter group W leaves the type invariant. Therefore, assume for the moment that W is modeled in A ∞ and x is a face of the positive chamber. Denote by w 0 ∶ A ∞ → A ∞ the maximal element of W . Then w 0 (y) is a face of the positive chamber and of the same type as y and hence y is of type −id ○ w 0 (x) = ιx. In particular all simplices opposite of elements in Flag τ are contained in Flag ιτ .
We denote for later use Opposition of simplices has the following important connection to biinfinite geodesics: Let z 1 , z 2 ∈ ∂ ∞ M and A ⊂ M an affine apartment with z 1 , z 2 ∈ ∂ ∞ A. Then one can show that there exists a bi-infinite geodesics joining z 1 and z 2 if and only if there exists one in A. From Euclidean geometry it follows that the z i can be joined by a bi-infinite geodesic in A if and only if z 1 = −id(z 2 ) with −id ∶ ∂ ∞ A → ∂ ∞ A as before. This can easily be seen to be equivalent to the unique simplices τ z i ∈ ∆ ∞ M containing the z i in its interior being opposite, i.e. τ z 1 op τ z 2 , and typ(z 1 ) = ιtyp(z 2 ).
We will call points z 1 , z 2 ∈ ∂ ∞ M opposite if they can be joined by a bi-infinite geodesic and denote this also by z 1 op z 2 . Given (x, y) ∈ Flag τ × Flag ιτ with x op y, for every ξ ∈ τ it follows that x ξ is opposite to y ιξ . In particular, it follows in this case that A τ and A op τ are open and dense subsets of (Flag τ × Flag ιτ ) 2 .
Every parabolic subgroup P x has a natural decomposition P x = K x A x N x called the Langlands decomposition, where K x is compact and N x is nilpotent. The group N x is called horospherical subgroup and is unique, while K x and A x are not. The horospherical subgroup has several important properties. It leaves the Busemann function with respect to Moreover, N x acts simply transitive on the simplices opposite to x. If x is a chamber, i.e. x ∈ Flag σ (M ), then N x acts simply transitive on the maximal flats containing x in its boundary. Let (x, y) ∈ Flag τ × Flag ιτ with x op y and denote by int(τ ) the interior of τ . Moreover, be ξ ∈ int(τ ). Then the parallel set with respect to x, y denoted by P (x, y) is the set of all points that lie on a bi-infinite geodesic joining x ξ to y ιξ .
The parallel sets split metrically as products, i.e. P (x, y) ≃ F xy × CS(x, y), where F xy is an isometrically embedded R n such that x, y ⊂ ∂ ∞ F xy and x, y are simplices of maximal dimension in the sphere ∂ ∞ F xy -in particular n − 1 equals the dimension of the spherical simplices x, y. Then it follows that the parallel set is independent of the choice of type ξ ∈ int(τ ), as for each type ξ ∈ int(τ ) geodesics in M joining x ξ , y ιξ are of the form (γ x ξ y ιξ (t), p) with γ x ξ y ιξ a geodesic in F xy joining x ξ , y ιξ and p is a point in The space CS(x, y) is called cross section. In case of a symmetric space X the cross section is itself a symmetric space without Euclidean de Rham factors, in case of a Euclidean building the cross section is again a Euclidean building. In both cases the rank is given by rk(CS(x, y)) = rk(M ) − dim F xy Let τ be a face of σ = a 1 . Then be a τ the subspace of a defined by τ , i.e. the smallest subspace of a containing τ and 0. Let ξ 1 , . . . , ξ k be the corners of the spherical simplex τ . Then a τ = span i=1,...,k ξ i . It is immediate that we can also identify P (x, y) ≃ a τ × CS(x, y). We can additionally impose that this identification is in such a way that x ≃ ∂ ∞ (a τ ∩ a + ).
Lemma 2.1. Let (x, y) ∈ Flag τ × Flag ιτ with x op y and be p, q ∈ P (x, y). Let π ∶ P (x, y) ≃ a τ × CS(x, y) → a τ be the projection to the first coordinate. Then for each ξ ∈ τ we have that b x ξ (p, q) = (b x ξ ) aτ (π(p), π(q)), i.e. the Busemann function is independent of the second coordinate of the product.
Proof. Let γ qx ξ denote the geodesic ray from q to x ξ . Moreover, be q = (q 1 , q 2 ) under the identification P (x, y) ≃ a τ × CS(x, y). Then we have Using a substitution t = s −1 and a Taylor series for the root expression below yields We will also need the following lemma.
x ξ and γ i (−∞) = y ιξ , which exists by assumption. The γ i are parallel and denote by C their distance. Then the Flat Strip Theorem (see e.g. [BH99]) implies that the convex hull of It follows that the level sets of the Busemann function b x ξ (⋅, p 2 ) in R × [0, C] are given by hyperplanes orthogonal to γ i , i.e. are of the form s × [0, C] and the same holds for b y ιξ (⋅, p 2 ). In addition, it follows directly Retracts [Par00]: Lastly we need to introduce the notion of retracts of M to affine apartments with respect to chambers at infinity. For the construction we will distinguish between Euclidean buildings and symmetric spaces.
Let E be a Euclidean building. Let A ⊂ E be an affine apartment and x ⊂ ∂ ∞ A a chamber of the building at infinity. Then there exists a 1-Lipschitz map ρ x,A ∶ E → A which is an isometry when restricted to any affine apartment A ′ with x ⊂ ∂ ∞ A ′ , i.e. any affine apartment that contains the chamber x in its boundary, and the identity on A [Par00, Prop.1.20]. We call this map (horospherical) retract with respect to x. Horospherical retracts have the following important property: Proof. To o ∈ E there exists an affine apartment A o containing o and x ⊂ ∂ ∞ A o . As mentioned, the horopsheres with repsect to x ξ in A o are hyperplanes orthogonal to the direction x ξ .
By construction, the two affine apartments A, A o have the same chamber in its boundary, which implies that they have a common subsector. Hence to the corresponding level set in A. The other equality follows for example form the symmetry b Let X be a symmetric space, A ⊂ X be a maximal flat (an affine apartment for us) and x ⊂ ∂ ∞ A a chamber at infinity. To any o ∈ X there exists a unique maximal flat For later reference: To every affine apartment A ⊂ M and chamber for all o, p ∈ M and ξ ∈ σ. Moreover, it is known that two opposite chambers x, y ∈ Flag σ are contained in an unique apartment A ∞ of ∆ ∞ M and this corresponds to an unique affine apartment A xy ⊂ M . Hence to x, y ∈ Flag σ with x op y we set ρ x,y ∶= ρ x,Axy .
where c x , c y ∈ Flag σ such that x is a face of c x , y is a face of c y and c x op c y .
We remark that x op y implies that such c x , c y ∈ Flag σ always exist. Namely, take an apartment containing x and y. Take c x ∈ Flag σ such that x is a face of c x . Take c y ∈ Flag σ the unique opposite chamber in the apartment. Then x op y implies that y is a face of c y .
Proof. For a symmetric space X this follows since ρ cx,cy is the same element of G for all points γ ox ξ (t) and that G < Iso(X). Hence ρ cx,cy (γ ox ξ (t)) is the image of a geodesic under an isometry.
Consider a Euclidean building E. Denote by A xy the unique affine apartment joining c x and c y . Let A be an affine apartment containing o and c x ⊂ ∂ ∞ A. Then it follows that γ ox ξ (t) ∈ A for all t ∈ R + . As ρ cx,cy is an isometry on affine apartments containing c x , it follows that ρ cx,cy (γ ox ξ (t)) ⊂ A xy is the image of a geodesic under an isometry. Since one of the endpoints is x ξ , we can extend the geodesic in A xy uniquely to a bi-infinite geodesic joining x ξ and y ιξ . Thus ρ cx,cy (γ ox ξ (t)) ⊂ P (x, y).

Cross ratios
Let M be a symmetric space of non-compact type or a thick Euclidean building. Let σ be the fundamental chamber of the associated spherical Coxeter complex and τ a face of σ. For any type ξ ∈ σ such that ξ ∈ int(τ ) and any o ∈ M we define a Gromov product for (x, y) ∈ Flag τ (M )×Flag ιτ (M ) and γ ox ξ (t), γ oy ιξ (t) the unit speed geodesics from o to x ξ , y ιξ , respectively. Using this we define the (additive) cross ratio By definition cr o,ξ has the following symmetries, whenever all factors are defined, 3 The last two symmetries are called cocycle identities.
Proposition 3.1. Let M be a symmetric space or thick Euclidean building, o ∈ M , (x, y) ∈ Flag τ × Flag ιτ with x op y and c x , c y ∈ Flag σ such that x is a face of c x , y is a face of c y and c x op c y . Then for every ξ ∈ τ Proof. In case of a symmetric space let N x be the horospherical subgroup of We define in the same way n y (o, x) ∈ N y and set γ xy (t) ∶= n x (o, y) ⋅ γ ox ξ (t) and γ yx (t) ∶= n x (o, y)⋅γ oy ιξ (t). Then γ xy , γ yx are geodesics in P (x, y) with the same (un-ordered) end points. Hence they are parallel. Moreover, The triangle inequality yields that (x y) o,ξ = lim t→∞ t− 1 2 d(γ xy (t), γ yx (t)). By construction γ xy , γ yx are parallel geodesics; hence by the Flat Strip Theorem (see e.g. [BH99]) the distance d(γ xy (t), γ yx (t)) decomposes into a part parallel to the geodesics and the distance of the images of the geodesics, which is a constant and will be denoted by C.
The part parallel to the geodesics is b x ξ (γ yx (t), γ xy (t)) -or equally b y ιξ (γ xy (t), γ yx (t)). Using that we have geodesics asymptotic to x ξ we derive while the second to last equality follows using Taylor series at s = 0 after substituting s = t −1 (see also the calculations in example 3.7).
In case of a Euclidean building E, let A o be an affine apartment contain- Moreover, be d y ∈ Flag σ a chamber opposite to d x such that y is a face of d y and let A xy be the unique affine apartment that d x and d y define.
Then the affine apartments A o and A xy have a common subsector. Hence there exists T x ≥ 0 such that for t ≥ T x the geodesic γ ox ξ (t) is parallel to a geodesic γ xy in the subsector -denote the distance of the geodesic rays by C x ; Extend γ xy bi-infinite in A xy such that it is in the same horosphere with respect to x ξ as γ ox ξ (t) for all (positive) time. That γ xy is in A xy with one endpoint being x ξ implies that γ xy joins x ξ and y ιξ and hence γ xy ⊂ P (x, y).
In the same way we construct γ yx ⊂ P (x, y) to γ oy ιξ such that those geodesics are parallel for t ≥ T y -denote the distance by C y . Since γ xy , γ yx join the same points at infinity, they are parallel -denote the distance by C 0 . Then the triangle inequality together with the Flat Strip theorem yields for t ≥ max{T x , T y } that d(γ ox ξ (2t), γ oy ιξ (2t)) is smaller or equal than Since γ xy and γ yx are are asymptotic to We substitute t = 1 s . Then a Taylor expansions for the root expressions at s = 0 yields that (x y) o,ξ ≥ − 1 2 b x ξ (γ yx (0), γ xy (0)) = 1 2 b x ξ (γ xy (0), γ yx (0)).

In a similar way it follows also
Proof. Let (x, y) ∈ Flag τ × Flag ιτ be such that x ✚ ✚ op y. Let A be an affine apartment containing x, y in its boundary. Let p ∈ A and γ px ξ , γ py ιξ be the unit speed geodesics joining p to x ξ , y ιξ , respectively. A straight forward argument in Euclidean geometry yields that d(γ px ξ (t), γ py ιξ (t)) = 2αt with α depending on the angle of the geodesics. Moreover x ✚ ✚ op y implies that γ px ξ (t) ≠ γ py ιξ (−t) and hence α < 1, i.e. (x y) p,ξ = ∞. Now let γ ox ξ , γ oy ιξ be the unit speed geodesics joining o to x ξ , y ιξ , respectively. Since γ ox ξ and γ px ξ define the same point in the ideal boundary, we can derive -by the convexity of the distance functions along geodesics in non-positive curvature -that d(γ ox ξ (t), γ px ξ (t)) ≤ d(o, p) for all t ≥ 0. Thus Let (x, y) ∈ Flag τ × Flag ιτ be such that x op y. Then by the above Remark 3.3. The above corollary implies that A ξ is the maximal domain of definition for cr o,ξ . As mentioned, in case of a symmetric space X, the set A ξ is an open and dense subset of (Flag τ (X) × Flag ιτ (X)) 2 , i.e. the cross ratio is generically defined.
Proof. If x ✚ ✚ op y, then by the above corollary (x y) o,ξ = ∞ = (x y)ô ,ξ . If x op y, let ρ x,y , ρ y,x be any horospherical retracts as in Proposition 3.1. Then By construction ρ y,x (o), ρ y,x (ô) ∈ P (x, y). Moreover x, y are opposite and hence by Lemma 2.2 and equation Together with Proposition 3.1 the claim follows.
Example 3.7. (see also [Kim10]) Consider the symmetric space X = H 2 × H 2 , where H 2 is the hyperbolic plane. The ideal boundary ∂ ∞ (H 2 × H 2 ) can be identified with S 1 × S 1 × [0, π 2 ] -this is in such a way that the unit-speed geodesic ray from a base-point . The types are exactly determined by the angle α and the opposition involution equals the identity. In particular every type is self opposite. Fix We substitute t = 1 s . Then a Taylor expansion for the root expression at s = 0 yields that Therefore cr α = cos(α) log cr ∂∞H 2 + sin(α) log cr ∂∞H 2 , where cr ∂∞H 2 is the usual multiplicative cross ratio on ∂ ∞ H 2 .
Proof. Since Flag τ (X), Flag ιτ (X) are manifolds it is enough to consider sequential continuity. Therefore let (x, y) ∈ Flag τ (X) × Flag ιτ (X) and let ) o with Gromov product on the right hand side the usual Gromov product on the metric space (X, d). As X is non-positively curved, the function t ↦ (x y) o,ξ (t) is monotone increasing. Let C > 0 be given. Then there is t C ∈ R + such that (x y) o,ξ (t C ) ≥ C +2. Since the topology on Flag τ (X) is induced by the cone topology, we have that (x i ) ξ → x ξ in the cone topology and similarly for y i and y. Hence we find L ∈ N such that d(γ o(x i ) ξ (t C ), γ ox ξ (t C )) < 1 and d(γ o(y i ) ιξ (t C ), γ oy ιξ (t C )) < 1 for all i ≥ L. Hence by the triangle inequality (x i y j ) o,ξ (t C ) > (x y) o,ξ (t C ) − 2 > C for all i, j ≥ L. As C was arbitrary, this yields lim i,j→∞ (x i y j ) o,ξ = ∞ -which proves continuity for x ✚ ✚ op y.
Assume x op y. Let K = stab G (o). We know that K acts transitively on Flag τ (X) and we have a K-equivariant and homeomorphic identification Flag τ (X) ≃ K K x . Therefore x i → x implies that we find k i ∈ K such that k i x i = x and k i → e ∈ G. Now, x op y and opposition being an open condition, together with y i → y and k i → e, imply that there exists L ∈ N such that k i y j op x for all i, j ≥ L. Thus there exists a unique n ij ∈ N x such that n ij k i y j = y for i, j ≥ L. From k i → e and y j → y it follows n ij → e ∈ G for i, j → ∞. We set g ij ∶= n ij k i and by construction g ij → e, g ij x i = x, Lemma 3.9. Let (x, y) ∈ Flag τ × Flag ιτ and x op y. Moreover, let ξ i ∈ τ be a sequence with Proof. Let c x , c y ∈ Flag σ such that c x op c y , x is a face of c x and y is a face of c y . Then Proposition 3.1 and equation and by A xy the unique affine apartment with c x , c y ⊂ ∂ ∞ A xy . Every affine apartment can be isometrically identified with R r where r is the rank of M . We identify A xy with R r such that 0 ≃ p x . Let v ξ ∈ A xy ≃ R r be of norm one and such that the line from 0 through v ξ is the goedesic ray in A xy from p x to x ξ . Then Euclidean geometry yields that b x ξ (p x , p y ) = ⟨v ξ , p y ⟩. In particular, we get Moreover ξ i → ξ 0 implies that v ξ i → v ξ 0 and hence the claim follows.
Remark 3.10. The assumption of opposition in the above lemma is needed, since there are (x, y) ∈ Flag τ × Flag ιτ with x ✚ ✚ op y but there are faces x 0 of x and y 0 of y with x 0 op y 0 . Then if ξ i ∈ int(τ ) converge to ξ 0 such that ξ 0 ∈ int(τ 0 ) and τ 0 is the type of x 0 , we get (x y) o,ξ i = ∞ ↛ (x 0 y 0 ) o,ξ 0 (as the latter is finite).
Proof. For the center of gravity ξ 0 ∈ σ we have g σ (ξ 0 ) = ξ 0 for all g ∈ Iso(M ), as g σ ∶ σ → σ is an isometry with respect to the angular metric. Then the first claim follows. In case of a symmetric space and g ∈ G, we know g σ = id σ , which implies the second claim.
Example 3.13. We want to determine the Gromov products and cross ratios of the symmetric spaces X(n) ∶= SL(n, R) SO(n, R). For a deeper description of the symmetric space X(n) see [Ebe96].
The "eigenvalues" λ in the eigenvalue flag pairs (λ, F ) determine the type of any point in the ideal boundary. A point in the boundary is contained in the interior of a chamber if and only if l = n which means λ consists of n different "eigenvalues"; equivalently m i = dim V i − dim V i−1 = 1 for all i. The action of the opposition involution is given by ι(λ 1 , . . . , λ l ) = (−λ l , . . . , −λ 1 ).
The space of flags F = (V 1 , . . . , V n ), i.e. m i = dim V i − dim V i−1 = 1, is called the space of full flags and equals Flag σ (X(n)). Let In the same way we fix basis (w 1 , . . . w n ), (y 1 , . . . y n ) and (z 1 , . . . z n ) for W, Y, Z, respectively. Additionally, fix an identification . . , w n−i j } -and in the same way for the other flags. Then the term can be shown to be independent of all choices for all j = 1, . . . , l − 1 -compare e.g. [MZ17].
Let V, W, Y, Z be as before and λ = (λ 1 , . . . , λ l ) a type with λ ∈ int(τ ). Then using the above conventions -see the appendix for a proof. We remark that some specific of those cross ratios are known already and have been used for analyzing Hitchin representations (see e.g. [Lab07], [MZ17]).
We remark that the unit-speed geodesic from some point (o 1 , . . . , o k ) ∈ M to x ξ is of the form (γ o 1 x ξ 1 (µ 1 t), . . . , γ o k x ξ k (µ k t)), where γ o i x ξ i denote the unit speed geodesics in the factors M i joining o i to (x i ) ξ i -cp. also Example 3.7.
Proof. Since the Gromov product was defined in terms of distances and Φ is an isometry, it is straight forward to show that for ξ ∈ τ , (x, y) ∈ Flag τ (X) × Flag ιτ (X) and o ∈ M we have that Hence cr ξ = (Φ τ × Φ ιτ ) * cr Φσ(ξ) and therefore the claim follows form the proposition above.

Vector valued cross ratios
So far, we have constructed families of cross ratios on subsets of the spaces (Flag τ × Flag ιτ ) 2 which are parametrized by ξ ∈ int(τ ). In this section we show that such a family gives rise to a single vector valued cross ratio containing all the information of the family. The vector valued cross ratio has the same symmetries as the usual cross ratios (cp. equations (3.1)) justifying the name cross ratio. We remind that σ = a + 1 ; hence every type can be viewed as vector in a of norm one.
Lemma 4.1. Let τ be a face of σ and ξ 0 , ξ 1 , . . . , ξ j ∈ τ such that there exist In particular it follows that cr ξ 0 (x, y, z, w) = ∑ j i=1 a i cr ξ i (x, y, z, w) for all (x, y, z, w) ∈ A op τ Proof. Let c x , c y ∈ Flag σ such that c x op c y , x is a face of c x and y is a face of c y . We recall the notation of the proof of Lemma 3.9: We denote p x ∶= ρ cx,cy (o), p y ∶= ρ cy,cx (o) and by A xy the unique apartment with c x , c y ⊂ ∂ ∞ A xy . Moreover, let A xy ≃ R r such that p x ≃ 0, in particular A xy inherits a inner product. Let v ξ ∈ A xy ≃ R r be of norm one and such that the line from p x ≃ 0 through v ξ is the geodesic ray in A xy from p x to x ξ . Then we know from equation we have the addition inherited to A xy under the identification with R r such that p x ≃ 0. Hence Let ξ 1 , . . . , ξ r be the corners of σ = a + 1 . Then every subset J ⊂ {1, . . . , r} defines a simplex in σ, i.e. a face τ of σ. In the same way every simplex τ ⊂ σ gives a subset J τ ⊂ {1, . . . , r}.
Given a simplex τ we recall that a τ = span j∈Jτ ξ j ⊂ a. Moreover, we define α τ j ∈ a τ for j ∈ J τ by ⟨α τ j , ξ i ⟩ = δ ij for all i ∈ J τ -the gives well defined vectors, as the ξ i with i ∈ J τ form a basis of a τ . We remind that a was naturally equipped with an inner product.
The ξ j correspond to normalized fundamental weights of the root system and the α σ j to possibly rescaled roots.
Here we set cr τ (x, y, z, w) It is straight forward to see that cr τ has the same symmetries as in equations (3.1), where the addition is now in the vector space a τ . The vector valued cross ratio contains the full information of the collection of cross ratios form the previous section: Lemma 4.3. Let ξ ∈ int(τ ). If (x, y, z, w) ∈ A op τ , then ⟨cr τ (x, y, z, w), ξ⟩ = cr ξ (x, y, z, w).
We remark that the above lemma also holds for ξ ∈ ∂τ as long as (x, y, z, w) ∈ A op τ , but does not hold for general (x, y, z, w) ∈ A τ -in this case cr ξ (x, y, z, w) might be finite while cr τ (x, y, z, w) is not (compare Remark 3.10).
The following corollary captures the topological properties of cr τ in case of symmetric spaces. It is an immediate consequence of the lemma above and Lemma 3.8.

Translation vectors and periods
We assume for this section that τ is self-opposite, i.e. τ = ιτ . Moreover denote by Iso e (M ) the subgroup of Iso(M ) such that g σ = id for all g ∈ Iso e (M ) -in particular G = Iso e (X) for a symmetric space X. Let g ∈ Iso e (M ) such that g stabilizes two points g ± ∈ Flag τ with g − op g + . Since g is an isometry, it maps every geodesic connecting points of the interior of g − and g + to another geodesic connecting the same points. In particular g stabilizes P (g − , g + ) set-wise.
In the preliminaries we have seen that P (g − , g + ) splits as a product a τ × CS(g − , g + ) such that g ± are identified with the positive and negative, respectively, maximal dimensional simplices in a τ , i.e. g + ≃ ∂ ∞ a + τ where a + τ ∶= a τ ∩ a + . Therefore, g being an isometry of a τ stabilizing each boundary point, yields that g acts as a translation on a τ . Let ℓ τ g denote the translation vector.
Proof. We remark that cr τ (g − , g ⋅ x, g + , x) is independent of the choice of x op g ± . This follows from the symmetries of cr τ together with Proposition 3.11. Therefore, we fix one x ∈ Flag τ with x op g ± .
Let o ∈ P (g − , g + ) and ξ i with i ∈ J τ the corners of τ . By assumption x op g ± and hence g ⋅ x op g ± . Then Proposition 3.4 yields (o, g⋅o). If we plug this in the definition of cr ξ i , several terms cancel and we get cr ξ i (g − , g ⋅ x, g + , x) . In particular Since o was arbitrary in P (g − , g + ) we can assume that its first coordinate under the identification P (g − , g + ) ≃ a τ × CS(g − , g + ) is 0 ∈ a τ . Moreover, we can use Lemma 2.1 to see that only the first factor matters for the Busemann functions b g ξ i , b g ιξ i . As g acts as a translation on a τ , we have that g ⋅ 0 = ℓ τ g . argumentation around equation (3.3)). By assumption τ = ιτ , hence ι restricts to an isometry ι ∶ a τ → a τ . Together with ι 2 = id, this yields ⟨ιξ i , ℓ τ g ⟩ = ⟨ξ i , ιℓ τ g ⟩. Altogether we derive It is an immediate consequence that ⟨cr τ (g − , g ⋅ x, g + , x), ξ i ⟩ = 1 2 (⟨ξ i , ℓ τ g ⟩ + ⟨ξ i , ιℓ τ g ⟩ for all i ∈ J τ . Since the ξ i with i ∈ J τ form a basis of τ , it follows that cr τ (g − , g ⋅ x, g + , x) = 1 2 (ℓ τ g + ιℓ τ g ).
Let g ∈ Iso e (M ) be as before. Then the term cr τ (g − , g ⋅ x, g + , x) is also called period -in analogy to rank one spaces. In particular, the periods give rise to the translation vector of the first factor of the parallel set if ι = id.

Geometric interpretation of the cross ratio
In this section we give an explicit geometric interpretation of the vector valued cross ratio cr τ . Let x, z ∈ Flag τ and y, w ∈ Flag ιτ with x, z op y, w. Pick c x , c z , d y , d w ∈ Flag σ such that x is a face of c x and accordingly the other chambers and that c x , c z op d y , d w . Then we use the following notations for the horospherical retracts ρ x ∶= ρ cx,dy , ρ w ∶= ρ dw,cx , ρ z ∶= ρ cz,dw and ρ y ∶= ρ dy,cz .
Lemma 4.7. Let (x, y, z, w) ∈ A op τ and let ρ x , ρ w , ρ z and ρ y as above. Moreover, be o in the unique affine apartment joining c x and d y . Then for all Proof. Denote by A xy the unique affine apartment joining c x and d y . Then ρ dy,cx restricted to A xy is the identity, i.e. ρ dy,cx (o) = o. Therefore Proposition 3.1 implies that 2( By definition ρ y (o) is contained in the unique affine apartment joining c z and d y . Then in the same way it follows that (z y) ρy (o),ξ i = 0. Moreover, We can use Proposition 3.4 and again equation (2.2) to derive that In a very similar way we get

Using that cr
Proposition 4.8. Let ρ x , ρ w , ρ z and ρ y as before. Let o be in the unique affine apartment joining c x , d y such that we have under the identification P (x, y) ≃ a τ × CS(x, y) that π(o) = 0 ∈ a τ , where π is the projection to the first factor (also assume x ≃ a + τ ). Then 2cr τ (x, y, z, w) = π(ρ x ρ w ρ z ρ y (o)).
Proof. By construction we have that o, ρ x ρ w ρ z ρ y (o) are in the unique affine apartment joining c x and d y . Then by Lemma 2.1 and from similar arguments as around equation (3.3) we can derive that b x ξ i (o, ρ x ρ w ρ z ρ y (o)) = ⟨ξ i , π(ρ x ρ w ρ z ρ y (o))⟩ for all i ∈ J τ . Together with Lemma 4.7 and the definition of cr τ we get The ξ i ∈ a τ for i ∈ J τ form a basis of a τ . Moreover, for all i ∈ J τ we have that ⟨2cr τ (x, y, z, w), ξ i ⟩ = ⟨ξ i , π(ρ x ρ w ρ z ρ y (o))⟩. Thus it follows that 2cr τ (x, y, z, w) = π(ρ x ρ w ρ z ρ y (o)).

Cross ratio preserving maps
We assume in this section that τ is self opposite, i.e. τ = ιτ .
Lemma 5.2. Let x, y ∈ Flag τ . Then there exists z ∈ Flag τ with z op x, y.
Proof. We take c x , c y ∈ Flag σ such that x is a face of c x and y is a face c y . Then there exists c z ∈ Flag σ with c z op c x , c y [AV00, 5.1]. Be z the face of c z which is of type τ . Then z ∈ Flag τ with z op x, y. Proof. Let x, y ∈ Flag τ 1 (M 1 ) be given. Choose z 1 , z 2 , z 3 ∈ Flag τ 1 (M 1 ) such that z 3 op x; z 2 op y, z 3 and z 1 op x, z 2 . By Corollary 3.2 such that for all x, y ∈ Flag τ 1 (M 1 ) it holds that x op y if and only if f (x) op f (y) is called opposition preserving.
Proof. Assume there exist x ≠ y ∈ Flag τ 1 (M 1 ) with f (x) = f (y). Take a ∈ Flag τ 1 (M 1 ) with a op x and a ✚ ✚ op y: For example take an apartment which contains x and y. Take a opposite of x in this apartment. Then x ≠ y implies that a ✚ ✚ op y -opposite points are unique in apartments. In addition, choose z, w ∈ Flag τ 1 (M 1 ) such that z op a and w op z, x. Then cr ξ 1 (x, a, z, w) ≠ ±∞ and cr ξ (y, a, z, w) = −∞ or is not defined. However cr ξ 1 (x, a, z, w) = f * cr ξ 2 (x, a, z, w) = f * cr ξ 2 (y, a, z, w) = cr ξ 1 (y, a, z, w), which contradicts cr ξ 1 (x, a, z, w) ≠ cr ξ (y, a, z, w).
When restricting to the full flag space we can apply the following result due to Abramenko and van Maldeghem.
Proof. That f is a building isomorphism implies that B and B ′ are modeled over the same spherical Coxeter complex, i.e. over the Coxeter group W = W 1 × . . . × W k , where W i are irreducible Coxeter groups. The irreducibility of the buildings B i , B ′ i yields then that k = k ′ . Assume without loss of generality that W 1 ≤ W i for all i = 1, . . . , k. Let x 1 be a chamber in B 1 . Then x 1 is a simplex in B. We know that Res(x 1 ) is a spherical building over the spherical Coxeter complex to W 2 × . . . × W k . As f is a building isomorphism, we derive that f (Res(x 1 )) = Res(f (x 1 )) is a spherical building over W 2 × . . . × W k . If f (x 1 ) would not correspond to a chamber in an irreducible factor B ′ i , then there would be a subgroup W ′ of W isomorphic to W 2 × . . . × W k such that the projection of W ′ to each W i is non-trivial (as W 1 is minimal). This would yield a decomposition of W 2 × . . . × W k into k Coxeter groups, which contradicts the irreducibility of the factors. In particular, up to reordering Res(f (x 1 )) is a spherical building over W 1 × W 3 × . . . × W k and W 1 is isomorphic to W 2 . Thus f (x 1 ) = y 2 for a chamber y 2 ∈ B ′ 2 . Since f is a building isomorphism it maps all simplices of the same type as x 1 to simplices of the same type as y 2 i.e. it maps the chambers of B 1 to chambers of B ′ 2 . In particular, f induces a building isomorphism naturally a subset of B, namely the set of simplices of B fully contained in B 1 ) and A straight forward induction yields the result.
2 ) a ξ i 1 -Moebius bijection andM i 1 is the space M i 1 with its metric rescaled (for the types ξ i 1 see the proof ).
Let T be a rank one thick Euclidean building; in particular T is a tree. Then every geodesic segment in T lies in an affine apartment, i.e. in a bi-inifinite geodesic. This means that T is geodesically complete (in the notation of [BS17]). Moreover, by definition of thickness for rank one Euclidean buildings we have that ∂ ∞ T ≥ 3.
We remark that rk(T ) = 1 implies that the positive chamber of the Coxeter complex σ T consists of a single point. Thus ∆ ∞ T = Flag σ (T ) = ∂ ∞ T . Hence there is a unique Gromov product (⋅, ⋅) o T for any o T ∈ T on ∂ ∞ T 2 and a unique cross ratio cr T on A T ⊂ ∂ ∞ T 4 .
Proposition 5.10. Let E 1 , E 2 be irreducible thick combinatorial Euclidean buildings. Then every Moebius bijection f ∶ Flag σ (E 1 ) → Flag σ (E 2 ) is the restriction of an isometry F ∶Ê 1 → E 2 to the boundary whereÊ 1 is E 1 with its metric rescaled. If E 1 is not the cone over a spherical building, then F is unique.
Proof. If the rank is one, then the result follows from the theorem above.
If the rank is 2, Struyve has shown in [Str16] that every isometry between ∂ ∞ E 1 and ∂ ∞ E 2 with respect to the Tits metric is induced by an isometry after rescaling the metric on E 1 . The isometry is unique if E 1 is not the cone over a spherical building. We know that f induces a building isomorphism f ∶ ∆ ∞ E 1 → ∆ ∞ E 2 and this yields an isometry f ∶ ∂ ∞ E 1 → ∂ ∞ E 2 with respect to the Tits metric when viewing simplices as subset of ∂ ∞ E i . Hence we can apply the result of Struyve.
Corollary 5.11. Let E 1 and E 2 be combinatorial Euclidean buildings and let f ∶ Flag σ (E 1 ) → Flag σ (E 2 ) Moebius bijection. Then one can rescale the metric of E 1 on irreducible factors -denote this space byÊ 1 -such that f is the restriction of an isometry F ∶Ê 1 → E 2 to the boundary. If none of the irreducible factors is a cone over a spherical building the isometry F is unique.
Proof. This follows from Lemma 5.8 together with the proposition above.

Symmetric spaces
We want to show that the above corollaries hold in a similar way for symmetric spaces. Therefore we essentially only need to show that Moebius bijections are homeomorphisms. Hence we analyze some topological properties of Moebius bijections for the case of symmetric spaces.
In this section we only consider symmetric spaces X. For r ∈ R, x 2 , y 1 , y 2 ∈ Flag τ (X) and ξ ∈ int(τ ) we define Those sets are open by the continuity of cr ξ and the fact that A ξ is open. However, it can happen that they are empty -which holds if x 2 ✚ ✚ op y 1 , y 2 .
Proposition 5.12. Let X be a symmetric space. The sets B − r,ξ (y 1 , x 2 , y 2 ) varying over all r ∈ R and all x 2 , y 1 , y 2 ∈ Flag τ form a subbase of the topology on Flag τ (X) Proof. As mentioned, those sets are open. Thus we need to show that any open neighborhood U of a point x ∈ Flag τ (X) contains a open neighborhood V won by finite intersection and arbitrary union of sets of the form B − r,ξ (y 1 , x 2 , y 2 ). Let x ∈ Flag τ (X) and let any neighborhood U of x be given. We set K ∶= Flag τ U . Then K is compact and x ∉ K.
For any a ∈ K, choose y a ∈ Flag τ (X) such that y a op a and y a ✚ ✚ op x. In addition, choose w a , z a ∈ Flag τ (X) such that w a op a, x and z a op y a , w a . This yields cr ξ (x, y a , z a , w a ) = −∞ and cr ξ (a, y a , z a , w a ) > r a for some r a ∈ R and hence x ∈ B − ra,ξ (y a , z a , w a ), x ∉ B + ra,ξ (y a , z a , w a ), a ∈ B + ra,ξ (y a , z a , w a ). Varying over all a ∈ K the sets B + ra,ξ (y a , z a , w a ) cover K and by compactness we find a finite number of points a i ∈ K, i = 1, . . . , l such that the according sets already cover K. We set V ∶= ⋂ a i ∶i=1,...,l B − ra i ,ξ (y a i , z a i , w a i ). As a finite intersection of open sets, V is open. Furthermore, x ∈ V and hence V is non-empty. By construction V ⊂ K C and hence V ⊂ U .
Proof. Since f leaves the cross ratio invariant and is a bijection, it is immediate that f (B − r,ξ 1 (y, z, w)) = B − r,ξ 2 (f (y), f (z), f (w)). This means that f yields a bijection of subbases of the topology and hence f is a homeomorphism.
As mentioned, for a symmetric space X the boundary Flag τ (X) can be identified homeomorphically with G P x for P x = stab(x) and x ∈ Flag τ (X). Hence Flag τ (X) can be given the structure of compact connected manifold (without boundary) -inherited from G P x . Using this there is a different way to characterize Moebius bijections captured in the following lemma.
We derive that Im(f ) is compact connected submanifold of Flag τ 2 (X 2 ) of the same dimension. However, Flag τ 2 (X 2 ) is a compact connected manifold without boundary and hence the only such submanifold is Flag τ 2 (X 2 ) itself, i.e. Im(f ) = Flag τ 2 (X 2 ) -which proves the claim.
Theorem 5.15. Let X 1 , X 2 be symmetric spaces of non-compact type of rank at least two with no rank one de Rham factors and let f ∶ Flag σ (X 1 ) → Flag σ (X 2 ) be a ξ 1 -Moebius bijection. Then one can multiply the metric of X 1 by positive constants on de Rham factors -denote this space byX 1 -such that f is the restriction of an unique isometry F ∶X 1 → X 2 to Flag σ (X 1 ).
Proof. We know that a ξ 1 -Moebius bijection f ∶ Flag σ (X 1 ) → Flag σ (X 2 ) can uniquely be extended to a building isomorphism f ∶ ∆ ∞ X 1 → ∆ ∞ X 2 . Moreover, f is a homeomorphism on the chamber sets Flag σ (X i ) by Lemma 5.13. Then for such maps the result is known [Ebe96,Sc.3.9].
Actually all we need for the above result is that f ∶ Flag σ (X 1 ) → Flag σ (X 2 ) is opposition preserving and a homeomorphism. However, when dealing also with rank one factors we really need Moebius maps.
Corollary 5.16. Let X 1 and X 2 be symmetric spaces of non-compact type and let f ∶ Flag σ (X 1 ) → Flag σ (X 2 ) be a Moebius bijection. Then one can rescale the metric of X 1 on de Rham factors -denote this space byX 1 -such that f is the restriction of an unique isometry F ∶X 1 → X 2 to the boundary.
Proof. This follows from Lemma 5.8 together with the theorem above and the fact that Moebius bijections of rank one symmetric spaces can be uniquely extended to isometries. For the latter result see [Bou96].
One should be able to derive the same (or at least similar) topological properties of the cross ratios and Moebius maps for Bruhat-Tits buildings, i.e. combinatorial Euclidean building associated to semi-simple algebraic groups over non-archimedian local field with finite residue field. However, since in this case Proposition 5.10 and Corollary 5.11 already yield the main result, we have not included these topological properties for simplicity.
It is immediate to see that f is induced by a map F ∶= , i ≠ j is the identity (under the natural identification with Flag σ (M 0 )). As F and hence the F i shall be isometries, it follows that F (p, q) = (q, p) and clearly F is an isometry only after rescaling on de Rham factors.
Let M 1 be a symmetric space or a combinatorial Euclidean building and assume that the image of cr σ,M 1 lies not in a proper subspace of a M 1 . Then the above situation is essentially the only possibility where rescaling can appear: Let M 1 , M 2 be irreducible -actually one of them being irreducible would be enough. In addition, be f ∶ Flag σ (M 1 ) → Flag σ (M 2 ) a ξ 1 -Moebius bijection, i.e. cr ξ 1 = f * cr ξ 2 . Then we know that we can rescale the metric on M 1 by some positive number µ 1 , such that f is induced by an isometry However, it follows from the assumption on cr σ,M 1 together with Lemma 4.3 that cr ξ ≠ αcr ξ ′ for ξ ≠ ξ ′ ∈ σ 1 and any α ∈ R. Therefore cr ξ 1 ,M 1 = f * cr ξ 2 = µ 1 cr ξ ′ 1 ,M 1 implies ξ 1 = ξ ′ 1 and µ 1 = 1 -in particular f is induced by an isometry without rescaling the metric.
We remark that for symmetric spaces with ι = id the image of cr σ is all of a. This follows from the fact that every vector of a can be realized as a translation vector of a hyperbolic element in G. Then the periods of those elements in G are exactly those translation vectors, as seen in Proposition 4.6. Hence the above discussion applies.
Corollary 5.17. Let M either be a symmetric space or a combinatorial Euclidean building with none of the irreducible factors being a cone over a spherical building. In addition, assume that the image of cr σ is not contained in a proper subspace of a. Let ξ 0 ∈ σ be the center of gravity of σ. Then there is a one-to-one correspondence between Iso(M ) and ξ 0 -Moebius bijections.
Proof. Let g ∈ Iso(M ) and g σ ∶ σ → σ the induced map. Then g σ is an isometry with respect to the angular metric, hence g σ stabilizes the center of gravity ξ 0 of σ. Therefore Proposition 3.11 yields a ξ 0 -Moebius bijections for each g ∈ Iso(M ).
On the other hand, by Corollaries 5.11 and 5.16, we know that each ξ 0 -Moebius bijections is induced by a unique isometry -after possible rescaling on irreducible factors. However, following the above discussion we can exclude rescaling of the metric: Let f be a ξ 0 -Moebius bijection and let f = f 1 × . . . × f k be the decomposition on irreducible factors M 1 , . . . , M k as in Lemma 5.8. Assume w.l.o.g. that f 1 ∶ Flag σ (M 1 ) → Flag σ (M 2 ), i.e. M 1 , M 2 are isometric after possibly rescaling the metric. From Proposition 3.14 we know cr ξ 0 = µ 1 cr ξ 1 ,M 1 + µ 2 cr ξ 2 ,M 2 + . . . + µ k cr ξ k ,M k . However, ξ 0 ∈ σ being the center of gravity of σ and M 1 , M 2 isometric after possibly rescaling the metric implies µ 1 = µ 2 and ξ 1 ≃ ξ 2 . Then f 1 is ξ 1 -Moebius bijection between irreducible spaces. From the above discussion it follows that it is induced by an isometry without rescaling the metrics. The same argument implies the result for all f i and hence the claim follows.

General Euclidean buildings
In this section we consider general Euclidean buildings, i.e. in particular nonlocally compact ones. The goal is again to show that Moebius bijections are induced by isometries. However, now we will need the vector valued cross ratio cr σ to derive such a result.
Let E be a thick Euclidean building considered with the complete apartment system. Let x ∈ Flag τ (E) and y ∈ Flag ιτ (E) with x op y and τ is a codimension 1 face of σ -in this case x, y are called panels of the building ∆ ∞ E. Then metrically we have the splitting P (x, y) ≃ a τ × CS(x, y), where CS(x, y) is a Euclidean building of rank rk(E) − dim a τ = 1, i.e. CS(x, y) is an R-tree. This tree is called wall tree and will be denoted by T xy . One can show that the isomorphism type of T xy does not depend on the choice of y ∈ Flag ιτ with y op x [KW14]; hence the isomorphism class of T xy will be denoted by T x .
We recall that the residue of an element z ∈ ∆ ∞ E is defined by Res(z) = {w ∈ ∆ ∞ E z ⊊ w}. In case of a panel x ∈ ∆ ∞ E we have that Res(x) consists of all the chambers in ∆ ∞ E containing x.
It is known that one can naturally identify Res(x) ≃ ∂ ∞ T x . For convenience of the reader we describe this identification: Fix y op x. Then T x ≃ T xy . Let o ∈ P (x, y). Then one can identify the chambers in Res(x) with (specific) Weyl sectors in P (x, y) with tip o [Par00, Cor. 1.9.]. Pick o ∈ P (x, y) such that we can identify P (x, y) ≃ a τ × T xy , o ≃ (0, o T ) and x ≃ ∂ ∞ (a τ ∩ a + ) -i.e. x corresponds to the positive chamber in a τ . Recall that a + τ = a τ ∩a + . Then the affine apartments in P (x, y) ≃ a τ ×T xy containing o are of the form a τ × γ, where γ is bi-infinite geodesic ray in T xy passing through o T (those are easily seen to be isometric to R r ). By definition every Weyl sector is contained in an affine apartment; hence we can derive that every Weyl sector with tip o and boundary chamber c ∈ Res(x) is contained in a + τ × γ o T z where γ o T z is a geodesic in T xy from o T to a boundary point z ∈ ∂ ∞ T xy . This yields a one-to-one correspondence of Res(x) with geodesic rays emanating from o T . As those rays are in one-to-one correspondence with ∂ ∞ T xy , we get Res(x) ≃ ∂ ∞ T x as claimed.
Remark 5.18. It follows that for z ∈ ∂ ∞ T xy , c ∈ Res(x) and d ∈ Res(y) we have that z ≃ c and z ≃ d under Res(x) ≃ ∂ ∞ T xy , Res(y) ≃ ∂ ∞ T xy respectively if and only if the Weyl sectors with By definition Res(x) is the set of chambers that contain x. Hence there is a unique corner ξ x of σ such that c ξx ∉ x for every chamber c ∈ Res(x). In the same way we get a type from y and it is immediate that this type equals ιξ x -following for example from the fact that x ∈ Flag τ implies that y ∈ Flag ιτ .
Lemma 5.19. Let x, y be opposite panels in ∆ ∞ E and T xy the associated tree. Let z c , z d ∈ ∂ ∞ T xy , c ∈ Res(x) such that c ≃ z c under Res(x) ≃ ∂ ∞ T xy and d ∈ Res(y) such that d ≃ z d under Res(y) ≃ ∂ ∞ T xy . Then (c d) o,ξx = sin(α)(z c z d ) o T where o ≃ (0, o T ) under P (x, y) ≃ a τ × T xy and α ∈ (0, π) does only depend on σ and the type of x.
Proof. Let γ c , γ d be the geodesics in P (x, y) from o to c ξx and d ξy , respectively. The splitting P (x, y) ≃ a τ × T xy gives geodesics γ x , γ y in a τ eminating from 0 and γ zc , γ z d in T xy eminating from o T such that γ c (t) = (γ x (t), γ zc (t)) and γ d (t) = (γ y (t), γ z d (t)) -while γ c , γ d are unit speed, the geodesics γ x , γ y , γ zc and γ z d are not. It is clear that the geodesics γ x , γ y do not depend on the choice of c, d and are in opposite directions (since the γ c , γ d are): The geodesics γ c , γ d are along those corners of Weyl sectors that are not contained in a τ . Since Weyl sectors are isometric to convex subsets of R r , it reduces to Euclidean geometry; for example γ x is the geodesic in a τ from 0 to π τx (ξ x ), where π τx is the orthogonal projection from σ to τ x and τ x is the type of x. In particular we have d(γ x (t), γ y (t)) = 2t.
Let from now on γ x , γ y , γ zc and γ z d be the geodesics as above but now parametrized such that they are unit speed. Let α be the angle of ξ x and π τx (ξ x ). Then we have γ c (t) = (γ x (cos(α)t), γ zc (sin(α)t)). Basic facts of trees imply that d(γ zc (t), γ z d (t)) = 2t − 2(z c z d ) o T for t ≥ (z c z d ) o T -see e.g. [BS17]. Together with d(γ x (t), γ y (t)) = 2t we get while the last equality follows from a Taylor series in the same way as we have seen several times before.
Corollary 5.20. The cross ratio on ∂ ∞ T xy is given by cr Txy (z 1 , w 1 , z 2 , w 2 ) = sin(α)cr ξx (c 1 , d 1 , c 2 , d 2 ) where ξ x ∈ σ is the corner not contained in τ x , the type of x, α is the angle between ξ x and τ x , The thickness of E means that ∆ ∞ E is thick and therefore we have for every panel x that ∂ ∞ T x ≥ 3 (as Res(x) ≃ ∂ ∞ T x ), i.e. T x is thick and geodesically complete. Therefore Theorem 5.9 implies that the whole isometry class T x has a natural cross ratio cr Tx .
In a similar way as before, we call a surjective map f ∶ Flag σ 1 (E 1 ) → Flag σ 2 (E 2 ) such that cr σ 1 (x, y, z, w) = f * cr σ 2 (x, y, z, w) for all (x, y, z, w) ∈ A σ 1 a σ 1 -Moebius bijection. We remark that to identify the image of cr σ 1 with the one of cr σ 2 it is already necessary that E 1 and E 2 are modeled over the same spherical Coxeter complex, i.e. σ 1 ≃ σ 2 =∶ σ.
It is immediate that such a map is a ξ 0 -Moebius map, for ξ 0 the center of gravity of σ. We assume f to be surjective, hence f is a ξ 0 -Moebius bijection and therefore can be extended uniquely to a building isomorphism f ∶ ∆ ∞ E 1 → ∆ ∞ E 2 by Proposition 5.6.
We recall that the affine Weyl groupŴ = W ⋉T W of the Coxeter complex over which a Euclidean building is defined gives a collection of hyperplanes, namely the hyperplanes of the finite reflection group W together with all its translates under T W . Each hyperplane defines two half spaces which we call affine half apartments. The image of an affine half apartment under a chart map is again called affine half apartment.
In spherical buildings the hyperplanes associated to the spherical Coxeter group define walls in apartments and those walls separate the apartments in two halfs, called half apartments. One can show that the boundary of an affine half apartment H ⊂ E defines a half apartment in H ∞ ⊂ ∆ ∞ E and to every half apartment in H ∞ ⊂ ∆ ∞ E we find an affine half apartment H ⊂ E which has H ∞ as its boundary.
Proof. Let z ∈ ∂ ∞ T xy , i.e. z is an equivalence class of geodesic rays. Every ray γ z in the class starting at a branching point defines an affine half apartment a τ × γ z in E 1 and thus (the equivalence class of rays) defines a half apartment H ∞ ⊂ ∆ ∞ E 1 . Then it follows form Remark 5.18 that c ≃ z with c ∈ Res(x) if and only if c is contained in the half apartment H ∞ and in the same way d ≃ z with d ∈ Res(y) if and only if d is contained in the half apartment H ∞ . By assumption, f is a building isomorphism, i.e. f (H ∞ ) ⊂ ∆ ∞ E 2 is a half apartment with f (x), f (y) ∈ f (H ∞ ). The metric splitting P (f (x), f (y)) ≃ a τ × T f (x)f (y) yields that we find an affine half apartment a τ × γ w with γ w a geodesic ray in T f (x)f (y) and boundary point w ∈ ∂ ∞ T f (x)f (y) such that the boundary of this affine half apartment is exactly f (H ∞ ). By definition f (c), f (d) ∈ f (H ∞ ). Hence from Remark 5.18 we get that f (c) ≃ w ≃ f (d). Therefore f x (z) = w and f y (z) = w.
Theorem 5.24. Let E 1 , E 2 be thick irreducible Euclidean buildings. Let f ∶ Flag σ (E 1 ) → Flag σ (E 2 ) be a σ-Moebius bijection. Then the induced isomorphism f ∶ ∆ ∞ E 1 → ∆ ∞ E 2 is ecological and hence can be extended to an isomorphism F ∶ E 1 → E 2 , i.e. an isometry after possibly rescaling the metric on E 1 Proof. What we need to show is, given a panel x ∈ ∆ ∞ E 1 , the induced map is the restriction of an isometry. This implies that f is ecological and therefore by the Theorem of Tits induced by an isomorphism.
We fix y op x to get a tree T xy in the class of T x . Since we are considering isometry classes of trees, it is enough to show that f xy ∶ ∂ ∞ T xy → ∂ ∞ T f (x)f (y) is induced by an isometry.
From Corollary 5.20 we derive for z 1 , w 1 , z 2 , w 2 ∈ ∂ ∞ T xy and c 1 , c 2 ∈ Res(x), d 1 , d 2 ∈ Res(y) with z i ≃ c i , w i ≃ d i that there is some α ∈ (0, π) with cr Txy (z 1 , w 1 , z 2 , w 2 ) = sin(α)cr ξx (c 1 , d 1 , c 2 , d 2 ) = sin(α)f * cr ξx (c 1 , d 1 , c 2 , d 2 ), while the last equality follows from f being a σ-Moebius bijection. By construction f xy ∶ ∂ ∞ T xy ≃ Res(x) → ∂ ∞ T f (x)f (y) ≃ Res(f (x)) is defined in the way that f (c 1 ) ≃ f xy (z 1 ) under ∂ ∞ T f (x)f (y) ≃ Res(f (x)) and similar for c 2 . In light of Lemma 5.23 we have that f (d i ) ≃ f xy (w i ). Applying again Corollary 5.20 this yields that sin(α)f * cr ξx (c 1 , d 1 , c 2 , d 2 ) = f * xy cr T f (x)f (y) (z 1 , w 1 , z 2 , w 2 ) -we remark that the α is the same as before as the simplices σ 1 and σ 2 coincide. Hence f xy is a Moebius bijection. Since T xy is a geodesically complete tree and the thickness of E 1 implies that ∂ ∞ T xy ≥ 3 we can apply Theorem 5.9 to derive that f xy is induced by an isometry.
Corollary 5.25. Let E 1 and E 2 be thick Euclidean buildings and let f ∶ Flag σ (E 1 ) → Flag σ (E 2 ) be a σ-Moebius bijection. Then we can rescale the metric on the irreducible factors of E 1 -denote this space byÊ 1 -such that f is the restriction of an isometry F ∶Ê 1 → E 2 to the boundary.
Proof. From Lemma 4.3 we know that for every type ξ ∈ σ we have that f * cr ξ = cr ξ , as f is a σ-Moebius bijection. Let σ = σ 1 ○ . . . ○ σ k be the decomposition of σ corresponding to the decomposition of E i into irreducible factors -the decompositions coincide as both buildings are thick and modeled over the same spherical Coxeter complex. Moreover, be f = f 1 × . . . × f k the decomposition from Lemma 5.8. Then f * cr ξ = cr ξ for all ξ ∈ σ implies that each f i is a σ i -Moebius bijection. Thus the above theorem yields the claim.
Corollary 5.26. Let E 1 , E 2 be thick irreducible Euclidean buildings. Moreover, assume that there exists a wall tree T x for a panel x ∈ ∆ ∞ E 1 which has more than one branching point. Let f ∶ Flag σ (E 1 ) → Flag σ (E 2 ) be a σ-Moebius bijection. Then f can be extended to an isometry F ∶ E 1 → E 2 (without rescaling the metric).
Moreover, if E 1 is not a Euclidean cone over a spherical building then every wall tree has more than one branching point.
Proof. From Theorem 5.24 we know that we can rescale the metric by some µ ∈ R + such that f is induced by an isometry F ∶ µE 1 → E 2 , where µE 1 is E 1 with the metric rescaled by µ. Let x ∈ ∆ ∞ E 1 be a panel such that the wall tree T x has more than one branching point. Then clearly the wall tree of x ∈ ∆ ∞ µE 1 is µT x . Let f x ∶ ∂ ∞ T x → ∂ ∞ T f (x) be the induced map from f on the wall tree. Since F restricted to the boundary is f , the map induced from F on ∂ ∞ µT x equals f x . Therefore we have cr Tx = f * x cr T f (x) = cr µTx = µcr Tx (the first equality follows from f being a σ-Moebius bijection, the second from f x = F ∂∞µTx ).
By assumption T x has two branching points. The distance of those two points can be given in terms of the cross ratio -i.e. let p, q ∈ T x be the branching points, then there exist z 1 , z 2 , w 1 , w 2 ∈ ∂ ∞ T x such that d(p, q) = cr Tx (z 1 , w 1 , z 2 , w 2 ) [BS17, Lem 4.2]. Since this distance d(p, q) is non-zero, we derive from cr Tx (z 1 , w 1 , z 2 , w 2 ) = µcr Tx (z 1 , w 1 , z 2 , w 2 ) that µ = 1. Hence F is an isometry without rescaling the metric on E 1 .
The second claim is a direct consequence of Proposition 4.21. and 4.26 in [KW14].
The second claim of Theorem B follows now from the fact that every σ-Moebius bijection splits as a product of σ i -Moebius bijections on irreducible factors, as in the proof of Corollary 5.25. The corollary above implies that those σ i -Moebius bijections induce isometries without the need of rescaling.

Appendix
Here, we determine the cross ratio for the symmetric spaces X(n) ∶= SL(n, R) SO(n, R). We will use the notation as in Example 3.13.
The map g ⋅ SO(n, R) ↦ gg t yields an identification of X(n) with P n = {A ∈ Mat(n × n, R) A = A t ∧ det(A) = 1 ∧ A is positive definit}.
The action of g ∈ SL(n, R) on A ∈ P n is given by g ⋅ A = gAg t . By the definition of the cross ratio, it will be enough to determine (⋅ ⋅) In ,λ with I n being the identity matrix in P n and λ = (λ 1 , . . . , λ l ) be identified with some type. We begin with considering types λ = (λ 1 , . . . , λ n ) ∈ int(σ): Let S = (V 1 , . . . , V n ) with V i = span{e 1 , . . . , e i } denote the standard flag -the e i being the standard base of R n . We know that K = SO(n, R) acts transitively on Flag σ , hence we need to determine (k 1 S k 2 S) In,λ for k 1 , k 2 ∈ SO(n, R).
Proof. Since (⋅ ⋅) In ,λ is invariant under the action of SO(n, R), we have (k ′ S hS) In,λ = (h −1 k ′ S S) In,λ , i.e. it reduces to determine (kS S) In,λ or in the same way (S kS) In,λ for any k ∈ SO(n, R).
Proposition 3.1 implies that (S kS) In,λ = 1 2 b S λ (I n , n kS (I n , S) ⋅ I n ), where S λ is point in the ideal boundary ∂ ∞ X(n) determined by the eigenvalue flag pair (λ, S) and n kS (I n , S) ∈ N kS , i.e. the element in the horospherical subgroup to kS such that n kS (I n , S) ⋅ I n ∈ P (kS, S).
We want to determine n kS (I n , S) ⋅ I n . First, we remark that N kS = k N S k −1 = k N S k t . The group N S consists of upper triangular matrices with ones on the diagonal. Lemma 6.2. Notations as before, in particular let λ = (λ 1 , . . . , λ l ) be a type and S τ , S ιτ the associated standard flags. Then (kS τ hS ιτ ) In,λ = n l−1 j=1 (λ j+1 − λ j ) log det(k 1 ⋯ k i j ĥ 1 ⋯ ĥ n−i j ) withĥ i denoting the i-th column of the matrix h and accordinglyk i .
Proof. This is a direct consequence of the lemma above together with Lemma 3.9.
Proof. As mentioned in Example 3.13, the term is independent of the choices made. By the transitivity of the SO(n, R) action, we know that every flag V ∈ Flag τ can be written as hS τ for S τ ∈ Flag τ the standard flag and some k ∈ SO(n, R). Then the columnsk i are such that V j = span{k 1 , . . . ,k i j }. In the same way every flag W ∈ Flag ιτ can be written as hS ιτ for S ιτ ∈ Flag ιτ the standard flag and some k, h ∈ SO(n, R).
We fix the identification ∧ n R n ≃ det. Let k, h ∈ SO(n, R) such that V = kS τ and W = hS ιτ . Then V j ∧ W l−j = det(k 1 ⋯ k i j ĥ 1 ⋯ ĥ n−i j ) .
Therefore the claim follows from the lemma above.