Auguste Hébert
email: auguste.hebert@univ-lorraine.fr

Distances on a masure

A masure (also known as an affine ordered hovel) I is a generalization of the Bruhat-Tits building that is associated to a split Kac-Moody group G over a nonarchimedean local field. This is a union of affine spaces called apartments. When G is a reductive group, I is a building and there is a G-invariant distance inducing a norm on each apartment. In this paper, we study distances on I inducing the affine topology on each apartment. We construct distances such that each element of G is a continuous automorphism of I and we study their properties (completeness, local compactness, ...).

5 Mixed distances 25 5.1 Comparison of positive and negative topologies . . . . . . . . . .

Introduction

If G is a split Kac-Moody group over a nonarchimedean local field, Stéphane Gaussent and Guy Rousseau introduced a space I on which G acts and they called this set a "masure" (or an "affine ordered hovel"), see [START_REF] Gaussent | Kac-Moody groups, hovels and Littelmann paths[END_REF], [START_REF] Rousseau | Almost split Kac-Moody groups over ultrametric fields[END_REF]. This construction generalizes the construction of the Bruhat-Tits building associated to a split reductive group over a field equipped with a nonarchimedean valuation made by François Bruhat and Jacques Tits, see [START_REF] Bruhat | Tits. Groupes réductifs sur un corps local[END_REF] and [START_REF] Bruhat | Tits. Groupes réductifs sur un corps local[END_REF]. A masure is an object similar to a building. It is a union of subsets called "apartments", each one having a structure of a finite dimensional real-affine space and an additional structure defined by hyperplanes (called walls) of this affine space. The group G acts transitively on the set of apartments. It induces affine maps on each apartment, sending walls on walls. We can also define sectors and retractions from I onto apartments with center a sector-germ, as in the case of Bruhat-Tits buildings. However there can be two points of I which do not belong to a common apartment. Studying I enables one to get information on G and this is one reason to study masures.

In this paper, we assume the valuation of the valued field to be discrete. Each Bruhat-Tits building BT associated to a split reductive group H over a field equipped with a discrete nonarchimedean valuation is equipped with a distance d such that H acts isometrically on BT and such that the restriction of d to each apartment is a euclidean distance. These distances are important tools in the study of buildings. We will show that we cannot equip masures which are not buildings with distances having these properties but it seems natural to ask whether we can define distances on a masure which:

• induce the topology of finite-dimensional real-affine space on each apartment,

• are compatible with the action of G,

• are compatible with retractions centered at a sector-germ.

We show that under the assumption of continuity of retractions, the metric space we have is never complete nor locally compact (see Subsection 3.3). We show that there is no distance on I such that the restriction to each apartment is a norm. However, for each sector-germ s of I, we construct distances having the following properties (Corollary 38, Lemma 27, Corollary 39 and Theorem 42):

• the topology induced on each apartment is the affine topology,

• each retraction with the center s is 1-Lipschitz,

• each retraction with center a sector-germ of the same sign as s is Lipschitz,

• each g ∈ G is Lipschitz when we regard it as an automorphism of I.

We call them distances of positive or of negative type, depending on the sign of s. We prove that all distances of positive type on a masure (resp. of negative type) are equivalent, where two distances d 1 and d 2 are said to be equivalent if there exist k, ∈ R >0 such that kd 1 ≤ d 2 ≤ d 1 (this is Theorem 37). We thus get a positive topology T + and a negative topology T -defined by distances of ± types. We prove (Corollary 46) that these topologies are different when I is not a building. When I is a building these topologies agree with the usual topology on a building (Proposition 43).

Let I 0 be the G-orbit in I of some special vertex. If I is not a building, I 0 is not discrete for both T -and T + . We also prove that if ρ is a retraction centered at a negative (resp. positive) sector-germ, ρ is not continuous for T + (resp. T -), see Proposition 45. For these reasons we introduce mixed distances, which are sums of a distance of positive type with a distance of negative type. We then have the following (Theorem 47): all the mixed distances on I are equivalent; moreover, if d is a mixed distance and I is equipped with d then:

• each g : I → I ∈ G is Lipschitz,

• each retraction centered at a sector-germ is Lipschitz,

• the topology induced on each apartment is the affine topology,

• the set I 0 is discrete.

The topology T m associated to mixed distances is the initial topology with respect to the retractions of I (see Corollary 51).

We prove that I is contractible for T + , T -and T m .

Let us explain how to define distances of positive or negative type. Let A be the standard apartment of I and C v f be the fundamental chamber of A. Let s be a sector-germ of I. After applying some g ∈ G to A, we may assume A = A and that s is the germ +∞ of C v f (or of -C v f but this case is similar). Fix a norm | . | on A. For every x ∈ I, there exists an apartment A x containing x and +∞ (which means that A x contains a sub-sector of C v f ). For u ∈ C v f , we define x + u as the translate of x by u in A x . If u is chosen to be sufficiently dominant, x + u ∈ C v f . Therefore, for all x, x ∈ I, there exist u, u ∈ C v f such that x + u = x + u . We then define d(x, x ) to be the minimum of the |u| + |u | for such couples u, u .

We thus obtain a distance for each sector-germ and for each norm | . | on A.

This paper is organized as follows.

In Section 2, we review basic definitions and set up the notation.

In Section 3, we show that if s is a sector-germ of I, we can write each apartment as a finite union of closed convex subsets each of which is contained in an apartment A containing s. The most important case for us is when A contains a sector-germ adjacent to s. We then can write A as the union of two half-apartments, each contained in an apartment containing s. We conclude Section 3 with a series of properties that distances on I cannot satisfy.

In Section 4, we construct distances of positive and negative type on I. We prove that all the distances of positive type (resp. negative type ) are equivalent. We then study them.

In Section 5, we first show that when I is not a building, T + and T -are different. Then we define mixed distances and study their properties.

In Section 6, we show that I is contractible for the topologies T + , T -and T m .

Masures

In this section, we review the theory of masures. We restrict our study to semi-discrete masures which are thick of finite thickness and such that there exists a group acting strongly transitively on them (we define these notions at the end of the section). These properties are satisfied by masures associated to split Kac-Moody groups over nonarchimedean local fields (see [START_REF] Rousseau | Groupes de Kac-Moody déployés sur un corps local II. Masures ordonnées[END_REF]). To avoid introducing too much notation, we do not treat the case of almost split Kac-Moody groups (see [START_REF] Rousseau | Almost split Kac-Moody groups over ultrametric fields[END_REF]). By adapting Lemma 4, one can prove that our results remain valid in the almost split case. We begin by defining the standard apartment. References for this section are [START_REF] Kac | Infinite-dimensional Lie algebras[END_REF], Chapter 1 and 3, [GR08] Section 2 and [GR14] Section 1.

Root generating system

A Kac-Moody matrix (or generalized Cartan matrix) is a square matrix C = (c i,j ) i,j∈I with integer coefficients, indexed by a finite set I and satisfying:

1. ∀i ∈ I, c i,i = 2 2. ∀(i, j) ∈ I 2 |i = j, c i,j ≤ 0 3. ∀(i, j) ∈ I 2 , c i,j = 0 ⇔ c j,i = 0. A root generating system is a 5-tuple S = (C, X, Y, (α i ) i∈I , (α ∨ i ) i∈I
) made of a Kac-Moody matrix C indexed by I, of two dual free Z-modules X (of characters) and Y (of co-characters) of finite rank rk(X), a family (α i ) i∈I (of simple roots) in X and a family (α ∨ i ) i∈I (of simple coroots) in Y . They have to satisfy the following compatibility condition: c i,j = α j (α ∨ i ) for all i, j ∈ I. We also suppose that the family (α i ) i∈I (resp. (α ∨ i ) i∈I ) freely generates a Z-submodule of X (resp. of Y )).

We now fix a Kac-Moody matrix C and a root generating system with the matrix C. Let A = Y ⊗ R. We equip A with the topology defined by its structure of a finitedimensional real-vector space. Every element of X induces a linear form on A. We will regard X as a subset of the dual A * of A: the α i , i ∈ I are viewed as linear forms on A. For i ∈ I, we define an involution r i of A by r

i (v) = v -α i (v)α ∨ i for all v ∈ A.
Its fixed points set is ker α i . The subgroup of GL(A) generated by the r i , i ∈ I is denoted by W v and is called the Weyl group of S. The system

(W v , {r i | i ∈ I}) is a Coxeter system. Let Q = i∈I Zα i and Q ∨ = i∈I Zα ∨ i .
The groups Q and Q ∨ are called the root lattice and the coroot-lattice.

One defines an action of the group W v on A * as follows:

if x ∈ A, w ∈ W v and α ∈ A * then (w.α)(x) = α(w -1 .x). Let Φ = {w.α i |(w, i) ∈ W v × I} be the set of real roots. Then Φ ⊂ Q. Let Q + = { i∈I n i α ∨ i |(n i ) ∈ Z ≥0 I } ⊂ Q, Q -= -Q + , Φ + = Φ ∩ Q + and Φ -= Φ ∩ Q -. Then Φ = Φ + ∪ Φ -.
The elements of Φ + (resp. Φ -) are called the real positive roots (resp. real negative roots). Let W a = Q ∨ W v ⊂ GA(A) be the affine Weyl group of S, where GA(A) is the group of affine automorphisms of A.

For α and k ∈ R, one sets

D(α, k) = {x ∈ A| α(x)+k = 0}, D • (α, k) = {x ∈ A| α(x)+k > 0} and M (α, k) = {x ∈ A| α(x) + k = 0}. One also sets D(α, +∞) = D • (α, +∞) = A and M (α, +∞) = ∅.
A wall (resp. a half-apartment) of A is a hyperplane (resp. a half-space) of the form M (α, k) (resp. D(α, k)) for some α ∈ Φ and k ∈ R. The wall (resp. half-apartment) is said to be a true wall (resp. a true half-apartment) if k ∈ Z and a ghost wall if k / ∈ Z. This choice of true walls means that the apartment (or the masure) is semi-discrete.

Vectorial faces and Tits preorder

Vectorial faces

Define C v f = {v ∈ A| α i (v) > 0, ∀i ∈ I}. We call it the fundamental chamber. For J ⊂ I, one sets F v (J) = {v ∈ A| α i (v) = 0 ∀i ∈ J, α i (v) > 0 ∀i ∈ J\I}. Then the closure C v f of C v
f is the union of the subsets F v (J) for J ⊂ I. The positive (resp. negative) vectorial faces are the sets w.F v (J) (resp. -w.F v (J)) for w ∈ W v and J ⊂ I. A vectorial face is either a positive vectorial face or a negative vectorial face. We call a positive chamber (resp. negative) every cone of the form w.C v f for some w ∈ W v (resp. -w.C v f ). By Section 1.3 of [START_REF] Rousseau | Masures affines[END_REF], the action of W v on the set of positive chambers is simply transitive. The Tits cone T is defined as the convex cone T = w∈W v w.C v f . We also consider the negative cone -T .

Tits preorder on A

One defines a W v -invariant relation ≤ on A by: x ≤ y ⇔ y -x ∈ T . This preorder need not be a partial order. For example, if W v is finite (i.e when the Kac-Moody matrix C defining S is a Cartan matrix), then T = A and thus x ≤ y for all x, y ∈ A. For an arbitrary Kac-Moody matrix C, every element x of i∈I ker(α i ) satisfies 0 ≤ x ≤ 0 and in particular when C is not invertible, ≤ is not a partial order. Let x, y ∈ A be such that x = y. The ray with the base point x and containing y (or an interval (x, y], (x, y), . . .) is called preordered if x ≤ y or y ≤ x and generic if y -x ∈ ± T , the interior of ±T .

Metric properties of W v

In this subsection we prove that when W v is infinite there does not exist a W v -invariant norm on A and we also establish a density property of the walls of A.

Two true walls M 1 and M 2 are said to be consecutive if they are of the form α -1 ({k}), α -1 ({k ± 1}) for some α ∈ Φ and some k ∈ Z.

Proposition 1.

1. Suppose that there exists a W v -invariant norm on A. Then W v is finite.

2. Let | . | be a norm on A, d be the induced distance on A and suppose that W v is infinite.

Then for every > 0 there exists a vectorial wall M 0 such that for all consecutive true walls M 1 and

M 2 parallel to M 0 , d(M 1 , M 2 ) < . Proof. Let B ⊂ Y dim A be a Z-basis of Y . Then the map W v → Y dim A sending each w to w.B is injective. Thus if W v is infinite, {w.B| w ∈ W v } is unbounded. Part 1 follows. Suppose that W v is infinite. Let (β n ) ∈ Φ Z ≥0 + be an injective sequence. Let > 0 and u ∈ C v f be such that |u| < . For n ∈ Z ≥0 , write β n = i∈I λ i,n α i , with λ i,n ∈ Z ≥0 for all (i, n) ∈ I × Z ≥0 . One has β n (u) = i∈I λ i,n α i (u) ≥ min i∈I α i (u) i∈I λ i,n → +∞. Let n ∈ Z ≥0 be such that β n (u) ≥ 1 and M 0 = β -1 n ({0}).
Then for all consecutive true walls M 1 and M 2 parallel to M 0 , d(M 1 , M 2 ) < , which proves the proposition.

Filters and enclosure

Filters A filter on a set E is a nonempty set F of nonempty subsets of E such that, for all subsets E, E of E, one has:

• E, E ∈ F implies E ∩ E ∈ F • E ⊂ E and E ∈ F implies E ∈ F .
If E is a set and F , F are filters on E, we define F F to be the filter

{E ∪ E |(E, E ) ∈ F × F }.
If F is a filter on a set E, and E is a subset of E, one says that F contains E if every element of F contains E. We denote it F E. If E is nonempty, the principal filter on E associated with E is the filter F E,E of subsets of E containing E.

A filter F is said to be contained in another filter

F : F F (resp. in a subset Z in E: F Z) if every set in F is in F (resp. if Z ∈ F ).
These definitions of containment are inspired by the following facts. Let E be a set, F be a filter on E and E, E ⊂ E. Then :

• E ⊂ E if and only if F E,E F E ,E , • E F if and only if F E,E F , • E F if and only if F E,E F .
If F is a filter on a finite-dimensional real-affine space E, its closure F (resp. its convex hull) is the filter of subsets of E containing the closure (resp. the convex hull) of some element of F . The support of a filter F on E is the minimal affine subspace containing F .

Enclosure of a filter

Let ∆ be the set of all roots of the root generating system S defined in Chapter 1 of [START_REF] Kac | Infinite-dimensional Lie algebras[END_REF]. We only recall that ∆ ⊂ A * and that ∆ ∩ RΦ = Φ.

Let F be a filter on A. The enclosure cl(F ) is the filter on A defined as follows. A set E is in cl(F ) if there exists (k α ) ∈ (Z ∪ {+∞}) ∆ satisfying:

E ⊃ α∈∆ D(α, k α ) F .
Suppose that we are in the reductive case, i.e that S is associated to a Cartan matrix or equivalently that Φ is finite. Then ∆ = Φ. Let E ⊂ A and E be the intersection of the true half-apartments containing E (E is the enclosure of E in the definition of [START_REF] Bruhat | Tits. Groupes réductifs sur un corps local[END_REF]). Then cl(E) = F E ,A .

Faces, sector-faces, chimneys and germs

Sector-faces, sectors

A sector-face f of A is a set of the form x + F v for some vectorial face F v and some x ∈ A. The point x is its base point and F v is its direction. The germ at infinity F = germ ∞ (f ) of f is the filter composed of all the subsets of A which contain an element of the form

x + u + F v , for some u ∈ F v .
When F v is a vectorial chamber, one calls f a sector. The intersection of two sectors of the same direction is a sector of the same direction. A sector-germ of A is a filter which is the germ at infinity of some sector of A. We denote by ±∞ the germ of ±C v f . The sector-face f is said to be spherical if F v ∩ ± T is nonempty. A sector-panel is a sector-face contained in a wall and spanning it as an affine space. Sectors and sector-panels are spherical.

Let q 1 and q 2 be two sector-germs of the same sign. Let C v 1 , C v 2 be the two vectorial chambers such that q 1 = germ ∞ (C v 1 ) and q 2 = germ ∞ (C v 2 ). We say that q 1 and q 2 are adjacent if C v 1 ∩ C v 2 contains some sector-panel. Let q, q be two sector-germs of the same sign. A gallery between q and q is a sequence of sector-germs Γ = (q 1 , . . . , q n ) such that n ∈ Z ≥0 , q 1 = q, q n = q and for all i ∈ 1, n -1 , q i and q i+1 are adjacent. The length of Γ is n. For every two sector germs q and q of the same sign, there exists a gallery joining q and q . Indeed, let C v and C v be the vectorial chambers

such that q = germ ∞ (C v ) and q = germ ∞ (C v ). Let w ∈ W v be such that C v = w.C v . Let w = r i 1 . . . r i k be a writing of w, with i 1 , . . . , i k ∈ I . Then germ ∞ (C v ), germ ∞ (r i 1 .C v ), germ ∞ (r i 1 r i 2 .C v ), . . . , germ ∞ (r i 1 . . . r i k .C v )
is a gallery from q to q . Faces Let x ∈ A and let F v be a vectorial face of A. The face F (x, F v ) is the filter defined as follows: a set E ⊂ A is an element of F (x, F v ) if, and only if, there exist

(k α ), (k α ) ∈ (Z ∪ {+∞}) ∆ and a neighborhood Ω of x in A such that E ⊃ α∈∆ D(α, k α ) ∩ D • (α, k α ) ⊃ Ω ∩ (x + F v ).
A face of A is a filter F that can be written as F = F (x, F v ), for some x ∈ A and some vectorial face F v .

A chamber is a face whose support is A. A panel is a face whose support is a wall. In the reductive case (i.e when Φ is finite), we obtain the usual notion of faces: the faces for the definition we gave are exactly the F F,A , where F is a face of A equipped with its structure of a simplicial complex.

Chimneys

Let F be a face of A and F v be a vectorial face of A. The chimney r(F, F v ) is the filter cl(F F +F v ,A ). A chimney r is a filter on A of the form r = r(F, F v ) for some face F and some vectorial face F v . The enclosure of a sector-face is thus a chimney. The vectorial face F v is uniquely determined by r (this is not necessarily the case of the face F ) and one calls it the direction of r.

Let r be a chimney and F v be its direction. One says that r is splayed if F v is spherical (or equivalently if F v contains a generic ray, see Subsection 2.2). One says that r is solid if the pointwise stabilizer in W v of the direction of the support of r is finite. A splayed chimney is solid.

Let r = r(F, F v ) be a chimney. A shortening of r is a chimney of the form r(F +u, F v ), for some u ∈ F v . The germ at infinity R = germ ∞ (r) of r is the filter composed of all subsets of A which contain a shortening of r. A sector-germ is an example of a germ of a splayed chimney.

Masure

Let α ∈ Φ. We can write α = w.α i for some i ∈ I and w ∈ W v . Then w.α ∨ i does not depend on the choice of w and one denotes it α ∨ . An automorphism of A is an affine bijection φ :

A → A stabilizing the set { M (α, k), α ∨ |(α, k) ∈ Φ × Z}. One has W a ⊂ W v Y ⊂ Aut(A),
where Aut(A) is the group of automorphisms of A.

An apartment of type A is a set A with a nonempty set Isom w (A, A) of bijections (called Weyl isomorphisms) such that if f 0 ∈ Isom w (A, A) then f ∈ Isom w (A, A) if and only if, there exists w ∈ W a satisfying f = f 0 • w. An isomorphism (resp. a Weyl isomorphism, a vectorially Weyl isomorphism) between two apartments φ : A → A is a bijection such that for every f ∈ Isom w (A, A) and f ∈ Isom w (A, A ), one has f

• φ • f -1 ∈ Aut(A) (resp. f • φ • f -1 ∈ W a , f • φ • f -1 ∈ (W v A) ∩ Aut(A)).
Each apartment A of type A can be equipped with the structure of an affine space by using an isomorphism of apartments φ : A → A. We equip each apartment with its topology defined by its structure of a finite-dimensional real-affine space.

We extend all the notions that are preserved by Aut(A) to each apartment. In particular, enclosures, sector-faces, faces, chimneys, germs of chimneys, ... are well defined in each apartment of type A. If A is an apartment of type A and x, y ∈ A, then we denote by [x, y] A the closed segment of A between x and y.

We say that an apartment contains a filter if it contains at least one element of this filter. We say that a map fixes a filter if it fixes at least one element of this filter.

We now give the definition of masures. These objects were introduced by Gaussent and Rousseau in [GR08] (they were initially called "hovels"). This axiomatic definition was introduced by Rousseau in [START_REF] Rousseau | Masures affines[END_REF].

Definition 1. A masure of type A is a set I endowed with a covering A by subsets called apartments such that:

(MA1) Each A ∈ A admits a structure of an apartment of type A.

(MA2) If F is a point, a germ of a preordered interval, a generic ray or a solid chimney in an apartment A and if A is another apartment containing F , then A ∩ A contains the enclosure cl A (F ) of F and there exists a Weyl isomorphism from A onto A fixing cl A (F ).

(MA3) If R is the germ of a splayed chimney and if F is a face or a germ of a solid chimney, then there exists an apartment that contains R and F .

(MA4) If two apartments A, A contain R and F as in (MA3), then there exists a Weyl isomorphism from A to A fixing cl A (R F ).

(MAO) If x, y are two points contained in two apartments A and A , and if x ≤ A y then the two segments [x, y] A and [x, y] A are equal.

We assume that there exists a group G acting strongly transitively on I, which means that:

• G acts on I,

• g.A is an apartment for every g ∈ G and every apartment A,

• for every g ∈ G and every apartment A, the map A → g.A is an isomorphism of apartments,

• all isomorphisms involved in the above axioms are induced by elements of G.

We choose in I a "fundamental" apartment, that we identify with A. As G acts strongly transitively on I, the apartments of I are the sets g.A for g ∈ G. The stabilizer N of A induces a group ν(N ) of affine automorphisms of A and we assume that ν(N ) = W v Y .

All the isomorphisms that we will consider in this paper will be vectorially Weyl isomorphisms and we will say "isomorphism" instead of "vectorially Weyl isomorphism".

Throughout the paper, we will only consider masures I which are thick of finite thickness, that is masures satisfying the following axiom:

(MAT) for each panel P , the number of chambers whose closure contains P is finite and greater than 2.

This definition coincides with the usual one when I is a building.

An example of such a masure I is the masure associated to a split Kac-Moody group over a field equipped with a nonarchimedean discrete valuation constructed in [START_REF] Gaussent | Kac-Moody groups, hovels and Littelmann paths[END_REF] and in [START_REF] Rousseau | Groupes de Kac-Moody déployés sur un corps local II. Masures ordonnées[END_REF].

A masure I is a building if and only if W v is finite, see [START_REF] Rousseau | Masures affines[END_REF] 2.2 6).

Retractions centered at sector-germs

If A and B are two apartments, and φ : A → B is an isomorphism of apartments fixing some filter X , one writes φ : A The map ρ A,s is a retraction from I onto A. It only depends on s and A and we call it the retraction onto A centered at s. We denote by I s → A the retraction onto A fixing s. We denote by ρ ±∞ the retraction onto A centered at ±∞.

Parallelism in I

Let us explain briefly the notion of parallelism in I. This is done in detail in [START_REF] Rousseau | Masures affines[END_REF] Section 3.

Let us begin with rays. Let δ and δ be two generic rays in I. By (MA3) and [Rou11] 2.2 3) there exists an apartment A containing sub-rays of δ and δ and we say that δ and δ are parallel, if these sub-rays are parallel in A. Parallelism is an equivalence relation. The parallelism class of a generic ray δ is denoted δ ∞ and is called its direction.

We now review the notion of parallelism for sector-faces. We refer to [Rou11], 3.3.4)) for the details.

Twin-building I ∞ at infinity

If f and f are two spherical sector-faces in I, there exists an apartment B containing their germs F and F . One says that f and f are parallel if F = germ ∞ (x + F v ) and F = germ ∞ (y + F v ) for some x, y ∈ B and for some vectorial face F v of B. Parallelism is an equivalence relation. The parallelism class of a sector-face germ F is denoted F ∞ and is called its direction. We denote by I ∞ the set of directions of spherical faces of I. If s is a sector, all the sectors having the germ at infinity s have the same direction. We denote it s by abuse of notation. If M is a wall of I, its direction M ∞ ⊂ I ∞ is defined to be the set of germs at infinity F ∞ such that F = germ ∞ (f ), with f a spherical sector-face contained in M .

Let F ∞ ∈ I ∞ (resp. let δ ∞ be the direction of a generic ray) and A be an apartment. One says that A contains F ∞ (resp. δ ∞ ) if A contains some sector-face f (resp. generic ray δ) whose direction is F ∞ (resp. is δ ∞ ).

Proposition 2.

1. Let x ∈ I and F ∞ ∈ I ∞ (resp. δ ∞ be a generic ray direction). Then there exists a unique sector-face x + F ∞ (resp. x + δ ∞ ) based at x and whose direction is F ∞ (resp. δ ∞ ).

2. Let A x be an apartment containing x and F ∞ (resp. δ ∞ ) (which exists by (MA3)). Let f (resp. δ ) be a sector-face (resp. a generic ray) of

A x whose direction is F ∞ (resp. δ ∞ ). Then x + F ∞ (resp. x + δ ∞ )
is the sector-face (resp. generic ray) of A x parallel to f (resp. δ ) and based at x.

3. Let B be an apartment containing F ∞ (resp. δ ∞ ). Then for all x ∈ B, x + F ∞ ⊂ B (resp. x + δ ∞ ⊂ B).
Proof. The points 1 and 2 for sector-faces are Proposition 4.7.1) of [START_REF] Rousseau | Masures affines[END_REF] and its proof. Point 3 is a consequence of 2. The statement for rays is analogous (see Lemma 3.2 of [START_REF] Hébert | Gindikin-Karpelevich finiteness for Kac-Moody groups over local fields[END_REF]).

Let f, f be sector-faces. One says that f dominates f (resp. f and

f are opposite) if germ ∞ (f ) = germ ∞ (x + F v ), germ ∞ (f ) = germ ∞ (x + F v ) for some x, x ∈ I and F v , F v two vectorial faces of a same apartment of I such that F v ⊃ F v (resp. such that F v = -F v ).
By Proposition 3.2 2) and 3) of [START_REF] Rousseau | Masures affines[END_REF], these notions extend to I ∞ .

Splitting of apartments

Splitting of apartments in two half-apartments

The aim of this section is to show that if A is an apartment, M is a wall of A, F is a sector-panel of M ∞ and s is a sector-germ dominating F ∞ , then there exist two opposite half-apartments D 1 and D 2 of A such that their common wall is parallel to M and such that for both i ∈ {1, 2}, D i and s are contained in some apartment. This is Lemma 9. This property is called "sundial configuration" in Section 2 of [START_REF] Bennett | On axiomatic definitions of non-discrete affine buildings[END_REF]. This section will enable us to show that for each choice of sign, the distances of positive types and of negative types are equivalent.

For simplicity, we assume that Φ is reduced. This assumption can be dropped with minor changes to the next lemma.

Lemma 3. Let α ∈ Φ and k ∈ R. Then cl(D(α, k)) = F D(α, k ),A . Proof. By definition of cl, D(α, k ) ∈ cl D(α, k) and hence cl D(α, k) F D(α, k ),A . Let E ∈ cl(D(α, k)). By definition, there exists (k β ) ∈ (Z ∪ ∞) ∆ such that E ⊃ β∈∆ D(β, k β ) ⊃ D(α, k). Let β ∈ ∆\{α}. As β / ∈ R + α, D(β, ) D(α, k) for all ∈ Z. Hence k β = +∞. As the family D(α, ) ∈R is ordered by inclusion, k α ≥ k . Therefore β∈∆ D(β, k β ) = D(α, k α ) ⊃ D(α, k ). Consequently, F D(α, k ),A cl D(α, k) and thus cl(D(α, k)) = F D(α, k ),A .
Lemma 4. Let A, B be two distinct apartments of I containing a half-apartment D. Then A ∩ B is a true half-apartment.

Proof. Using isomorphisms of apartments, we may assume A = A. Let α ∈ Φ and k ∈ R be such that D = D(α, k). Set M 0 = α -1 ({0}). Let S be a sector of A based at 0 and dominating some sector-panel f ⊂ M 0 . Let f = -f and s, F ∞ and F ∞ be the directions of S, f and f . Let

x ∈ A ∩ B. Then by Proposition 3 (3), A ∩ B ⊃ x + s and A ∩ B ⊃ x + F ∞ . As germ ∞ (x + s), germ ∞ (x + F ∞ )
are the germs of splayed chimneys, we can apply (MA4) and we get that

A ∩ B cl germ ∞ (x + s) germ ∞ (x + F ∞ ) . But cl germ ∞ (x + s) germ ∞ (x + F ∞ ) = cl Conv germ ∞ (x + s) germ ∞ (x + F ∞ ) ,
where Conv denotes the closure of the convex hull. Therefore

cl germ ∞ (x + s) germ ∞ (x + F ∞ ) = cl D(α, -α(x) = F D(α, -α(x) ),A (by Lemma 4). Thus A ∩ B ⊃ D(α, -α(x) ) x. Consequently, A ∩ B ⊃ x∈A∩B D(α, -α(x) ) ⊃ A ∩ B.
Hence A ∩ B = D(α, ), where = max x∈A∩B -α(x) ∈ Z, and the lemma follows.

From now on, unless otherwise stated, "a half-apartment" (resp. "a wall") will implicitly refer to "a true half-apartment" (resp. "a true wall").

Lemma 5. Let M be a wall of A and w ∈ W v Y be an element fixing M . Then w ∈ {Id, s}, where s is the reflection of W v Y with respect to M . Proof. One writes w = τ • u, with u ∈ W v and τ a translation of A. Then u(M ) is a wall parallel to M . Let M 0 be the wall parallel to M containing 0. Then u(M 0 ) is a wall parallel to M 0 and containing 0: u(M 0 ) = M 0 . Let C be a vectorial chamber adjacent to M 0 . Then u(C) is a chamber adjacent to C: u(C) ∈ {C, s 0 (C)}, where s 0 is the reflection of W v with respect to M 0 . After composing u with s 0 , we may assume that u(C) = C and thus u = Id (because the action of W v on the set of chambers is simply transitive).

If A is an apartment and D, D are half-apartments of A, we say that D and D are opposite if D ∩ D is a wall and one says that D and D have opposite directions if their walls are parallel and D ∩ D is not a half-apartment. Lemma 6. Let A 1 , A 2 , A 3 be distinct apartments. Suppose that A 1 ∩A 2 , A 1 ∩A 3 and A 2 ∩A 3 are half-apartments such that A 1 ∩ A 3 and A 2 ∩ A 3 have opposite directions. Let M be the wall of A 1 ∩ A 3 .

1. One has A 1 ∩A 2 ∩A 3 = M where M is the wall of A 1 ∩A 3 , and for all (i, j, k) ∈ {1, 2, 3} 3 such that {i, j, k} = {1, 2, 3}, A i ∩ A j and A i ∩ A k are opposite.

2. Let s : A 3 → A 3 be the reflection with respect to M , φ 1 : A 3

A 1 ∩A 3 → A 1 , φ 2 : A 3 A 2 ∩A 3 → A 2 and φ 3 : A 2 A 1 ∩A 2 → A 1 .
Then the following diagram is commutative:

A 3 φ 2 s / / A 3 φ 1 A 2 φ 3 / / A 1
Proof. Point 1 is a consequence of "Propriété du Y" and of its proof (Section 4.9 of [START_REF] Rousseau | Masures affines[END_REF]).

Let φ = φ -1 1 • φ 3 • φ 2 : A 3 → A 3 . Then φ fixes M . Let D 1 = A 2 ∩ A 3 , D 2 = A 1 ∩ A 3 and D 3 = A 1 ∩ A 2 . One has φ 3 (A 2 ) = A 1 = D 2 ∪ D 3 and thus φ 3 (D 1 ) = D 2 . One has φ -1 1 (D 2 ) = D 2 . Thus φ(D 1 ) = D 2 .
We conclude with Lemma 6. Lemma 7. Let s, s be two opposite sector-germs of I. Then there exists a unique apartment containing s and s .

Proof. The existence is a particular case of (MA3). Let A and A be apartments containing s s . Let x ∈ A ∩ A . Then by Proposition 3 (3), A = y∈x+s y + s ⊂ A ∩ A , thus A ⊂ A and the lemma follows by symmetry.

Recall the definition of I ∞ and of the direction M ∞ of a wall M from Subsection 2.8. The following lemma is similar to Proposition 2.9.1) of [START_REF] Rousseau | Masures affines[END_REF]. This is analogous to the sundial configuration of Section 2 of [START_REF] Bennett | On axiomatic definitions of non-discrete affine buildings[END_REF].

Lemma 8. Let A be an apartment, M be a wall of A and M ∞ be its direction. Let F ∞ be the direction of a sector-panel of M ∞ and s be a sector-germ dominating F ∞ and not contained in A. Then there exists a unique pair {D 1 , D 2 } of half-apartments of A such that:

• D 1 and D 2 are opposite with the common wall M parallel to M

• for all i ∈ {1, 2}, D i and s are in some apartment A i .

Moreover:

• D 1 and D 2 are true half-apartments

• such apartments A 1 and A 2 are unique and if D is the half-apartment of A 1 opposite to D 1 , then D ∩ D 2 = D 1 ∩ D 2 = M and A 2 = D 2 ∪ D.
Proof. Let us first show the existence of D 1 and D 2 . Let F ∞ be the sector-panel of M ∞ opposite to F ∞ . Let s 1 and s 2 be the sector-germs of A containing F ∞ . For i ∈ {1, 2}, let A i be an apartment of I containing s i and s, which exists by (MA3). Let i ∈ {1, 2} and x ∈ A ∩ A i . Then by Proposition 3 (3), x + s i ⊂ A ∩ A i and the open half-apartment

E i = y∈x+s i y + F ∞ ⊂ A ∩ A i is contained in A and A i . Suppose A 1 = A 2 . Then A 1 ⊃ x∈E 1 x + s 2 = A and thus A 1 = A s . This is absurd and thus A 1 = A 2 .
The apartments A 1 , A 2 contain F ∞ and s. Take x ∈ A 1 ∩ A 2 . Then by Proposition 3 (3), A 1 ∩ A 2 contains the open half-apartment y∈x+s y + F ∞ . By Lemma 5, A 1 ∩ A 2 is a halfapartment. Thus we can apply Lemma 7:

A 1 ∩ A 2 ∩ A = M , where M is a wall of A parallel to M . Set D i = A ∩ A i for all i ∈ {1, 2} . Then {D 1 , D 2 } fulfills the requirements of the lemma.
Let D 1 , D 2 be another pair of opposite half-apartments of A such that for all i ∈ {1, 2}, D i and s are contained in some apartment A i and such that D 1 ∩ D 2 is parallel to M .

We can assume D i s i for both i ∈ {1, 2}. Let s be the sector-germ of A i opposite to s. Then s dominates F ∞ and is contained in D i . Therefore s = s i . By Lemma 8, A i = A i , which proves the uniqueness of {D 1 , D 2 } and {A 1 , A 2 }.

Moreover, by Proposition 2.9 2) of [START_REF] Rousseau | Masures affines[END_REF], D ∪ D 2 is an apartment. As D ∪ D 2 s s 2 , one has D ∪ D 2 = A 2 , which concludes the proof of the lemma.

Splitting of apartments

In this subsection we mainly generalize Lemma 9. We show that if s is a sector-germ of I and if A is an apartment of I, then A is the union of a finite number of convex closed subsets P i of A such that for all i, P i and s are contained in some apartment. This is Proposition 10. Let s, s be two sector-germs of the same sign. Let A be an apartment containing s and s , which exists by (MA3). Let d(s, s ) be the length of a minimal gallery from s to s (see Subsection 2.5 for the definition of a gallery). By (MA4), d(s, s ) does not depend on the choice of A.

Let s be a sector-germ and A be an apartment of I. Let d s (A) be the minimum of the d(s, s ), where s runs over the sector-germs of A of the same sign as s. Let D A be the set of half-apartments of A. One sets P A,0 = {A} and for all n ∈ Z ≥1 ,

P A,n = { n i=1 D i |(D i ) ∈ (D A ) n }.
The following proposition is very similar to Proposition 4.3.1 of [START_REF] Charignon | Immeubles affines et groupes de Kac-Moody[END_REF].

Proposition 9. Let A be an apartment of I, s be a sector-germ of I et n = d s (A). Then there exist P 1 , . . . , P k ∈ P A,n , with k ≤ 2 n such that A = k i=1 P i and for each i ∈ 1, k , P i and s are contained in some apartment A i such that there exists an isomorphism f i :

A i P i → A.
Proof. We prove the proposition by the induction on n. This is clear if n = 0. Let n ∈ Z ≥0 >0 . Suppose this is true for every apartment B such that d s (B) ≤ n -1.

Let B be an apartment such that d s (B) = n. Let t be a sector-germ of B such that there exists a minimal gallery t = s 0 , . . . , s n-1 = s from t to s. By Lemma 9, there exist opposite half-apartments D 1 , D 2 of B such that for both i ∈ {1, 2}, D i and s 1 are contained in an apartment B i . Let i ∈ {1, 2}. One has d s (B i ) = n -1 and thus

B i = k i j=1 P (i) j , with k i ≤ 2 n-1 , for all j ∈ 1, k i , P (i) j ∈ P B i ,n-1 and s, P (i) j is contained in some apartment A (i) j . One has B = D 1 ∪ D 2 = B 1 ∩ D 1 ∪ B 2 ∩ D 2 = i∈{1,2},j∈ 1,k i P (i) j ∩ D i . Let i ∈ {1, 2}, j ∈ 1, k i and φ i : B i B∩B i → B. Then P (i) j ∩ D i = φ i (P (i) j ∩ D i ) ∈ P B,n and B i ⊃ (P (i) j ∩ D i ), s. Let f (j) i : A (j) i P (j) i → B i and f = φ i • f (j) i . Then f : A (i) j P (i) j ∩D i
→ B and the proposition follows.

We deduce from the previous proposition a corollary which was already known for masures associated to split Kac-Moody groups over fields equipped with a nonarchimedean discrete valuation by Section 4.4 of [GR08]:

Corollary 10. Let s be a sector-germ, A be an apartment and x, y ∈ A. Then there exists

x = x 1 , . . . , x k = y ∈ [x, y] A such that [x, y] A = k-1 i=1 [x i , x i+1
] A and such that for every i ∈ 1, k -1 , s and [x i , x i+1 ] A are contained in an apartment A i such that there exists an

isomorphism f i : A [x i ,x i+1 ] A i → A i .

Restrictions on the distances

In this subsection, we show that some properties cannot be satisfied by distances on masures.

If A is an apartment of I, we show that there exist apartments branching at every wall of A (this is Lemma 12). This implies that if I is not a building the interior of each apartment is empty for the distances we study. We write I as a countable union of apartments and then use Baire's Theorem to show that under a rather weak assumption of regularity for retractions, a masure cannot be complete nor locally compact for the distances we study.

Let us show a slight refinement of Corollaire 2.10 of [START_REF] Rousseau | Masures affines[END_REF]:

Lemma 11. Let A be an apartment of I and D be a half-apartment of A. Then there exists an apartment B such that A ∩ B = D.

Proof. Let M be the wall of D, P be a panel of M and C be a chamber whose closure contains P and which is not contained in A. By Proposition 2.9 1) of [START_REF] Rousseau | Masures affines[END_REF], there exists an apartment B containing D and C. By Lemma 5, A ∩ B = D, which proves the lemma.

Proposition 12. Assume that there exists a distance d I on I such that for every apartment A, d I | A 2 is induced by some norm. Then I is a building and Let M be a wall of A, D 1 and D 2 be the half-apartments defined by M and s ∈ W a be the reflection with respect to M . Let A 2 be an apartment of I such that A ∩ A 2 = D 1 , which exists by Lemma 12. Let D 3 be the half-apartment of B opposite to D 1 . Then

d I | A 2 is W a -invariant.
D 3 ∩ D 2 ⊂ D 3 ∩ A ⊂ M and thus D 2 ∩ D 3 = M . By Proposition 2.9 2) of [Rou11], D 3 ∪ D 2 is an apartment A 1 of I. Let φ 2 : A A∩A 1 → A 1 , φ 1 : A A∩A 2 → A 2 and φ 3 : A 1 A 1 ∩A 2 → A 2 .
Then by Lemma 7, the following diagram is commutative:

A φ 2 s / / A φ 1 A 1 φ 3 / / A 2 .
By the first part of the proof, s is an isometry of A and thus W a is a group of isometries for d I | A 2 . By Proposition 1 (1), W v is finite and by [START_REF] Rousseau | Masures affines[END_REF] 2.2 6), I is a building. Proof. For all λ ∈ I 0 , ρ -∞ (λ), ρ +∞ (λ) ∈ I 0 and thus ρ -∞ (λ), ρ +∞ (λ) ∈ Y . Therefore

I 0 = (λ,µ)∈Y 2 ρ -1 -∞ ({λ}) ∩ ρ -1 +∞ ({µ}). By Theorem 5.6 of [Héb17], ρ -1 -∞ ({λ}) ∩ ρ -1 +∞ ({µ}) is finite for all (λ, µ) ∈ Y 2 , which completes the proof.
Let s be a sector-germ of I. For λ ∈ I 0 choose an apartment A(λ) containing λ + s. Let x ∈ I and A be an apartment containing x and s. Then there exists λ ∈ I 0 ∩ A such that x ∈ λ + s and thus x ∈ A(λ). Therefore I = λ∈I 0 A(λ).

Proposition 17. Let d be a distance on I. Suppose that there exists a sector-germ s such that every apartment containing s is closed and with empty interior. Then (I, d) is incomplete and the interior of every compact subset of I is empty.

Proof. One has I = λ∈I 0 A(λ), with I 0 countable by Lemma 17. Thus by Baire's Theorem, (I, d) is incomplete.

Let K be a compact subset of I. Then K = λ∈I 0 K ∩ A(λ) and thus K has empty interior.

Distances of positive type and of negative type

Translation in a direction

Let s be a sector-germ. We now define a map + s such that for all x ∈ I and u ∈ C v f , x + s u is the "translate of x by u in the direction s". Let sgn(s) ∈ {-, +} be the sign of s.

Definition/Proposition 18. Let s be a sector-germ. Let x ∈ I. Let A 1 be an apartment containing x + s. Let (x + s) A 1 be the closure of x + s in A 1 . Then (x + s) A 1 does not depend on the choice of A 1 and we denote it by x + s.

Proof. Let A 2 be an apartment containing x + s and φ : A 1 A 1 ∩A 2 → A 2 . By (MA4), φ fixes the enclosure of x + s, which contains (x + s) A 1 . Therefore (x + s) A 1 ⊃ (x + s) A 2 and by symmetry, (x + s) A 1 = (x + s) A 2 . Proposition follows.

If A and B are apartments and ψ : A → B is an isomorphism, then ψ induces a bijection still denoted ψ between the sector-germs of A and those of B.

Definition/Proposition 19. Let s be a sector-germ. Let x ∈ I and A 1 be an apartment containing x + s. Let sgn(s) ∈ {-, +} be the sign of s. Let u ∈ C v f and ψ 1 : A → A 1 be an isomorphism such that ψ 1 (sgn(s)∞) = s. Then ψ 1 ψ -1 1 (x + sgn(s)u) does not depend on the choice of A 1 and of ψ 1 and we denote it x+ s u. Moreover x+ s C v f = x+s and x+ s C v f = x+s. Proof. As the case where s is negative is similar, we assume that s is positive.

We first prove the independence of the choice of isomorphism. Let ψ 1 : A → A 1 be an isomorphism such that ψ 1 (+∞) = s. Then ψ -1

1 • ψ 1 ∈ W v Y fixes the direction +∞ and thus ψ -1 1 • ψ 1 is a translation of A. Therefore ψ -1 1 • ψ 1 ψ -1 1 (x) + u = ψ -1 1 • ψ 1 ψ -1 1 (x) + u = ψ -1 1 (x) + u, and thus ψ 1 ψ -1 1 (x + u) = ψ 1 ψ -1 1 (x + u) .
Let now A 2 be an apartment containing x + s and ψ 2 : A → A 2 be an isomorphism such that ψ 2 (+∞) = s. From what has already been proved, we can assume that

ψ 2 • ψ -1 1 = φ, where φ : A 1 A 1 ∩A 2 → A 2 . As x ∈ A 1 ∩ A 2 , φ(x) = x and thus ψ -1 1 (x) = ψ -1 2 (x). Let i ∈ {1, 2}. Then ψ i ψ -1 i (x) + C v f
is a sector with the base point x and with the direction s:

ψ i ψ -1 i (x) + C v f = x + s (see Proposition 3). Moreover ψ i ψ -1 i (x) + C v f is the closure of ψ i ψ -1 i (x) + C v f = x + s in A i and thus ψ i ψ -1 i (x) + C v f = x + s. Consequently ψ 1 ψ -1 1 (x + u) ∈ x + s ⊂ A 1 ∩ A 2 . Thus φ ψ 1 ψ -1 1 (x + u) = ψ 1 ψ -1 1 (x + u) = ψ 2 ψ -1 1 (x) + u = ψ 2 ψ -1 2 (x) + u ,
which is our assertion.

Through the end of this section, we fix a sector-germ s. As the case when s is negative is similar to the case when it is positive, we assume that s is positive.

Lemma 20. Let x ∈ I and u, u ∈ C v f . Then (x + s u) + s u = x + s (u + u ). Proof. Let A be an apartment containing x + s and ψ : A → A be such that ψ(+∞) = s. One has (x + s u)

+ s u , x + s (u + u ) ∈ A. By definition, ψ -1 (x + s u) = ψ -1 (x) + u, thus (x + s u) + s u = ψ(ψ -1 (x + s u) + u ) = ψ(ψ -1 (x) + u + u ) = x + s (u + u ),
which proves the lemma.

For x, x ∈ I, we set U s (x, x ) = {(u, u ) ∈ C v f 2 | x + s u = x + s u }.
Lemma 21. Let x, x ∈ I. Then U s (x, x ) is nonempty.

Proof. Let A be an apartment containing s. Choose a ∈ (x+s)∩A and a ∈ (x +s)∩A. Then a+s and a +s are sectors of A of the same direction and thus there exists b ∈ (a+s)∩(a +s). By Definition/Proposition 20, there exist u, u , v, v

∈ C v f such that a = x + s u, a = x + s u and b = a + s v = a + s v . By Lemma 21, (u + v, u + v ) ∈ U s (x, x
) and the lemma is proved.

Definition of distances of positive type and of negative type

Let Θ + (resp. Θ -) be the set of pairs (| . |, s) such that s is a positive (resp. negative) sector-germ and | . | is a norm on A.

We now define the distance on I associated to (| • |, s). When I is a building the distance on I is usually defined as follows. One equips A with a euclidean W v -invariant norm | • |. For x, y ∈ A, one chooses g ∈ G such that g.x, g.y ∈ A and one sets d(x, y) = |g.y -g.x|. The fact | . | is W v invariant implies that this distance is well defined. When I is a masure which is not a building, there exists no W v -invariant norm on A (by Proposition 1) and there can exist pairs of points which are not contained in a common apartment. Thus we have to find another method to define a distance. For each pair of points, there exists a piecewise linear path joining them (where a piecewise linear path is a map γ : [0, 1] → I such that there exists n ∈ Z ≥0 and t 0 = 0 < t 1 < . . . < t n = 1 such that for all i ∈ 0, n-1 , γ| [t i ,t i+1 ] takes its values in some apartment A i and is an affine parametrization of the segment [γ(t i ), γ(t i+1 )] A i ). Thus we can try to define the distance between two points as the minimal length of a piecewise linear path joining them. However to this end, we need to define a notion of a length for a path and the nonexistence of a W v -invariant norm on A makes it difficult (because we cannot simply define it on A and transport it to each apartment by using isomorphism of apartments). To avoid this problem, we fix a sector s, which enables us to define a length on every apartment containing s. As the paths described above are difficult to handlefor example we do not really understand the segments that are not increasing or decreasing for the Tits preorder -we impose very strict conditions on the paths that we measure. In particular, we require n = 2 (i.e at most one break-point in the paths) and we require the directions of the segments to be contained in s, which leads to the definition below. Let us show the triangle inequality. Let x, x , x ∈ I. Let > 0 and let (u, u

) ∈ U s (x, x ), (v , v ) ∈ U s (x , x ) be such that |u| + |u | ≤ d θ (x, x ) + and |v | + |v | ≤ d θ (x , x ) + . One has x+ s u = x + s u and x + s v = x + s v . Thus x+ s u+ s v = x + s v + s u = x + s v + s u (by Lemma 21) and hence (u + v , v + u ) ∈ U s (x, x ). Consequently, d θ (x, x ) ≤ |u + v | + |v + u | ≤ |u| + |v | + |v | + |u | ≤ d θ (x, x ) + d θ (x , x ) + 2 ,
which proves the triangle inequality.

Let x, x ∈ I be such that d θ (x, x ) = 0. Then there exists

(u n , u n ) n∈Z ≥0 ∈ U s (x, x ) Z ≥0 such that u n → 0 and u n → 0. Let n ∈ Z ≥0 . One has x + s ⊃ x + s u n + s = x + s u n + s and thus x + s ⊃ n∈Z ≥0 x + u n + s = x + s. By symmetry, x + s ⊃ x + s and hence x + s = x + s.
Let B be an apartment containing x and s. By (MA2), B cl(x + s) = cl(x + s) and thus x ∈ B. Therefore x = x .

Thus we have constructed a distance d θ for all θ ∈ Θ + ∪ Θ -. A distance of the form d θ + (resp. d θ -) for some θ + ∈ Θ + (resp. θ -∈ Θ -) is called a distance of positive type (resp. distance of negative type). When I is a tree, we obtain a distance proportional to the distance of a Euclidean building on I: there exists k ∈ R such that if x, y ∈ A = R, then d θ (x, y) = k|y -x| and the action of G on I is isometric.

The choice of the norm has an influence on the metric on I but not on the topology defined on I (see Theorem 37). Independently of the choice of the norm every pair of points is joined by a geodesic and there exists pairs of points joined by infinitely many geodesics (when dim A ≥ 2, see Proposition 31).

Examples

We suppose that A is two-dimensional and that i∈I ker α i = {0} (thus the root generating system defining A is associated with a size 2 Kac-Moody matrix). We determine the restriction of d θ to A when θ = (| • |, +∞) for different choices of a norm on A.

Write

I = {1, 2}. Let u 1 , u 2 ∈ A be such that α 1 (u 1 ) = 1, α 2 (u 1 ) = 0, α 1 (u 2 ) = 0 and α 2 (u 2 ) = 1. Then C v f = {x 1 u 1 + x 2 u 2 |(x 1 , x 2 ) ∈ (R + ) 2 }. We begin by determining d θ | A 2 , when θ is associated to any Euclidean norm. For x = (x 1 , x 2 ) ∈ A, one sets |x| 2 = x 2 1 + x 2 2 . Let θ 2 = (| . | 2 , +∞)
. We now determine the restriction of d θ 2 to A.

Proposition 23. Let x, y ∈ A. Then:

d θ 2 (x, y) = |y -x| 2 if y -x ∈ ±C v f |α 1 (y -x)u 1 | 2 + |α 2 (y -x)u 2 | 2 if y -x / ∈ ±C v f . Moreover, if α 1 (x) ≤ α 1 (y) and α 2 (y) ≤ α 2 (x), then (α 1 (y -x)u 1 , α 2 (x -y)u 2 ) ∈ U +∞ (x, y) and d θ 2 (x, y) = |α 1 (y -x)u 1 | 2 + |α 2 (x -y)u 2 | 2 .
For x ∈ A and u ∈ A, we denote by x + R + u the set {x

+ tu|t ∈ R + }. Lemma 24. Let x, y ∈ A. Let f : A → R + be defined by f (z) = |z -x| 2 + |y -x| 2 , for z ∈ A.
Then the set of points at which f admits a local minimum is the segment [x, y] and

min f = |y -x| 2 = f (z), for all z ∈ [x, y].
Proof. Using a translation and a rotation, we may assume that x = 0 and y = (y 1 , 0) for some y 1 ∈ R. The lemma follows by straightforward computations of ∂f ∂z 2 (z 1 , z 2 ) and z 1 → f (z 1 , 0).

We now prove Proposition 24. Let x, y ∈ A and f : A → R + be defined by

f (z) = |z -x| 2 + |y -x| 2 , for z ∈ A. Suppose y -x ∈ C v f . Then y = x + y -x and thus d θ 2 (x, y) ≤ |0| 2 + |y -x| 2 = |y -x| 2 = min f ≤ d θ 2 (x, y), which proves that d θ 2 (x, y) = |y -x| 2 . Suppose y -x / ∈ ±C v f . Then [x, y] ∩ (x + C v f ) ∩ (y + C v f ) is empty and thus min{f (z)|z ∈ (x + C v f ) ∩ (y + C v f )} > |y -x| 2 . By Lemma 25, the minimum of f on (x + C v f ) ∩ (y + C v f ) is attained on the boundary ∂ (x+C v f )∩(y+C v f ) of (x+C v f )∩(y+C v f ). Suppose for example that α 1 (x) ≤ α 1 (y) and α 2 (y) ≤ α 2 (x). Let z ∈ A be such that α 1 (z) = α 1 (y) and α 2 (z) = α 2 (x). Then ∂ (x+C v f )∩(y +C v f ) = (z +R + u 1 )∪(z +R + u 2 ). Let z ∈ z +R + u 1 . Write z = z +tu 1 . Then f (z ) = |z -x| 2 + |z -y| 2 = |z -x| 2 + t|u 1 | 2 + |z + tu 1 -y| 2 ≥ f (z). By symmetry we deduce that min{f (z )| z ∈ (x + C v f ) ∩ (y + C v f )} = f (z) = α 1 (y -x)|u 1 | 2 + α 2 (x -y)|u 2 | 2
, and the proposition follows.

We now determine d θ | A 2 , when θ is associated to a certain norm 1 on A.

Proposition 25. Define | • | 1 : A → R + by |x| 1 = |α 1 (x)| + |α 2 (x)| for x ∈ A. Let θ 1 = (| • | 1 , +∞). Then d θ 1 (x, y) = |y -x| 1 for all x, y ∈ A. Proof. Let x, y ∈ A. Let (v, v ) ∈ U +∞ (x, y). Then for i ∈ {1, 2}, α i (v) ≥ max 0, α i (y -x) and α i (v ) ≥ max 0, α i (x -y) . Thus |v| 1 + |v | 1 ≥ |y -x| 1 and hence d θ 1 (x, y) ≥ |y -x| 1 . Suppose y -x ∈ C v f . Then (y -x, 0) ∈ U +∞ (x, y), thus d θ 1 (x, y) ≤ |y -x| 1 and hence d θ 1 (x, y) = |y -x| 1 . Suppose y -x / ∈ ±C v f . Suppose for example α 1 (x) ≥ α 1 (y) and α 2 (x) ≤ α 2 (y). Then α 2 (y -x)u 2 , α 1 (x -y)u 1 ∈ U +∞ (x,

Study on the apartments containing s

We now study the d θ , for θ ∈ Θ + ∪ Θ -. In order to simplify the notation and by symmetry, we will mainly take θ ∈ Θ + .

Fix θ ∈ Θ + . Write θ = (| . |, s), where | . | is a norm and s is a sector-germ.

Lemma 26. Let A and B be two apartments containing s. Set ρ :

I s → A and φ : A A∩B → B. Then: 1. the distance d θ | A 2 is induced by some norm on A, 2. for all x ∈ I and u ∈ C v f , ρ(x + s u) = ρ(x) + s u, 3. the retraction ρ : (I, d θ ) → (A, d θ | A 2 ) is 1-Lipschitz, 4. the map φ : (A, d θ | A 2 ) → (B, d θ | B 2 ) is an isometry. Proof. Let us prove 1. Let ψ : A → A be such that ψ(+∞) = s. Let | . | : A → R + be defined by |a| = d θ ψ(a), ψ(0) for a ∈ A. For a 1 , a 2 ∈ A, set V (a 1 , a 2 ) = {(u 1 , u 2 ) ∈ C v f 2 |a 1 -a 2 = u 2 -u 1 }. Let (a 1 , a 2 ) ∈ A. Let i ∈ {1, 2} and u i ∈ C v f . Then a i + s u i = ψ(ψ -1 (a i ) + u i ) and thus U s (a 1 , a 2 ) = V ψ -1 (a 1 ), ψ -1 (a 2 ) . Let a 1 , a 2 . Then U s (a 1 , a 2 ) = V ψ -1 (a 1 ), ψ -1 (a 2 ) = V ψ -1 (a 1 ) -ψ -1 (a 2 ), 0 . Conse- quently d θ (a 1 , a 2 ) = |ψ -1 (a 1 ) -ψ -1 (a 2 )| . It remains to prove that | . | is a norm on A. Let a 1 , a 2 ∈ A. Then |a 1 + a 2 | = d θ ψ(a 1 + a 2 ), ψ(0) ≤ d θ ψ(a 1 + a 2 ), ψ(a 1 ) + d θ ψ(a 1 ), ψ(0)
by Definition/Proposition 23. As V (a 1 + a 2 , a 1 ) = V (a 2 , 0), we deduce that d θ ψ(a 1 + a 2 ), ψ(a 1 ) = d θ ψ(a 2 ), ψ(0) and hence

|a 1 + a 2 | ≤ |a 1 | + |a 2 | .
Let t ∈ R and a ∈ A. As V (0, ta) = tV (0, a), we deduce that |ta| = |t||a| , which proves 1. Let us prove 2. Let x ∈ I and A x be an apartment containing x + s. Let φ :

A x Ax∩A → A. Let ψ x : A → A x be such that ψ x (+∞) = s and ψ A = φ • ψ x . Then ψ A (+∞) = s. Let u ∈ C v f .
Then by Definition/Proposition 20, A x x + s u and A ρ(x) + s u. Therefore

ρ(x + s u) = φ(x + s u) = φ • ψ x (ψ -1 x (x) + u) = ψ A (ψ -1 x (x) + u) and ρ(x) + s u = ψ A ψ -1 A φ(x) + u = ψ A ψ -1 x (x) + u = ρ(x + s u),
which proves 2. By 2, for all x, x ∈ I, U s ρ(x), ρ(x ) ⊃ U s (x, x ), which proves 3. By 3, φ -1 : (B,

d θ | B 2 ) → (A, d θ | A 2 ) is 1-Lipschitz. By symmetry, φ : (A, d θ | A 2 ) → (B, d θ | B 2 ) is 1-Lipschitz, which proves 4.
Lemma 27. Let d be a distance on A induced by some norm on A. Define d θ,d :

I×C v f → R + by d θ,d (x, u), (x , u ) = d θ (x, x ) + d (u, u ) for (x, u), (x , u ) ∈ I × C v f . Then the map (I × C v f , d θ,d ) → (I, d θ ) continuous defined by (x, u) → x + s u is Lipschitz.
Proof. Using isomorphisms of apartments, we may assume that s is contained in A. Since all norms on A are equivalent, it suffices to prove the assertion for a particular choice of d . We choose d = d θ | A 2 , which is possible by Lemma 27 (1). We regard C v f as a subset of I.

Let (x, u), (x , u ) ∈ I × C v f . Let > 0. Let (u, u ) ∈ U s (x, x ) and (v, v ) ∈ U s (u, u ) be such that |u| + |u | ≤ d θ (x, x ) + and |v| + |v | ≤ d θ (u, u ) + . By Lemma 21, (u + v, u + v ) ∈ U s (x + s u, x + s u ), thus d θ (x + u, x + u ) ≤ |u| + |v| + |u | + |v | ≤ d θ (x, x ) + d θ (u, u ) + 2 and hence d θ (x + u, x + u ) ≤ d θ (x, x ) + d θ (u, u ) = d θ,d (x, u), (x , u ) . Lemma follows.
Lemma 28. For all x, x ∈ I, there exists (u, u

) ∈ U s (x, x ) such that d θ (x, x ) = |u| + |u |. Proof. Let x, x ∈ I and let (u n , u n ) ∈ U s (x, x ) Z ≥0 be such that |u n | + |u n | → d θ (x, x ).
Then (|u n |), (|u n |) are bounded and thus after extraction we can assume that (u n ) and

(u n ) converge in (C v f , | . |). Lemma 28 implies that (lim u n , lim u n ) ∈ U s (x, x
), which proves our assertion.

Geodesics in I

Fix θ = (| . |, s) ∈ Θ + . We now prove that for all x 1 , x 2 ∈ I, there exists a geodesic for d θ between x 1 and x 2 . However we prove that when dim A ≥ 2, such a geodesic is not unique (even if θ is associated with a Euclidean norm). The non-uniqueness already appeared on the examples of Subsection 4.2 (see Proposition 24 and Proposition 26). As a comparison, Euclidean buildings equipped with their usual metrics are CAT(0) and thus uniquely geodesic.

Using isomorphisms of apartments, we may assume that s = +∞. For all x ∈ A and

u ∈ C v f , x + +∞ u = x + u.
To simplify the notation we write + instead of + +∞ .

Lemma 29.

1. Let x 1 , x 2 ∈ I and let (u 1 , u 2 ) ∈ U +∞ (x 1 , x 2 ) be such that d θ (x 1 , x 2 ) = |u 1 | + |u 2 |.
Then for both i ∈ {1, 2} and all t, t ∈ [0, 1],

d θ (x i + tu i , x i + t u i ) = |t -t||u i | and d θ (x 1 + tu 1 , x 2 + t u 2 ) = (1 -t)|u 1 | + (1 -t )|u 2 |. 2. Let x ∈ A and (u 1 , u 2 ) ∈ U +∞ (0, x) be such that d θ (0, x) = |u 1 | + |u 2 |. Then for all t 1 , t 1 , t 2 , t 2 ∈ [0, 1] such that t 1 ≤ t 1 and t 2 ≤ t 2 , d θ (t 1 u 1 -t 2 u 2 , t 1 u 1 -t 2 u 2 ) = (t 1 -t 1 )|u 1 | + (t 2 -t 2 )|u 2 |.
Proof. Let t, t ∈ [0, 1]. We assume t ≤ t . Let i ∈ {1, 2} and let j be such that {i, j} = {1, 2}.

As x i + u i = x j + u j , d θ (x 1 , x 2 ) ≤ d θ (x i , x i + tu i )+d θ (x i + tu i , x i + t u i ) + d θ (x i + t u i , x i + u i ) + d θ (x j + u j , x j ).
By the definition of

d θ , d θ (x i , x i + tu i ) ≤ t|u i |, d θ (x i + tu i , x i + t u i ) ≤ (t -t)|u i |, d θ (x i + t u i , x i + u i ) ≤ (1 -t )|u i | and d θ (x j + u j , x j ) ≤ |u j |. As d θ (x 1 , x 2 ) = |u 1 | + |u 2 | = t|u i | + (t -t)|u i | + (1 -t )|u i | + |u j |, we deduce that d θ (x i , x i + tu i ) = t|u i |, d θ (x i + tu i , x i + t u i ) = (t -t)|u i |, d θ (x i + t u i , x i + u i ) = (1 -t )|u i | and d θ (x j + u j , x j ) = |u j |.
We no longer assume t ≤ t . One has

d θ (x 1 + tu 1 , x 2 + t u 2 ) ≥ d θ (x 1 , x 2 ) -d θ (x 1 , x 1 + tu 1 ) -d θ (x 2 , x 2 + t u 2 ) = (1 -t)|u 1 | + (1 -t )|u 2 |.
Moreover

d θ (x 1 + tu 1 , x 2 + t u 2 ) ≤ d θ (x 1 + tu 1 , x 1 + u 1 ) + d θ (x 2 + u 2 , x 2 + t u 2 ) = (1 -t)|u 1 | + (1 -t )|u 2 |,
which proves 1. A similar argument proves 2.

Remark 1. When we are in the situation of Proposition 24, then the u 1 and u 2 of Lemma 30 are in the boundary of C v f . Proposition 30. Equip I with d θ . For all x 1 , x 2 ∈ I, there exists a geodesic from x 1 to x 2 . Moreover, if dim A ≥ 2, there exists a pair (x 1 , x 2 ) ∈ I 2 such that there are infinitely many geodesics from x 1 to x 2 .

Proof. Let x 1 , x 2 ∈ I be such that

x 1 = x 2 . Let (u 1 , u 2 ) ∈ U +∞ (x 1 , x 2 ) be such that |u 1 | + |u 2 | = d θ (x 1 , x 2 ). Let a 1 = |u 1 | |u 1 |+|u 2 | and a 2 = 1 -a 1 . Set 1 0 u 1 = 1 0 u 2 = 0. Let γ : [0, 1] → I be defined by γ(t) = x 1 + t a 1 u 1 if t ∈ [0, a 1 ] and γ(a 1 +t) = x 2 +(1-t a 2 )u 2 if t ∈ [0, a 2 ]. Then by Lemma 30 (1), for all t, t ∈ [0, 1], d θ (γ(t), γ(t )) = |t -t|(|u 1 | + |u 2 |) and hence γ is a geodesic from x 1 to x 2 . Let now x ∈ A\(C v f ∪ -C v f ).
Let us construct infinitely many geodesics joining 0 to x. Let (u 1 , u 2 ) ∈ U +∞ (0, x) be such that d θ (0, x) = |u 1 | + |u 2 |. One has x = u 1 -u 2 and thus u 1 , u 2 = 0. By Lemma 30, for z ∈ [0, 1], one has d θ (0, zu 1 ) + d θ (zu 1 , x) = d θ (0, x). Thus our idea is to concatenate a geodesic from 0 to zu 1 and a geodesic from zu 1 to x.

Let z ∈ [0, 1]. Set t z = z|u 1 | |u 1 |+|u 2 | . Let γ z : [0, 1] → A be defined by:

γ z (t) =        t |u 1 | + |u 2 | |u 1 | u 1 for t ∈ [0, t z ] zu 1 + |u 1 | + |u 2 | (1 -z)|u 1 | + |u 2 | (t -t z ) (1 -z)u 1 -u 2 for t ∈ [t z , 1]. Let t, t ∈ [0, 1]. First assume 0 ≤ t ≤ t ≤ t z .
Then by Lemma 30 (2),

d γ z (t), γ z (t ) = d(t |u 1 | + |u 2 | |u 1 | u 1 , t |u 1 | + |u 2 | |u 1 | u 1 ) = (t -t)(|u 1 | + |u 2 |). Assume t z ≤ t ≤ t ≤ 1. Let t = |u 1 |+|u 2 | (1-z)|u 1 |+|u 2 | (t -t z ) and t = |u 1 |+|u 2 | (1-z)|u 1 |+|u 2 | (t -t z ). Then by Lemma 30 (2), d γ z (t), γ z (t ) = d ( t(1 -z) + z)u 1 -tu 2 , ( t (1 -z) + z)u 1 -t u 2 = ( t -t)((1 -z)|u 1 | + |u 2 |) = (t -t)(|u 1 | + |u 2 |).
Assume t ≤ t z ≤ t . Then by Lemma 30 (2),

d γ z (t), γ z (t ) = d t |u 1 | + |u 2 | |u 1 | u 1 , ( t (1 -z) + z)u 1 -t u 2 ) = t (1 -z) + z -t |u 1 | + |u 2 | |u 1 | |u 1 | + t |u 2 | = (t -t)(|u 1 | + |u 2 |).
Therefore, γ z is a geodesic from 0 to x. Moreover, as

x / ∈ C v f ∪ -C v f , Ru 1 = Ru 2 , thus γ z ([0, 1]) = γ z ([0, 1]
) for all z = z and the proposition is proved.

Equivalence of the distances of positive type

The aim of this section is to show that if θ 1 , θ 2 ∈ Θ + , then d θ 1 and d θ 2 are equivalent. Lemma 31. There exists 0 ∈ R >0 such that for every apartment B containing s and s , for all x, x ∈ B, d θ (x, x ) ≤ 0 d θ (x, x ).

Proof. By Lemma 27 (1) and the fact that all the norms on A are equivalent, there exists 0 ∈ R >0 such that for all x, x ∈ A, d θ (x, x ) ≤ 0 d θ (x, x ). Let B be an apartment containing s and s . Let x, x ∈ B. By Lemma 27 (4), d θ (x, x ) = d θ (ρ s (x), ρ s (x )) and d θ (ρ s (x), ρ s (x )) = d θ (x, x ). Moreover ρ s |B = ρ s|B , which proves the lemma.

We now fix an apartment B 0 containing s but not s . Let F ∞ be the sector-panel direction dominated by s and s . Using Lemma 9, one writes B 0 = D 1 ∪ D 2 , where D 1 and D 2 are two opposite half-apartments whose wall contains F ∞ and such that D i ∪ s is contained in some apartment B i for both i ∈ {1, 2}. We assume that D 1 ⊃ s.

Let M 0 be a wall of A 0 containing F ∞ and t 0 : A 0 → A 0 be the reflection with respect to M 0 .

Lemma 32. One has:

ρ s (x) = ρ s (x) if x ∈ D 1 ρ s (x) = t • ρ s (x) if x ∈ D 2
, where t = τ • t 0 , for some translation τ of A 0 .

Proof. Let ρ s,B 1 :

I s → B 1 and ρ s ,B 1 : I s → B 1 . Let φ i : B 0 B 0 ∩B i → B i , for i ∈ {1, 2} and φ : B 2 B 1 ∩B 2 → B 1 .
Let t be the reflection of B 1 with respect to D 1 ∩ D 2 . By Lemma 7, the following diagram is commutative:

B 0 φ 1 φ 2 / / B 2 φ B 1 t / / B 1 . Let x ∈ D 1 . Then ρ s,B 1 (x) = x = ρ s ,B 1 (x). Let φ 3 : B 1 B 1 ∩A 0 → A 0 . Then ρ s (x) = φ 3 (ρ s,B 1 (x)) = φ 3 (ρ s ,B 1 (x)) = ρ s (x).
Let x ∈ D 2 . One has ρ s,B 1 (x) = φ 1 (x) and ρ s ,B 1 (x) = φ(x) and thus ρ s,B 1 (x) = t•ρ s ,B 1 (x). Let t be such that the following diagram commutes: Corollary 39. Let A, B be two apartments of I. Then A ∩ B is a closed subset of A (seen as an affine space).

B 1 φ 3 t / / B 1 φ 3 A 0 t / / A 0 .
Proof. By Lemma 14, A and B are closed for T + and thus A ∩ B is closed for T + . Consequently it is closed for the topology induced by T + on A, and Corollary 38 completes the proof.

Remark 2. Suppose that I is not a building. Then by Proposition 18, for all θ + ∈ Θ + , (I, d θ + ) is incomplete.

Let s be a positive sector-germ of I, θ + = (| . |, s ) and (S n ) be an increasing sequence of sectors with the germ s . One says that (S n ) is converging if there exists a retraction onto an apartment ρ : I s → ρ(I) such that (ρ(x n )) converges, where x n is the base point of S n for all n ∈ Z ≥0 and we call limit of (S n ) the set n∈Z ≥0 S n . One can show that the incompleteness of (I, d θ ) implies the existence of a converging sequence of the direction s whose limit is not a sector of I. To prove this one can associate to each Cauchy sequence (x n ) a sequence (x n ) such that d θ + (x n , x n ) → 0 and such that x n + s ⊂ x n+1 + s for all n ∈ Z ≥0 . Then we show that (x n ) converges in (I, d θ ) if, and only if the limit of (x n + s ) is a sector of I.

Study of the action of G

In this subsection, we show that for every g ∈ G, the induced map g : I → I is Lipschitz for the distances of positive type.

Lemma 40. Let g ∈ G and s be a sector-germ of I. Then for every x ∈ I and u ∈ C v f , g.(x + s u) = g.x + g.s u.

Proof. Let x ∈ I and u ∈ C v f . Let A be an apartment containing x + s. Let A = g.A. Then A contains s = g.s. Let ψ : A → A be an isomorphism such that ψ(+∞) = s. Let f : A → A be the isomorphism induced by g.

Set ψ = f • ψ. Then ψ (+∞) = s . As x + s u ∈ A, g.(x + s u) = f (x + s u) = f • ψ ψ -1 (x) + u = ψ ψ -1 • f -1 f (x) + u = g.x + g.s u.

Case of a building

In this subsection we assume that I is a building. We show that the distances of positive type are equivalent to the usual distance.

Let d A be a distance on A induced by some W v -invariant euclidean norm | . | on A. Let x, x ∈ I, A be an apartment containing x, x and f : A → A be an isomorphism of apartments. One sets d I (x, x ) = d A f (x), f (x ) . Then d I : I → R + is well defined and is a distance on I (see [START_REF] Brown | Buildings[END_REF]

d I , d θ (x, x ) = d θ (ρ +∞ (x), ρ +∞ (x )) ≥ kd I (ρ +∞ (x), ρ +∞ (x )) = kd I (x, x ). From Lemma 35 we deduce that Id : (I, d θ ) → (I, d I ) is 1 k -Lipschitz.
Let x, x ∈ I. By Corollary 11 there exist n ∈ Z ≥0 >0 and x 0 = x, x 1 , . . . ,

x n = x ∈ [x, x ] such that [x, x ] = n-1 i=0 [x i , x i+1
] and such that [x i , x i+1 ] ∪ +∞ is contained in an apartment for all i ∈ 0, n -1 . By Lemma 27 (4),

d θ (x, x ) ≤ n-1 i=0 d θ (x i , x i+1 ) = n-1 i=0 d θ (ρ +∞ (x i ), ρ +∞ (x i+1 )) ≤ n-1 i=0 d I (ρ +∞ (x i ), ρ +∞ (x i+1 )) = n-1 i=0 d I (x i , x i+1 ) = d I (x, x ),
which proves the proposition.

Mixed distances

In this section, we begin by proving that unless I is a building, if s -is a negative sector-germ, then every retraction centered at s -is not continuous for T + see Subsection 5.1). To prove this we show that the set of vertices I 0 contains no isolated points when I is not a building and then we apply finiteness results of [START_REF] Hébert | Gindikin-Karpelevich finiteness for Kac-Moody groups over local fields[END_REF].

This implies that T + = T -and motivates the introduction of mixed distances, each of which is the sum of a distance of positive type with a distance of negative type. We then study them. Proof. Assume that I is a building. By Proposition 43, we can replace d θ by the usual distance d I on I. By Lemma 16, I 0 ∩ A = Y , which is a lattice in A. Let η > 0 be such that for all λ, λ ∈ Y , d I (λ, λ ) < η implies λ = λ . Let λ, λ ∈ I 0 be such that d I (λ, λ ) < η. Let A be an apartment of I containing λ and λ and g ∈ G be such that g.A = A. Then d I (g.λ, g.λ ) = d I (λ, λ ) < η, thus λ = λ and hence I 0 is discrete in I.

Comparison of positive and negative topologies

Assume now that I is not a building and, thus, W v is infinite. By Theorem 37, we can assume that θ = (| . |, +∞). Let > 0. Let us show that there exists λ ∈ I 0 such that d θ (λ, 0) < 2 and λ = 0. Let M 0 be a wall of A containing 0 such that for all consecutive walls M 1 and M 2 of the direction M 0 , d θ (M 1 , M 2 ) < (such a direction exists by Proposition 1 (2)). Let M be a wall such that d θ (0, M ) < and such that 0 / ∈ D, where D is the half-apartment of A delimited by M and containing +∞. By Lemma 12, there exists an apartment A such that A ∩ A = D. Let φ : A A∩A → A and µ = φ(0). Let x ∈ M be such that d θ (0, x) < . Then by Lemma 27 (4), d θ (λ, x) = d θ (0, x) and thus d(λ, 0) < 2 . As λ / ∈ A, λ = 0 and we get the proposition.

Remark 3. In fact, by Theorem 42 and since G acts transitively on I 0 , we proved that when I is not a building, every point of I 0 is a limit point.

If B is an apartment and (x n ) ∈ B Z ≥0 , one says that (x n ) diverges to ∞, if for some isomorphism f : B → A, |f (x n )| → +∞.
Proposition 44. Assume that I is not a building. Let s -be a negative sector-germ of I and θ ∈ Θ + . Equip I with d θ . Let ρ -be a retraction centered at s -and (λ n ) ∈ I

Mixed distances

In this section we define and study mixed distances.

Let Ξ = Θ + × Θ -. Let ξ = (θ + , θ -) ∈ Ξ. Set d ξ = d θ + + d θ -.

Theorem 46. Let ξ ∈ Ξ. We equip I with d ξ . Then:

1. For each ξ ∈ Ξ, d ξ and d ξ are equivalent.

2. For each g ∈ G, the induced map g : I → I is Lipschitz.

3. The topology induced on every apartment is the affine topology.

4. Every retraction of I centered at a sector-germ is Lipschitz.

5. The set I 0 is discrete.

Proof. The assertions 1 to 4 are consequences of Theorem 37, Theorem 42, Corollary 38 and Corollary 39. Let us prove (5). Let λ ∈ I 0 and set λ + = ρ +∞ (λ) and λ -= ρ -∞ (λ). By Theorem 5.6 of [START_REF] Hébert | Gindikin-Karpelevich finiteness for Kac-Moody groups over local fields[END_REF], ρ -1 +∞ ({λ + }) ∩ ρ -1 -∞ ({λ -}) is finite and thus there exists r > 0 such that B(λ, r) ∩ ρ -1 +∞ ({λ + }) ∩ ρ -1 -∞ ({λ -}) = {λ}, where B(λ, r) is the open ball of the radius r and the center λ. Let k ∈ R >0 be such that ρ +∞ and ρ -∞ are k-Lipschitz. Let η > 0 be such that for all µ, µ ∈ Y , µ = µ implies d ξ (µ, µ ) ≥ η. Let r = min(r, η k ). Let us prove that B(λ, r ) ∩ I 0 = {λ}. Let µ ∈ B(λ, r ) ∩ I 0 . Suppose ρ σ∞ (µ) = λ σ , for some σ ∈ {-, +}. Then kd ξ (µ, λ) ≥ d ξ (ρ σ∞ (µ), ρ σ∞ (λ)) ≥ η, thus λ / ∈ B(λ, r ), a contradiction. Therefore ρ +∞ (µ) = λ + and ρ -∞ (µ) = λ -, hence λ = µ by choice of r, which completes the proof of the theorem.

We denote by T m the topology on I induced by any d ξ , ξ ∈ Ξ.

Link with the initial topology with respect to the retractions

In this subsection, we prove that the topology T m agrees with the initial topology with respect to the family of retractions centered at sector-germs (see Corollary 51). To this end, for each u ∈ C v f we introduce a map T u : I → R + which, for each x ∈ I, measures the distance along the ray x + (R + u) ∞ between x ∈ I and A. We then use the fact that for all λ ∈ Y ∩ C v f , T λ ≤ (ρ +∞ -ρ -∞ ), for some ∈ R + (see Lemma 48). Fix a norm | . | on A.

Definition of y u and T u

We now review briefly the results of the paragraph "Definition of y ν and T ν " of Section 3 of [START_REF] Hébert | Gindikin-Karpelevich finiteness for Kac-Moody groups over local fields[END_REF]. Let u ∈ C v f and σ ∈ {-, +}. Let δ + = R + u ⊂ A and δ -= R -u ⊂ A. Then δ + and δ -are generic rays. Let x ∈ I, then there exists a unique y σu (x) ∈ A such that x + δ σ,∞ ∩ A = y σu (x) + σR + u ⊂ A and there exists a unique T σu (x) ∈ R + such that y σu (x) = x + σ∞ T σu (x).u = ρ σ∞ (x) + σT σu (x).u.

Then for each x ∈ I, x ∈ A if and only if y u (x) = x if and only if T u (x) = 0. 1. X is bounded.

2. For every retraction ρ centered at a sector-germ of I, ρ(X) is bounded.

3. There exist two opposite sector-germs s + and s -such that if ρ s -and ρ s + are retractions centered at s -and s + , ρ s -(X) and ρ s + (X) are bounded.

Moreover every bounded subset of I 0 is finite.

Proof. By Theorem 47, (1) implies (2) which clearly implies (3). The fact that (3) implies (1) is a consequence of Lemma 49. The last assertion is a consequence of (3) and of Theorem 5.6 of [START_REF] Hébert | Gindikin-Karpelevich finiteness for Kac-Moody groups over local fields[END_REF].

Corollary 50. The sets ρ -1 + (V ) ∩ ρ -1 -(V ) such that V is an open set of an apartment A and ρ -, ρ + are retractions onto A centered at opposite sector-germs of A form a basis of T m . In particular T m is the initial topology with respect to the retractions centered at sector-germs.

Proof. This is a consequence of Lemma 49.

A continuity property for the map + +∞

The aim of this subsection is to prove the theorem below, which will be useful to prove the contractibility of I for T m . To simplify the notation, we write + instead of + +∞ .

Theorem 51. Let ξ ∈ Ξ and u ∈ C v f . Equip I with d ξ . Then the map I × R + → I defined by (x, t) → x + tu is continuous.

To prove this theorem we prove that if a sequence (x n ), (t n ) ∈ (I × R + ) Z ≥0 converges to some (x, t) ∈ I × R + , then (x n + t n u) converges to x + tu. We first treat the case where t = 0.

Fix ξ ∈ Ξ and write ξ = (θ + , θ -).

X→

  B. If A and B share a sector-germ s, there exists a unique isomorphism of apartments φ : A → B fixing A ∩ B. Indeed, by (MA4), there exists an isomorphism ψ : A → B fixing s. Let x ∈ A ∩ B. By (MA4), A ∩ B contains the convex hull Conv(x, s) in A of x and s and there exists an isomorphism of apartments ψ : A → B fixing Conv(x, s). Then ψ -1 • ψ : A → A is an isomorphism of affine spaces fixing s: ψ = ψ. By definition ψ (x) = x and thus ψ fixes A ∩ B. The uniqueness is a consequence of the fact that the only affine morphism fixing some nonempty open set of A is the identity. One denotes by A A∩B → B or by A s → B the unique isomorphism of apartments from A to B fixing s. Fix a sector-germ s of I and an apartment A containing s. Let x ∈ I. By (MA3), there exists an apartment A x of I containing x and s. Let φ : A x s → A fixing s. By [Rou11] 2.6, φ(x) does not depend on the choices we made and thus we define ρ A,s (x) = φ(x).

Proof.

  Let s be a sector-germ and A, B be two apartments containing s. Let φ : A A∩B → B. Let us first prove that φ : (A, d I ) → (B, d I ) is an isometry. Let d : A × A → R + be defined by d (x, y) = d I (φ(x), φ(y)) for all x, y ∈ A. Then d is induced by some norm. Moreover d | (A∩B) 2 = d I | (A∩B) 2 . As A ∩ B has nonempty interior, we deduce that d = d I and thus φ : (A, d I ) → (B, d I ) is an isometry.

Lemma 13 .

 13 Let s be a sector-germ of I and d be a distance on I inducing the affine topology on each apartment and such that there exists a continuous retraction ρ of I centered at s. Then each apartment containing s is closed.Proof. Let A be an apartment containing s and B = ρ(I). Let φ : B s → A and ρ A :I s → A. Then ρ A = φ • ρ is continuous because φ is an affine map. Let (x n ) ∈ A Z ≥0 be a converging sequence for d and x = lim x n . Then x n = ρ A (x n ) → ρ A (x) and thus x = ρ A (x) ∈ A.Proposition 14. Suppose I is not a building. Let d be a distance on I inducing the affine topology on each apartment. Then the interior of each apartment of I is empty. Proof. Let V be a nonempty open set of I. Let A be an apartment of I such that A ∩ V = ∅. By Proposition 1 (2), there exists a wall M of A such that M ∩ V = ∅. Let D be a halfapartment delimited by M . Let B be an apartment such that A ∩ B = D, which exists by Lemma 12. Then B ∩ V is an open set of B containing M ∩ V and thus E ∩ V = ∅, where E is the half-apartment of B opposite to D. Therefore V \A = ∅ and we get the proposition. One sets I 0 = G.0 where 0 ∈ A. This is the set of vertices of type 0. Recall that ±∞ = germ ∞ (±C v f ) and that ρ ±∞ : I ±∞ → A. Lemma 15. One has I 0 ∩ A = Y . Proof. Let λ ∈ I 0 ∩ A. Then λ = g.0 for some g ∈ G. By (MA2), there exists φ : g.A → A fixing λ. Then λ = φ(g.0) and φ • g| A : A → A is an automorphism of apartments. Let h ∈ G inducing φ on g.A. Then h.g ∈ N , hence (h.g)| A ∈ ν(N ) = W v Y (by the end of Subsection 2.6) and thus λ = h.g.0 ∈ Y.Lemma 16. The set I 0 is countable.

Definition/Proposition 22 .

 22 Let θ = (| . |, s) ∈ Θ + ∪ Θ -. Let d θ : I 2 → R + be defined by d θ (x, x ) = inf{|u| + |u | | (u, u ) ∈ U s (x, x )} for x, x ∈ I. Then d θ is a distance on I.Proof. By Lemma 22, d θ is well defined. Moreover it is clearly symmetric.

  y) and thus d θ 1 (x, y) ≤ |y -x| 1 . Therefore d θ 1 (x, y) = |y -x| 1 and the proposition follows.

  Fix a norm | . | on A. Fix two adjacent positive sector-germs s and s and set θ = (| . |, s) and θ = (| . |, s). We begin by proving the existence of ∈ R + such that d θ ≤ d θ (see Lemma 36). Fix an apartment A 0 containing s and s , which exists by (MA3). Let ρ s : I s → A 0 and ρ s : I s → A 0 .

  2. the map ρ|B : (B, d B ) → (A, d A ) is Lipschitz.Proof. By Theorem 37 we may assume θ = (| . |, t), where t is the center of ρ. Then by Lemma 27 (3), ρ : (I, d θ ) → (A, d θ ) is Lipschitz and Lemma 27 (1) completes the proof.

Theorem 41 .

 41 Let g ∈ G and θ ∈ Θ + . Then g : (I, d θ ) → (I, d θ ) is Lipschitz. Proof. Write θ = (| . |, s). Let θ = (| . |, g.s). By Theorem 37, it suffices to prove that g : (I, d θ ) → (I, d θ ) is Lipschitz.Let x, x ∈ I. By Lemma 41, U g.s (g.x, g.x ) ⊃ U s (x, x ), thus d θ (g.x, g.x ) ≤ d θ (x, x ), which proves the theorem.

  VI.3 for example). Recall that ρ +∞ : I +∞ → A. Proposition 42. Let θ ∈ Θ + . Then d I and d θ are equivalent. Proof. By Theorem 37, one can assume that θ = (| . |, +∞). Let k, ∈ R >0 be such that kd I | A 2 ≤ d θ | A 2 ≤ d I | A 2 , which exists by Lemma 27 (1). Let us first show that Id : (I, d θ ) → (I, d I ) is 1 k -Lipschitz. Let A be an apartment containing +∞. Let x, x ∈ A. Then by Lemma 27 (4) and the fact that the restriction of ρ +∞ to A is an isometry for

Fix

  a norm | . | on A. Proposition 43. Let θ ∈ Θ. Then I 0 is discrete in (I, d θ ) if and only if I is a building.

  converging sequence. Then ρ -(λ n ) → ∞ in ρ -(I). In particular ρ -is not continuous.Proof. Let A = ρ -(I) and s + be the sector-germ of A opposite to s -. Using Theorem 37, we may assume that θ = (| . |, s + ). Let ρ + :I s + → A. Let λ = lim λ n and µ = ρ + (λ). Then by Corollary 39, ρ + (λ n ) → µ. Let Y A = I 0 ∩ A. Then Y A is a lattice in A by Lemma 16. As ρ + (λ n ) ∈ Y A for all n ∈ Z ≥0 , ρ + (λ n ) = µ for n large enough. For all n ∈ Z ≥0 , ρ -(λ n ) ∈ Y A . By Theorem 5.6 of [Héb17], for all λ ∈ Y A , ρ -1 + ({λ }) ∩ ρ -1 -({µ})is finite, and the proposition follows. Corollary 45. If I is not a building, T + and T -are different. Remark 4. Proposition 45 shows that if θ, θ ∈ Θ have opposite signs, then every open subset of (I, d θ ) containing a point of I 0 is unbounded with respect to d θ .

Lemma 47 .

 47 Let λ ∈ Y ∩ C v f . Then there exists | . | ∈ R >0 such that for all x ∈ I,T λ (x), T -λ (x) ≤ | . | |ρ +∞ (x) -ρ -∞ (x)|.Proof. By Corollary 4.2 and Remark 4.3 of[START_REF] Hébert | Gindikin-Karpelevich finiteness for Kac-Moody groups over local fields[END_REF], there exists a linear map h :A → R such that T λ (x), T -λ (x) ≤ h ρ -∞ (x) -ρ +∞ (x)for all x ∈ I, which proves the existence of| . | .Lemma 48. Let ξ ∈ Ξ and a ∈ I. Let A be an apartment containing a. Let s -, s + be two opposite sector-germs of A and ρ + :I s + → A, ρ -: I s - → A. Then there exists k ∈ R >0 such that for all x ∈ I, d ξ (a, x) ≤ k d ξ a, ρ -(x) + d ξ a, ρ + (x) .Proof. Using isomorphisms of apartments, we may assume A = A, s + = +∞ and s -= -∞. By Theorem 47 (1) we may assume ξ = (| . |, s + ), (| . |, s -) . Let λ ∈ C v f . Let T + = T λ : I → R + and T -= T -λ : I → R + . By Lemma 48 and Lemma 27 (1), there exists ∈ R >0 such that T σ (x) ≤ d ξ (ρ -(x), ρ + (x)) for all x ∈ I and both σ ∈ {-, +}.Setd + = d (| . |,+∞) and d -= d (| . |,-∞) . Let x ∈ I and σ ∈ {-, +}. One has x + σ∞ T σ (x)u = ρ σ (x) + σT σ (x)u. Thus d σ (ρ σ (x), x) ≤ 2T σ (x)|u| ≤ 2 |u|d σ (ρ -(x), ρ + (x)) ≤ 2 |u| d σ (ρ -(x), a) + d σ (ρ + (x), a) . As d σ (a, x) ≤ d a, ρ σ (x) + d ρ σ (x), x we deduce that d ξ (a, x) = d -(a, x) + d + (a, x) ≤ (4 |u| + 2) d ξ (a, ρ -(x)) + d ξ (a, ρ + (x)) .Corollary 49. Let ξ ∈ Ξ. Equip I with d = d ξ . Then, for X ⊂ I the following assertions are equivalent:

  Fix a norm | . | on A. Lemma 52. Let u ∈ C v f . Then T u : I → R + and y u : I → (A, | . |) are Lipschitz for d θ + and d ξ .

Acknowledgements I would like to thank Stéphane Gaussent, Michael Kapovich and Guy Rousseau for their comments on a previous version of this paper.

Then ρ s (x) = t • ρ s (x). Moreover t fixes φ 3 (D 1 ∩ D 2 ), which contains F ∞ . Thus t = τ • t 0 for some translation τ of A 0 (by Lemma 6).

By Lemma 27 (1) and since every affine map on A 0 is Lipschitz, there exists 1 ∈ R + such that t 0 : (A 0 , d θ ) → (A 0 , d θ ) is 1 -Lipschitz. As t 0 is an involution, 1 ≥ 1.

Lemma 33. Let 0 be as in Lemma 32. Then for all x, x ∈ B 0 , d θ (x, x ) ≤ 0 1 d θ (x, x ).

Proof. Let i ∈ {1, 2} and x, x ∈ D i . By Lemma 27 (4), d θ (x, y) = d θ (ρ s (x), ρ s (x )) and d θ (x, x ) = d θ (ρ s (x), ρ s (x )). By Lemma 33, for all x, x ∈ D i , d θ (x, x ) ≤ 0 1 d θ (x, x ).

Let x, x ∈ B 0 . Assume that x ∈ D 1 and x

Proof. One implication is clear. Assume that for every apartment A containing s, Proof. For i ∈ {1, 2}, write θ i = (| . | i , s i ). As all the norms on A are equivalent, we may assume

. . , t n = s 2 be a gallery between s 1 and s 2 . For i ∈ 0, n set θ i = (| . |, t i ). By an induction using Lemma 36, there exists ∈ R >0 such that

Theorem follows by symmetry.

We thus obtain (at most) two topologies on I: the topology T + induced by d θ + , for each θ + ∈ Θ + and the topology T -induced by d θ -, for each θ -∈ Θ -. We will see that when I is not a building, these topologies are different (see Corollary 46). Corollary 37. Let A be an apartment of I. Then the topology on A induced by T + is the affine topology on A.

Proof. By Theorem 37, this topology is induced by d (| . |,t) for some positive sector-germ t of A. Then Lemma 27 (1) concludes the proof.

Corollary 38. Let ρ be a retraction centered at a positive sector-germ, A = ρ(I), B be an apartment and d A (resp. d B ) be a distance on A (resp. B) induced by a norm. Then:

Proof. By Theorem 37 and Theorem 47, we can assume

. By symmetry we deduce that T u is -Lipschitz for d θ + . The fact that y u is Lipschitz for d θ + is a consequence of the continuity of the map + (Lemma 28) and of the fact that y u = ρ +∞ + T u .u. As d θ + ≤ d ξ , the lemma is proved.

. By the continuity of the map + for d θ + (Lemma 28), x n + t n u → x + tu for d θ + . As the topologies induced by d θ + and d ξ on A agree with the topology induced by its structure of a finite-dimensional real vector-space (by Corollary 38 and Theorem 47 (3)), we deduce that x n + u → x + u for d ξ .

We no longer assume that x ∈ A. Let A be an apartment containing x + ∞. Let φ :

It remains to prove that if a sequence (x n , t n ) ∈ (I × R + ) converges to (x, 0), for some x ∈ I, then (x n + t n u) converges to x. In order to prove this we first study the map t → ρ -∞ (x + tu).

Tits preorder on I, vectorial distance on I and paths

Recall the definition of the Tits preorder ≤ on A from Subsection 2.2. As ≤ is invariant under the action of the Weyl group W v , ≤ induces a preorder ≤ A on every apartment A. Let A be an apartment and x, y ∈ A be such that x ≤ A y. Then by Proposition 5.4 of [START_REF] Rousseau | Masures affines[END_REF], if A is an apartment containing x, y, x ≤ A y. This enables to define the following relation ≤ on I: if x, y ∈ I, one says that x ≤ y if there exists an apartment A containing x, y and such that x ≤ A y. By Théorème 5.9 of [START_REF] Rousseau | Masures affines[END_REF], this defines a preorder on I and one calls ≤ the Tits preorder.

Let x, x ∈ I be such that x ≤ x . Let A be an apartment containing x, x and f : A → A be an isomorphism of apartments. Then f (x ) -f (x) is in the Tits cone T . Therefore there exists a unique

A u-path is a piecewise linear continuous map π : [0, 1] → A such that each (existing) tangent vector π (t) belongs to W v .u. Let x, x ∈ I be such that x ≤ x , A be an apartment containing them and f : A → A. We define π x,x : [0, 1] → I by t → f (1 -t)f -1 (x) + tf -1 (x) . By Proposition 5.4 of [START_REF] Rousseau | Masures affines[END_REF], π x,x does not depend on the choice of A.

Let x ∈ I and u ∈ C v f . Then

Proof. This is a weak version of Theorem 6.2 of [START_REF] Gaussent | Kac-Moody groups, hovels and Littelmann paths[END_REF] (a Hecke path of the shape u is a u-path satisfying some conditions, see Section 5 of [START_REF] Gaussent | Kac-Moody groups, hovels and Littelmann paths[END_REF] for the definition).

Recall that the α ∨ i , for i ∈ I, denote the simple roots. Let

and the lemma follows.

The following lemma completes the proof of Theorem 52.

be such that (x n ) converges for d ξ and (t n ) converges to 0. Then (x n + t n u) converges to lim x n for d ξ .

Proof. By Theorem 47, we can assume ξ = (| . | 0 , +∞), (| . | 0 , -∞). Let x = lim x n . By the same reasoning as in the proof of Lemma 54:

-we can assume x ∈ A,

By Lemma 57, for all n ∈ Z ≥0 ,

By continuity of the map + (Lemma 28) and the continuity of ρ +∞ (Corollary 39) for d θ + , ρ +∞ (x n + t n u) → ρ +∞ (x). Using Lemma 49 we deduce that (x n + t n u) converges to x, which is the desired conclusion.

where we set 1 0 = +∞ > t for all t ∈ R. Then χ u is a strong deformation retract on A for d θ and d ξ .

Proof. Let x ∈ A and t ∈ [0, 1]. Then T u (x) = 0 and thus χ u (x, t) = y u (x) = x. Let x ∈ I. Then χ u (x, 0) = x and χ u (x, 1) = y u (x) ∈ A. It remains to show that χ u is continuous for d θ and d ξ . Let (x n , t n ) ∈ (I × [0, 1]) Z ≥0 be a converging sequence for d θ or d ξ and (x, t) = lim(x n , t n ). Suppose for example that t 1-t < T u (x) (the case t 1-t = T u (x) and t 1-t > T u (x) are analogous). Then by the continuity of T u (Lemma 53), tn 1-tn < T u (x n ) for n large enough and thus by the continuity of the map + (Lemma 28 for d θ and Theorem 52 for d ξ ), χ u (x n , t n ) = x n + tn 1-tn u → x + t 1-t u = χ u (x, t). Therefore, χ u is continuous, which concludes the proof.

Corollary 59. The masure I is contractible for T + , T -and T m .

Proof. Let u ∈ C v f . We define Υ u : I × [0, 1] → I by

Then Υ u is a strong deformation retract on {0} for d θ and d ξ , which proves that (I, T + ) and (I, T m ) are contractible. By symmetry, (I, T -) is contractible.