Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 42 (2001), No. 2, 341-357.

Moufang Buildings and Twin Buildings

To André Vermeulen

Valery Vermeulen

Universiteit Gent, Vakgroep Zuivere Wiskunde en Computeralgebra Galglaan 2, B-9000 Gent, Belgium

1. Introduction

The “Moufang Condition” for spherical buildings was introduced by J. Tits in the appendix of [9], as a tool to give more structure to the classification of spherical buildings of rank at least three (which are automatically “Moufang”). More recently, also the spherical buildings of rank 2 satisfying the Moufang Condition are classified [13]. Hence one could say that, on the geometric level, spherical buildings axiomatize the situation of a simple group of Lie type, while on the group-theoretic level, the Moufang Condition characterizes the groups themselves as automorphism groups. A few years ago, a similar phenomenon occured after the discovery of Kac-Moody algebras and Kac-Moody groups. In [11] Tits gave a group theoretical definition of a “Moufang Condition” intrinsically generalizing the notion of “Moufang spherical building”. This definition was formally translated into geometrical language by Ronan in his book [7]. The main motivation was an attempt to characterize the Kac-Moody groups as automorphism groups of certain buildings (namely, the Moufang buildings). However, this equivalence is not yet established. On the geometric level, Ronan and Tits introduced the so-called “twin buildings” (see [12]), and these axiomatize the situation of a Kac-Moody group geometrically. Again, the equivalence of (simple) Kac-Moody groups and twin buildings (possibly under some additional hypotheses) has not been established yet. However, the work of Tits (see [11], [12], [9]), Mühlherr (see [4], [5], [6]) and Ronan points in the direction that a classification of 2-spherical twin buildings (i.e., twin buildings with a diagram containing no edges labelled ∞) is feasible. Moreover, Mühlherr and Ronan show in [3] that, under some mild restrictions, both combinatorial buildings of a 2-spherical twin building satisfy the Moufang Condition. Hence, in view of the analog for the spherical case, and in view of a possible classification of Moufang buildings (still under some additional,

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

natural, conditions), the question of whether every Moufang building corresponds to a twin building is very interesting. In fact, Proposition 4 of [12] says that this is indeed the case. For a proof of that proposition, Tits refers the reader to the paper [11] without any additional specification or hint. The paper [11] indeed contains a rough outline of a possible proof of the proposition, and with some effort, one can reconstruct in detail the arguments and (non-trivial technical) computations. However, in the present paper, we want to present a proof of Tits’ proposition partly following his ideas, but mainly using alternative geometric arguments. In my opinion this gives better insight into the geometry of twin buildings and the relation with groups, as a generalization of the connection between geometries and groups in the theory of (semi-simple) algebraic groups.

The paper is organized as follows. In Section 2 we give some definitions and facts to introduce the setting for the Moufang building Δ. In Section 3 we construct a chamber system C^{−} using the groups which come from the Moufang structure on Δ. At the end of this section a covering κ between C^{−} and Δ is defined. Using universal properties of Δ this implies that κ is an isomorphism. In the last section we give a proof that Δ is the half of a twin building.

Acknowledgment. I am very grateful to Bernhard Mühlherr for some helpful discussions on the subject of the paper.

2. Preliminaries

By a Coxeter matrix M over the finite index set I we will mean a symmetric matrix M = (m_{ij})_{i,j∈I} with entries in ℕ ∪ ∞ such that m_{ii} ≥ 1 and m_{ij} ≥ 2 if i ≠ j. With every such Coxeter matrix one can associate a group W with presentation

W = ⟨s_{i}|(s_{i}s_{j})^{m_{ij}}⟩.

This is the Coxeter group or Weyl group of type M. We sometimes write (W, S) instead of W where S = {s_{i}|i ∈ I}. This couple is called a Coxeter system of type M. This notation is used when we have a particular set S of generators of W in mind. The length of an element w ∈ W with respect to this generating set is denoted by l(w). An expression w = s_{i_{1}} … s_{i_{m}} with l(w) = m is called reduced or minimal.

Definition 1. Let M be a Coxeter matrix and (W, S) a Coxeter system of type M. A building of type M is a quadruple (Δ, W, S, d) with Δ a set whose elements are called chambers, (W, S) the given Coxeter system and d a distance function going from Δ × Δ to W satisfying 3 axioms: Bu1 w = 1 if and only if x = y. Bu2 If z ∈ Δ is a chamber such that d(y, z) = s with s ∈ S then d(x, z) = w or ws.

Moreover if l(ws) > l(w) then d(x, z) = ws. Bu3 If s ∈ S then there exists a chamber z of Δ such that d(x, z) = ws.

This definition was taken from [12]. When there is no confusion possible, or when (W, S) is not important a building (Δ, W, S, d) will also briefly be written as (Δ, d) or as Δ.

An obvious example of a building of type M is given by the Coxeter group itself. Chambers are elements of W and distance between two chambers x and y is defined as x^{−1}y. In the sequel we sometimes view the Coxeter group either as group or as a building. Which approach is used will be clear from the context unless stated otherwise.

If the associated Coxeter group is finite, then the building Δ is called a spherical building. For more information about spherical buildings we refer to [9], where they are classified in case |S| ≥ 3. As a generalisation of the idea of spherical building the concept of twin buildings was introduced by M.Ronan and J.Tits. The paper [12] can be seen as a standard reference on this subject. We give the definition of [12] of a twin building.

Definition 2. For a certain Coxeter matrix and associated Weyl group (W, S) a twinned pair of buildings or a twin building of type M is a pair of buildings (Δ_{+}, W, S, d_{+}) and (Δ_{−}, W, S, d_{−}) with a codistance function d^{∗} going from Δ_{+} × Δ_{−} ⊔ Δ_{−} × Δ_{+} to W satisfying (ϵ ∈ {−1, 1}, x ∈ Δ_{ϵ}, y ∈ Δ_{−ϵ} and d^{∗}(x, y) = w): Tw1 d^{∗}(y, x) = w^{−1}. Tw2 If z ∈ Δ_{−ϵ} is such that d_{−ϵ}(y, z) = s ∈ S and l(ws) < l(w) then d^{∗}(x, z) = ws. Tw3 For every s ∈ S there exists at least one chamber z ∈ Δ_{−ϵ} with d^{∗}(x, z) = ws.

Apart from the geometrical description of buildings there is also a nice group theoretical counterpart. This is the notion of a BN-pair or Tits system.

Definition 3. Let G be a group with two subgroups B and N. Then (G, B, N, S) forms a BN-pair or Tits system if the following axioms are satisfied: BN0 ⟨B, N⟩ = G. BN1 H = B ∩ N ⊴ N and N/H is a Coxeter group with generating set

S = {s_{i}|i ∈ I}.

BN2 BsBwB ⊂ BswB ∪ BwB whenever w ∈ W and s ∈ S. BN3 sBs ≠ B for s ∈ S.

More information about BN-pairs can be found in [9].

Remains to define what a Moufang building is. To do this we follow the approach in Paragraph 4 of Chapter 6 in [7]. First we need some additional notions like panel, apartment and root. Let (Δ, W, S, d) be a building then two chambers x and y are called s-adjacent for s ∈ S whenever d(x, y) = s or x = y. An s-panel P is a maximal set of mutually s-adjacent chambers. When we don’t have a specific s in mind an s-panel will also be called a panel.

To relate buildings one needs a notion of morphisms.

Definition 4. Given two buildings (Δ, W, S, d) and (Δ′, W, S, d′) of the same type a morphism from (Δ, W, S, d) to (Δ′, W, S, d′) is a mapping θ going from Δ to Δ′ such that if x and y are s-adjacent then θ(x) and θ(y) are also s-adjacent where x, y ∈ Δ. An isomorphism is also called an isometry.

A nice example of an isometry of the Coxeter group onto itself is given by left multiplication with a fixed element.

Definition 5. Given a Coxeter group viewed as a building then for an element s ∈ S the fundamental root defined by s is the set α_{s} = {w ∈ W |l(sw) > l(w)}. All other roots in W are subsets of the form w(α_{s}) for some w ∈ W. The opposite root of a root α is the root −α having an empty intersection with α. For every root α the boundary of α denoted by ∂α, is the set of panels that have non-empty intersection with both α and −α. Roots are called positive or negative according whether they contain 1 or not. If a root α is positive, this is denoted by α > 0. Similarly α < 0 means that the root α is a negative root. The set of all roots in W is denoted by Ψ.

We fix some notation. If s_{i}, i ∈ I denotes a fundamental reflection then the root associated with s_{i} is written as α_{i}.

Definition 6. An apartment Σ in a building Δ is an isometric copy of the Coxeter group W viewed as a building in Δ. A root in a building Δ is defined as an isometric copy of a root α in W. The boundary of a root in Δ, and the notion of positive and negative roots are defined in a similar way.

It can be proven that apartments always exist in buildings and that they characterize the geometry (cf. Theorem 3.11 of [7]).

Definition 7. Two roots α and β in W are called prenilpotent if and only if α ∩ β ≠ ∅ and (−α) ∩ (−β) ≠ ∅. If two roots α and β in W are prenilpotent then the interval [α, β] is defined as the set

{γ ∈ Ψ | α ∩ β ⊂ γ and (−α) ∩ (−β) ⊂ (−γ)}.

The notation (α, β), with α, β a prenilpotent pair of roots, stands for [α, β] \ {α, β}.

Starting with a building (Δ, W, S, d) of a certain type M and the set Φ of all roots in a fixed apartment Σ_{0} (called the standard apartment) of Δ we call the building Moufang if there exists a set of automorphism groups (U_{α})_{α∈Φ} also called root groups such that: Mo1 Every element u ∈ U_{α} fixes all chambers of α. If π is a panel on ∂α and c is the chamber

of π lying in α then U_{α} fixes c and acts regularly on all the chambers of π \ {c}. Mo2 If {α, β} is a pair of prenilpotent distinct roots then

[U_{α}, U_{β}] ⊂ U_{(α,β)}.

Mo3 For each u_{α} ∈ U_{α} \ {1} there exists an element m(u_{α}) ∈ U_{−α}u_{α}U_{−α} stabilizing Σ. Mo4 If n = m(u_{α}) then for every root β we have nU_{β}n^{−1} = U_{s_{α}(β)}.

Given a Moufang building Δ with root groups (U_{α})_{α∈Φ}, we define the group G = ⟨U_{α}⟩_{α∈Φ}, N the group generated by all m(u_{γ}) with u_{γ} ∈ U_{γ} for a root γ in Φ. The standard chamber c_{+} is defined as the image of 1 under the isometry going from the Coxeter group W to the standard apartment Σ_{0}. It follows from the construction that c_{+} is the intersection of all positive roots in Σ_{0}. The subgroup of elements of N that fix Σ_{0} is denoted by H, the torus in the classical sense. It is easy to check that H ⊂ N_{G}(U_{α}) for all root groups U_{α}. The group B_{+} stands for ⟨H, U_{α}⟩_{α>0}, B_{−} = ⟨H, U_{α}⟩_{α<0} and B_{α} = ⟨H, U_{α}⟩ for every α ∈ Φ. The group B_{+} also has a geometrical meaning: it is the full stabilizer in G of the standard chamber c_{+} in Δ and N is the stabilizer of the apartment Σ_{0} in G. The first fact is not obvious to show. It follows mainly from Lemma 4 in Section 5 of [11]. In fact this lemma yields that G = ∪(B_{+}wB_{+})_{w∈W}. With this setup we thus get a system (G, (U_{α})_{α∈Φ}) of groups. In the paper [11] J.Tits considers similar systems satisfying 5 axioms, namely (RD1) till (RD5). It is not hard to check that (G, (U_{α})_{α∈Φ}) satisfies (RD1), (RD2), (RD3) and (RD4). However axiom (RD5) cannot be proved by elementary techniques. It states that ∀i ∈ I and α_{i} ∈ Φ with α_{i} > 0, B_{α_{i}} ⊄ B_{−} and B_{−α_{i}} ⊄ B_{+}. That B_{−α_{i}} ⊄ B_{+} is easy to check. This follows essentially from the equality Stab_{G}(c_{+}) = B_{+}. Namely if B_{−α_{i}} ⊂ B_{+} then every u_{−α_{i}} ∈ U_{−α_{i}} would fix c_{+} contradicting the regular action of U_{−α_{i}}. To exclude that B_{α_{i}} is contained in B_{−} one cannot use the same argument as for the other case. The difference here is that B_{−} has no interpretation in terms of the building geometry. For this we will have to look deeper into the structure of Δ. Moreover the following results are true. For proofs we refer to [11].

Theorem 1. Given a Moufang building (Δ, W, S, d) of type M (with notations as above) then there is a unique homorphism ν : N ↦ W such that for n ∈ N and α ∈ Φ

nB_{α}n^{−1} = B_{ν(n)(α)}.

The kernel of ν is H. This implies that N/H ≅ W and N/H is generated by a set s̃_{i}H where {s̃_{i}} is a set of m(u_{α_{i}}) with u_{α_{i}} ≠ 1 and {α_{i} | i ∈ I} a fundamental root system in Φ.

Proof. This is a restatement of Lemma 3(i),(iii) in Paragraph 5 of [11]. The only thing one has to check are axioms (RD2), (RD3) and (RD4) of loc. cit. for the system (G, (B_{α})_{α∈Φ}).

□

In what follows we will also consider elements of w as they were in the group G and write for example wB_{α}w^{−1} = B_{w(α)} if there is no confusion. If we consider w as being an element of G we have a representative of w in N/H in mind.

Theorem 2. Given a Moufang building (Δ, W, S, d) of type M (with notations as above) then G acts transitively on Δ and the system (G, B_{+}, N, S) forms a BN-pair where S is a set of generators of the group N/H.

Proof. One checks that axiom (RD1) of RD-systems (cf. Paragraph 5 in [11]) is satisfied for (G, (B_{α})_{α∈Φ}) and that B_{−α} ⊆ B_{+} for all α > 0. Then the proof of Lemma 4 in Paragraph 5 of loc. cit. is still valid. Following the strategy of Proposition 4(i) of loc. cit. one deduces that (G, B_{+}, N, S) is a BN-pair. □

We also mention the following property which we will use later on.

Lemma 1. Given a Moufang building (Δ, W, S, d) as above then for every i ∈ I the group B_{α_{i}} with α_{i} > 0 has two double cosets in the group it generates with B_{−α_{i}}, these are B_{α_{i}} and B_{α_{i}}s_{i}B_{α_{i}}. Hence we can write

⟨B_{α_{i}}, B_{−α_{i}}⟩ = B_{α_{i}} ∪ B_{α_{i}}s_{i}B_{α_{i}}.

Proof. If u_{−α_{i}} ∈ B_{−α_{i}} and u_{−α_{i}} ∉ H then there exist elements u_{α_{i}} and u′_{α_{i}} in U_{α_{i}} such that m(u_{−α_{i}}) = u_{α_{i}}u_{−α_{i}}u′_{α_{i}}. From the properties of the Moufang building Δ it follows that u_{α_{i}}u_{−α_{i}}u′_{α_{i}}s_{i}^{−1} ∈ H, hence u_{−α_{i}} ∈ B_{α_{i}}s_{i}B_{α_{i}}. This proves the claim. □

3. The chamber system C^{−}

In this paragraph we construct a chamber system C^{−} using the groups. First we need some lemmas.

Lemma 2. Given a negative root α_{i} with i ∈ I then

B_{α}B_{α_{i}} s̃_{i}B_{−} ⊂ B_{α_{i}} s̃_{i}B_{−}

for every negative root α ∈ Φ.

Proof. For the proof we refer to Lemma 4 in Section 5 of [11]. One replaces all positive roots by negative roots. □

Lemma 3. Let w ∈ W (with (W, S) a Weyl group) , and s_{i_{1}} … s_{i_{m}} a reduced expression of w. Set for j ∈ {1, …, m} w_{j} = s_{i_{1}} … s_{i_{j}}, w_{0} = 1 and β_{j} = w_{j−1}(α_{j}) then {β_{1}, …, β_{m}} is the set of all positive roots sent by w^{−1} to a negative root.

Proof. This lemma is a restatement of Proposition 3(i) of [11], Section 5. The proof can be found there. □

Lemma 4. Given any w ∈ W and a reduced expression s_{i_{1}} … s_{i_{m}} of w then the set

U_{−β_{1}} … U_{−β_{m}}

is a group U_{−w} only depending on w. The group B_{−w} satsifies B_{−w}wB_{−} = B_{−}wB_{−}.

Proof. The statement of this lemma is analogous to the statement of Proposition 3(ii), (iv) in Section 5 of [11]. The only difference is that the groups here are parametrized by negative roots. One can easily check that the proof given in loc. cit. remains partially valid if positive roots are replaced by negative roots. □

Using the groups U_{−w} we construct the following chamber system C^{−}. Let U_{−} = ⟨U_{−α}⟩_{α>0}. For a given w ∈ W the group U_{−w} defines a coset structure on U_{−}. We define C_{w}^{−} as the set of all right cosets of U_{−w} in U_{−}. The set of chambers of C^{−} is the disjoint union ⊔C_{w}^{−}. As we want the chamber system C^{−} to be defined over the set I we have to define an i-adjacency relation for every i ∈ I. To do this we first fix some terminology which is introduced in [11] in Section 5.11.

Given J ⊆ I such that W_{J} = ⟨s_{i}|i ∈ J⟩ is finite and an element w ∈ W, then w is called right J-anti-reduced if l(w) = max{l(u)|u ∈ wW_{J}}. For w ∈ W and i ∈ I, w^{i} stands for the unique right {i}-anti-reduced element in the i-panel in W containing w. For adjacency we state the following rule: Two chambers xU_{−w} and yU_{−v} are i-adjacent if and only if

(1) w^{i} = v^{i},

(2) xU_{−w^{i}} = yU_{−w^{i}}.

It is easily checked that C^{−} equipped with this adjacency relation is indeed a chamber system over I in the sense of [7], Chapter 1. We also remark that the group U_{−} acts on the chamber system C_{−} by left multiplication. It is easily checked that under this action i-adjacent chambers are sent to i-adjacent chambers. This means that the group U_{−} acts as a group of type preserving automorphisms of the chamber system C^{−}. The next step is to construct a chamber systems morphism between C^{−} and (Δ, W, S, d).

Lemma 5. The map κ between C^{−} and (Δ, W, S, d) that sends xU_{w} to xw(c_{+}) is a type preserving morphism between the chamber systems C^{−} and (Δ, W, S, d) (i.e. it sends i-adjacent chambers to i-adjacent chambers).

Proof. We have to check that κ is well defined and that if xU_{−w} and yU_{−v} are i-adjacent, then also κ(xU_{−w}) and κ(yU_{−v}) are i-adjacent. To see this we rely on the following property:

U_{−w} ⊂ Stab_{G}(w(c_{+})).  (∗)

Let’s first check this property. By Theorem 1 and Lemma 2 the group w^{−1}U_{−w}w is contained in B_{+}. As Stab_{G}(c_{+}) = B_{+} formula (∗) is clear. Because of property (∗) the map κ is well defined, i.e. if xU_{−w} = x′U_{−w} then x(w(c_{+})) = x′(w(c_{+})).

Suppose that xU_{−w} and yU_{−v} are i-adjacent, i.e. w^{i} = v^{i} and xU_{−w^{i}} = yU_{−w^{i}}. From w^{i} = v^{i} it follows that w(c_{+}) and v(c_{+}) are i-adjacent and belong to the i-panel containing w^{i}. From y^{−1}x ∈ U_{−w^{i}} we deduce that y^{−1}x stabilizes w^{i}(c_{+}), hence also stabilizes the i-panel through w^{i}(c_{+}). This means that y^{−1}x(w(c_{+})) and v(c_{+}) are i-adjacent, hence also x(w(c_{+})) and y(v(c_{+})) are i-adjacent. This completes the proof of the lemma. □

4. Properties of κ

In this paragraph we show that κ is a 2-covering from C^{−} onto (Δ, W, S, d). We start by showing that κ is surjective. For this we need an additional property of Moufang buildings.

Proposition 1. Given a Moufang building (Δ, W, S, d) with standard apartment Σ_{0}, then the orbit of Σ_{0} (as a set of chambers) under B_{−} is the full building Δ, i.e. B_{−}(Σ_{0}) = Δ.

Proof. The proposition follows from the decomposition G = B_{−}W B_{+} regarded the fact that {w(c_{+})|w ∈ W } = Σ_{0}. First we show that G = ∪(B_{−}wB_{−})_{w∈W}.

Using Lemma 4 we write:

B_{−}s_{i}B_{−}wB_{−} = B_{−}s_{i}B_{−w}wB_{−}.

Two cases occur:

(1) l(s_{i}w) > l(w). Then s_{i}B_{−w}s_{i} ⊂ B_{−} and

B_{−}s_{i}B_{−}wB_{−} = B_{−}(s_{i}B_{−w}s_{i})s_{i}wB_{−} = B_{−}s_{i}wB_{−}.

(2) l(s_{i}w) < l(w). Hence

B_{−}s_{i}B_{−}wB_{−} = B_{−}s_{i}B_{−}s_{i}s_{i}wB_{−}
⊂ {B_{−}s_{i}B_{−}, B_{−}}s_{i}wB_{−}
⊂ B_{−}wB_{−} ∪ B_{−}s_{i}wB_{−}.

From this one deduces that ∪(B_{−}wB_{−})_{w∈W} = G.

Then we show that for every w ∈ W and s_{i} ∈ S

B_{−}s_{i}B_{−}wB_{+} ⊆ B_{−}swB_{+} ∪ B_{−}wB_{+}.

As above again two cases can occur:

(1) l(s_{i}w) < l(w). This means that the root w^{−1}(α_{i}) is negative, hence

B_{−}s_{i}B_{−}wB_{+} = B_{−}s_{i}B_{−α_{i}}wB_{+}
= B_{−}s_{i}w(w^{−1}B_{−α_{i}}w)B_{+}
= B_{−}s_{i}wB_{+}.

(2) l(s_{i}w) > l(w). Then we use the above equation and calculate:

B_{−}s_{i}B_{−}wB_{+} = B_{−}s_{i}B_{−}s_{i}s_{i}wB_{+}
⊂ B_{−}{1, s_{i}}B_{−}s_{i}wB_{+}
= B_{−}s_{i}wB_{+} ∪ B_{−}wB_{+}.

It follows that B_{−}W B_{+} = G. □

Corollary 1. The morphism κ is surjective.

Proof. Consider an arbitrary chamber a in Δ. Then by Proposition 1 we have a = b_{−}v(c_{+}) for some b_{−} ∈ B_{−} and v ∈ W. As for every root α, H ⊂ Stab_{G}(U_{α}) we can write b_{−} as u_{−}h for u_{−} ∈ U_{−} and h ∈ H. Because H fixes every chamber of Σ_{0} we can write a = u_{−}v(c_{+}). If we consider the element u_{−}U_{−v} of C^{−} then clearly κ(u_{−}U_{−v}) = a. □

The only problem that remains to prove is that κ is a 2-covering.

Theorem 3. The map κ is 2-covering from C^{−} to Δ, i.e. it sends spherical rank 2 residues isomorphically onto spherical rank 2 residues.

Proof. To prove this we remark that the action of U_{−} on C_{−} and Δ is compatible with κ, i.e. for all xU_{−w} ∈ C_{−} and u_{−} ∈ U_{−} we have κ(u_{−}xU_{−w}) = u_{−}κ(xU_{−w}). In order to prove that κ is a 2-covering, it will then be enough to show that κ induces an isomorphism between every {i, j} residue containing a chamber U_{−w}, with w an {i, j}-anti-reduced element in W, and its image in Δ. To see this we remark that every rank 2 residue in C_{−} always contains a chamber xU_{−w} where w is {i, j}-anti-reduced and x ∈ U_{−}. The morphism determined by x^{−1} will then send the given rank 2 residue to another rank 2 residue that contains U_{−w}.

Fix a certain rank 2 residue in C_{−} of spherical type {i, j} (hence m_{ij} < ∞). Call this residue R_{−}^{ij}. Suppose that R_{−}^{ij} contains a chamber U_{−w} with w {i, j}-anti-reduced. As U_{−w} ∈ R_{−}^{ij}, we see that w(c_{+}) ∈ κ(R_{−}^{ij}). If we denote by R^{ij} the {i, j}-residue in Δ which contains w(c_{+}) then we have to show that κ induces an isomorphism between R_{−}^{ij} and R^{ij}.

(1) The map κ induces a surjection between R_{−}^{ij} and R^{ij}. This will follow from the fact that κ induces a surjection between rank 1 residues. Consider a fixed i ∈ I and a chamber a in Δ. Using Proposition 1 and the action of U_{−} on Δ we can assume that a = v(c_{+}), v ∈ W. Then every chamber of the i-residue containing a can be written under the form v(u_{α_{i}}s_{i}(c_{+})) with u_{α_{i}} ∈ U_{α_{i}} and α_{i} > 0.

Two cases occur:

(i) l(vs_{i}) < l(v). Then vu_{α_{i}} = vu_{α_{i}}v^{−1}v with vu_{α_{i}}v^{−1} ∈ U_{v(α_{i})}. Granted the condition on v, one has U_{v(α_{i})} ⊂ U_{−v^{i}}. If we consider in C^{−} the chamber vu_{α_{i}}v^{−1}U_{−vs_{i}}, then this chamber is i-adjacent to U_{−v} and κ(vu_{α_{i}}v^{−1}U_{−v}) = a. (ii) l(vs_{i}) > l(v). Using Lemma 1, one starts by rewriting u_{α_{i}} as u_{−α_{i}}s_{i}b_{−α_{i}} with u_{−α_{i}} ∈ U_{−α_{i}} and b_{−α_{i}} ∈ B_{−α_{i}}. As we also know that s_{i}b_{−α_{i}}s_{i} ⊂ B_{α_{i}} the chamber a coincides with vu_{−α_{i}}(c_{+}). Because of the condition on v we have that vu_{−α_{i}}v^{−1} ∈ U_{−v(α_{i})} ⊂ U_{−v^{i}}. Hence the chamber vu_{−α_{i}}v^{−1}U_{−vs_{i}} is i-adjacent to U_{−v} and κ(vu_{−α_{i}}v^{−1}U_{−vs_{i}}) = a. This completes the proof that κ induces a surjection between rank 1 residues in C^{−} and Δ. Because rank 2 residues are connected it is clear that κ induces a surjection of R_{−}^{ij} onto R^{ij}.

(2) The morphism κ induces an injection of R_{−}^{i,j} into R^{ij}. Suppose that we have two chambers u′_{−}U_{−w′} and u′′_{−}U_{−w′′} in R_{−}^{ij} such that κ(u′_{−}U_{−w′}) = κ(u′′_{−}U_{−w′′}). This means that u′_{−}w′(c_{+}) = u′′_{−}w′′(c_{+}) and both w′ and w′′ belong to the {i, j}- residue in W determined by w. Because of the conditions on w it is easy to check that both u′_{−} and u′′_{−} belong to U_{−w}. We rewrite the above equality as

(w^{−1}u′_{−}w)w^{−1}w′(c_{+}) = (w^{−1}u′′_{−}w)w^{−1}w′′(c_{+}).

As both u′_{−} and u′′_{−} belong to U_{−w} the elements w^{−1}u′_{−}w and w^{−1}u′′_{−}w belong to B_{+}. Call the first one b′_{+} and the second one b′′_{+}, then we find

b′_{+}w^{−1}w′(c_{+}) = b′′_{+}w^{−1}w′′(c_{+}).

But this implies by the Bruhat decomposition of the group G (as we have a BN-pair in G) that w^{−1}w′ = w^{−1}w′′, yielding w′ = w′′. There remains to show that u′_{−}U_{−w′} = u′′_{−}U_{−w′′}. From the equality u′_{−}w′(c_{+}) = u′′_{−}w′(c_{+}) one deduces that w^{′−1}u_{−}^{′′−1}u′_{−}w′ ∈ B_{+}. The element u_{−}^{′′−1}u′_{−} is contained in U_{−w} and we call it u_{−w}. Then u_{−w} satisfies w^{′−1}u_{−w}w′ ∈ B_{+}. Consider the set of positive roots sent by w^{−1} into negative roots, namely {γ_{1}, …, γ_{n}}. Because of the properties of w we can divide this set into two subsets (after possibly reordering) {γ_{1}, …, γ_{l−1}} ⊔ {γ_{l}, …, γ_{n}}. Here {γ_{1}, …, γ_{l−1}} is the set of positive roots sent by w′ to a negative root and {γ_{l}, …, γ_{n}} is the set of remaining roots. With this notation in mind we write u_{−w} as u_{−w′}u_{−r} with u_{−w′} ∈ U_{−w′} and u_{−r} = u_{−γ_{l}} … u_{−γ_{n}}. We rewrite the formula w^{′−1}u_{−w}w′ ∈ B_{+} as

w^{′−1}u_{−r}w′ = w^{′−1}u_{−w′}^{−1}w′b̃_{+}

for a b̃_{+} ∈ B_{+}. The element w^{′−1}u_{−r}w′ apparently belongs to B_{+}. Suppose that w = w′ \underbrace{s_{j}s_{i} … s_{j}}_{m terms} with l(w) = l(w′) + m. Then

w^{′−1}{γ_{l}, …, γ_{n}} = {α_{j}, s_{j}(α_{i}), …, s_{j}s_{i} … s_{i}(α_{j})}.

Hence we can write w^{′−1}u_{−r}w′ as u_{−α_{j}}u_{−s_{j}(α_{i})} … u_{−s_{j}s_{i}...s_{i}(α_{j})} yielding that w^{−1}u_{−r}w′ ∈ U_{−} ∩ B_{+}. Now we look at the rank 2 building Γ_{ij} determined by B_{α_{i}}, B_{−α_{i}}, B_{α_{j}} and B_{−α_{j}} (i.e. the rank 2 building we get by considering the group ⟨B_{α_{i}}, B_{α_{j}}, B_{−α_{i}}, B_{−α_{j}}⟩ and the induced BN-pair in it). It follows that w^{′−1}u_{−r}w′ is inside the group generated by these four groups. But w^{′−1}u_{−r}w′ fixes the fundamental chamber c_{+}^{ij} in this polygon. Hence this element is inside U_{−}^{ij} ∩ B_{+}^{ij} where the groups B_{+}^{ij} and B_{−}^{ij} are similarly as above. The proof that κ is a 2-covering will be done if we show the following lemma.

Lemma 6. If we are given a spherical rank 2 building with Weylgroup ⟨s_{1}, s_{2}|(s_{1}s_{2})^{m_{12}}⟩ then

B_{+} ∩ B_{−} = H.

Proof. If we consider a spherical rank 2 Moufang building, the groups B_{+} and B_{−} both have a geometric meaning. Indeed, in the standard apartment Σ there will be two chambers c_{+} and c_{−} such that the l(d(c_{+}, c_{−})) is maximal in the Weylgroup. The group B_{+} will then be the stabilizer of c_{+} in G, B_{−} will be the stabilizer of c_{−} and H will be the stabilizer of the standard apartment in G. This implies in particular that B_{−} ∩B_{+} ⊆ H and as H ⊆ B_{+} ∩B_{−} we have

H = B_{+} ∩ B_{−}.  □

This lemma implies that w′u_{−r}w^{′−1} lies in H. Moreover by properties of spherical Moufang buildings explained in [7] on pages 75 and 76 it follows that u_{r}^{−1} = 1. This yields u_{−} ∈ U_{−w′} or u_{−}^{′′−1}u′_{−} ∈ U_{−w′}, hence u′′_{−}U_{−w′} = u′_{−}U_{−w′} what we wanted to show. This completes the proof of Theorem 3. □

As already mentioned the group U_{−} acts on both C^{−} and Δ in a way compatible with κ. This implies that Stab_{U_{−}}(a) = Stab_{U_{−}}(κ(a)) with a ∈ C^{−}. If we do this for c_{+} then κ(1) = c_{+} and Stab_{U_{−}}(1) = 1 and Stab_{U_{−}}(c_{+}) = U_{−} ∩ B_{+}. This gives us U_{−} ∩ B_{+} = {1}, which is a very strong condition. Consider B_{−} ∩ B_{+}. Every element in this intersection can be written as hu_{−} for h ∈ H and u_{−} ∈ U_{−}. But then u_{−} = 1 and the element is contained in H. As also H ⊆ B_{−} ∩ B_{+} we find

B_{−} ∩ B_{+} = H.

Using the universal properties of buildings we get the following corollary.

Corollary 2. The chamber system C^{−} is a building of type M isomorphic to (Δ, W, S, d) under κ.

Proof. This follows from the results in [10]. It is shown in this paper that every building is a universal object with respect to 2-coverings. This means that if we cover a building by a chamber system, then this covering is necessarily an isomorphism. □

Corollary 3. The pair (G, B_{−}, N, S) is a BN-pair.

Proof. The proof is similar to the proof of Theorem 2 as B_{α} ⊄ B_{−}, ∀α > 0. □

5. The relation O

We start from a Moufang building (Δ, W, S, d). The set of all roots in W is given by Φ = {α}. The rootgroups are denoted by U_{α}. We use notation as before. Then we know that there are two BN-pairs involved, (G, B_{+}, N, S) and (G, B_{−}, N, S). The first BN-pair yields a building (Δ_{+}, W, S, d_{+}) isomorphic to (Δ, W, S, d). From the second the building (Δ_{−}, W, S, d_{−}) is constructed. As the chambers of Δ_{+} and Δ_{−} correspond to cosets of B_{+} respectively B_{−} the group G acts in a natural way on both buildings. Let c_{+} and c_{−} be the chambers of Δ_{+} and Δ_{−} corresponding to B_{+} and B_{−}. We define the relation O ⊂ Δ_{+} × Δ_{−} ∪ Δ_{−} × Δ_{+} by the following rules: ( (x_{+}, y_{−}) ∈ Δ_{+} × Δ_{−}, (y_{−}, x_{+}) ∈ Δ_{−} × Δ_{+})

(x_{+}, y_{−}) ∈ O

⇕
∃g ∈ G such that (g(x_{+}), g(y_{−})) = (c_{+}, c_{−})

\begin{matrix}(y_{−}, x_{+}) ∈ O \\ ⇕ \\ (x_{+}, y_{−}) ∈ O \end{matrix}

We describe the relation O for rank 2 Moufang buildings.

Theorem 4. Suppose that (Δ, W, S, d) is a rank 2 Moufang building of spherical type then the relation O defines a twinning between Δ_{+} and Δ_{−}.

Proof. As the building Δ is of spherical type there exists a unique element w_{0} in W such that l(w_{0}) > l(w) ∀w ∈ W. We make the following construction. Set (Δ_{1}, W, S, d_{1}) = (Δ, W, S, d), (Δ_{2}, W, S, d_{2}) = (Δ, W, S, w_{0}dw_{0}). Define a codistance function d^{∗} between Δ_{1} and Δ_{2} by: ( (x_{1}, x_{2}) ∈ Δ_{1} × Δ_{2}, (x_{2}, x_{1}) ∈ Δ_{2} × Δ_{1})

d^{∗}(x_{1}, x_{2}) = w_{0}d(x_{1}, x_{2})
d^{∗}(x_{2}, x_{1}) = d(x_{1}, x_{2})w_{0}.

It follows from Proposition 1 in [12] that the couple ((Δ_{1}, W, S, d_{1}), (Δ_{2}, W, S, d_{2})) with the codistance function d^{∗} is a twin building. It can be shown that this is the only possible twinning on Δ. We know that two BN-pairs (G, B_{+}, N, S) and (G, B_{−}, N, S) can be constructed. Each of these BN-pairs has an associated building. Denote them by (Δ_{+}, W, S, d_{+}) and (Δ_{−}, W, S, d_{−}).

We give a short description of (Δ_{+}, W, S, d_{+}). The set of chambers Δ_{+} is given by the set {gB_{+} | g ∈ G}. Let s ∈ S then g_{1}B_{+} is s-adjacent to g_{2}B_{+} if and only if B_{+}g_{1}^{−1}g_{2}B_{+} = B_{+}sB_{+}. To define the distance between two chambers one uses the Bruhat decomposition of the group G. This means that the group G has a decomposition

G = ⊔(B_{+}wB_{+})_{w∈W} .

Moreover if B_{+}w′B_{+} = B_{+}w′′B_{+} then it follows that w′ = w′′. For two chambers g_{1}B_{+} and g_{2}B_{+} of Δ_{+} the distance d(g_{1}B_{+}, g_{2}B_{+}) is defined as the unique element v ∈ W such that

B_{+}g_{1}^{−1}g_{2}B_{+} = B_{+}vB_{+}.

Using standard arguments it follows that (Δ_{+}, W, S, d_{+}) is a building. The same can be done for (G, B_{−}, N, S). This gives the building (Δ_{−}, W, S, d_{−}). From the construction of (Δ_{+}, W, S, d_{+}) it can be proved that it is isomorphic to (Δ, W, S, d). The isomorphism is given by

φ_{1} : Δ_{+} → Δ
φ_{1}(gB_{+}) ↦ g(c_{+}).

In a similar way (Δ_{−}, W, S, d_{−}) is isomorphic to (Δ_{2}, W, S, d_{2}). Consider the group B_{−}. As we work in a spherical building it follows that B_{−} = ⟨U_{−w_{0}}, H⟩. Hence w_{0}B_{+}w_{0} = B_{−}. By this we can map every chamber of Δ_{−} to a chamber of Δ_{+}. Namely every hB_{−} can be written as hw_{0}B_{+}w_{0}. If we send every hB_{−} to hw_{0}B_{+} this is well defined. Call this map Opp_{w_{0}}. The composition φ_{2} = Opp_{w_{0}} ∘ φ_{1} defines a bijection of Δ_{−} to Δ. Moreover φ_{2} sends s-adjacent chambers to w_{0}sw_{0}-adjacent chambers. This implies that (Δ_{−}, W, S, d_{−}) is isomorphic to (Δ_{2}, W, S, w_{0}dw_{0}) under φ_{2}. The explicit formula for φ_{2} is given by

φ : Δ_{−} → Δ_{2}
hB_{−} ↦ hw_{0}(c_{+}).

To finish the proof we show the following equivalence: ((x_{+}, y_{−}) ∈ Δ_{+} × Δ_{−})

(x_{+}, y_{−}) ∈ O ⇔ d^{∗}(φ_{1}(x_{+}), φ_{2}(y_{−})) = 1.

(1) If (x_{+}, y_{−}) ∈ O then x_{+} = gB_{+} and y_{−} = gB_{−}, with g ∈ G. Hence φ_{1}(x_{+}) = g(c_{+}) and φ_{2}(y_{−}) = gw_{0}(c_{+}). We calculate

d(g(c_{+}), gw_{0}(c_{+})) = d(c_{+}, w_{0}(c_{+}))
= d(c_{+}, w_{0}c_{+})
= w_{0}.

This implies that d^{∗}(φ_{1}(x_{+}), φ_{2}(y_{−})) = 1.

(2) Suppose gB_{+} and hB_{−} are such that d^{∗}(φ_{1}(gB_{+}), φ_{2}(hB_{−})) = 1. This means that d(g(c_{+}), hw_{0}(c_{+})) = w_{0}. Using the isomorphism φ_{1} and the Bruhat decomposition of G if follows that

hb_{−} = gb_{+}

for appropriate b_{−} ∈ B_{−} and b_{+} ∈ B_{+}. This means that (gB_{+}, hB_{−}) ∈ O. □

Remains to prove the same result for non-spherical rank 2 Moufang buildings. Let (Γ, W, S, d) be such a building. We consider a graph whose vertex set V is the set of all residues in Γ. Two vertices are joined by an edge if and only if they lie in a chamber. In this way we get a bipartite graph (V, E), which turns out to be a tree. It can also be easily checked that every isomorphism of Γ as building induces an isomorphism of the tree (V, E). For more information about non-spherical rank 2 Moufang buildings we refer to [8]. The result we will prove is:

Theorem 5. Given a non-spherical rank 2 Moufang building (Γ, W, S, d) then the relation O defines the opposition relation of a twinning between Δ_{+} and Δ_{−}.

Proof. First we fix some notations and terminology. Denote W = {s, t}. The chambers of Γ will be considered as doubletons {x, x′}, where x and x′ stand for the simplices in the chamber {x, x′}. We assume that the standard chamber is given by c_{0} = {x_{0}, x_{1}} and the standard apartment Σ_{0} is the sequence … c_{−2} ∼^{t} c_{−1} ∼^{s} c_{0} ∼^{t} c_{1} ∼^{s} c_{2} . . .. Write c_{i} = {x_{i}, x_{i+1}}, ∀i. Then the standard apartment Σ_{0} corresponds to a sequence … x_{−2} ∼ x_{−1} ∼ x_{0} ∼ x_{1} ∼ x_{2} … in the tree (V, E). As to the Moufang structure on Γ we keep the notations from above. Let α_{i}^{+} be the positive root of Σ_{0} such that x_{i} lies on its boundary. Similarly α_{i}^{−} is the negative root of Σ_{0} such that x_{i} lies on ∂α_{i}^{−}. By calculations already made there are two BN-pairs involved: (G, B_{+}, N, S) and (G, B_{−}, N, S). They give rise to two buildings (Δ_{+}, W, S, d_{+}) and (Δ_{−}, W, S, d_{−}). To prove that O is the opposition relation of a twinning between Δ_{+} and Δ_{−} we refer to Proposition 5.4. of [2]. In order to use this proposition we show the following:

(i) The relation O defines a 1-twinning between Δ_{+} and Δ_{−}. (ii) For any four chambers y_{−}, c_{−}^{1} and c_{−}^{2} in Δ_{−} and e_{+} ∈ Δ_{+} such that (e_{+}, c_{−}^{1}) ∈ O, (e_{+}, c_{−}^{2}) ∈ O and

l(d_{−}(c_{−}^{1}, y_{−})) = l(d_{−}(c_{−}^{2}, y_{−}))
= min{l(d_{−}(a_{−}, y_{−}))|(e_{+}, a_{−}) ∈ O}

we have d_{−}(c_{−}^{1}, y_{−}) = d_{−}(c_{−}^{2}, y_{−}). (iii) For any four chambers y_{−} ∈ Δ_{−}, y_{+}^{1}, y_{+}^{2}, e_{+} ∈ Δ_{+} such that (y_{+}^{1}, y_{−}) ∈ O, (y_{+}^{2}, y_{−}) ∈ O and

l(d_{+}(e_{+}, y_{+}^{1})) = l(d_{+}(e_{+}, y_{+}^{2}))
= min{l(d_{+}(a_{+}, c_{+}))|(a_{+}, y_{−}) ∈ O}

we have d_{+}(y_{+}^{1}, e_{+}) = d_{+}(y_{+}^{2}, e_{+}).

(iv) Given chambers y_{−}, a_{−} ∈ Δ_{−}, e_{+} and b_{+} ∈ Δ such that a_{−} is as in (ii), l(d(a_{−}, y_{−})) is minimal, b_{+} is as in (iii) and l(d(d_{+}, b_{+})) is minimal then

d_{+}(e_{+}, b_{+}) = d(a_{−}, y_{−}).

If (i), (ii), (iii) and (iv) are satisfied we define for every x ∈ Δ_{ϵ} (ϵ ∈ {1, −1}) a codistance function d_{x} : Δ_{−} ↦ W. For every z ∈ Δ_{−ϵ}, d_{x}(z) equals d_{−ϵ}(x_{−ϵ}, z) with (x, x_{−ϵ}) ∈ O such that l(d(x_{−ϵ}, z) is mimimal as in (ii) or (iii). One easily checks this defines a codistance funtion for every x. Remains to check these 4 properties:

(1) Because of the definition of O it suffices to check that O defines a 1-twinning between the s-residue R_{+}^{s} in Δ_{+} containing c_{+} and the s-residue R_{−}^{s} in Δ_{−} containing c_{−}. We check that for all the chambers x_{−} of R_{−}^{s} satisfy (x_{−}, c_{+}) ∈ O except s(c_{−}). As the stabilizer of R_{+}^{s} and R_{−}^{s} acts transitively on the chambers of these residues this will be enough to ensure that O defines a 1-twinning between R_{+}^{s} and R_{−}^{s}. Every element of R_{−}^{s} has the form u_{−α_{s}}s(c_{−}) for u_{−α_{s}} ∈ U_{−α_{s}}. Suppose that u_{−α_{s}} ≠ 1. Granted the properties of the BN-pair (G, B_{−}, N, S) we can write u_{−α_{s}}sc_{−} = u_{α_{s}}su′_{α_{s}}sc_{−} for appropriate u_{α_{s}} and u′_{α_{s}} ∈ U_{α_{s}}. But then u_{−α_{s}}s(c_{−}) = u_{α_{s}}(c_{−}). And (c_{+}, u_{−α_{s}}(c_{−})) = (u_{α_{s}}(c_{+}), u_{α_{s}}(c_{−})). Hence (c_{+}, u_{−α_{s}}(c_{−})) ∈ O. Consider the chamber s(c_{−}). If (c_{+}, s(c_{−})) ∈ O then there would exist a g ∈ G such that g(c_{+}) = c_{+} and g(s(c_{−})) = c_{−}. But then g ∈ B_{+} and gs ∈ B_{−} or s = b_{+}b_{−} for b_{+} ∈ B_{+} and b_{−} ∈ B_{−}. This contradicts the fact that s stabilizes the standard apartment Σ_{0}. Hence (s(c_{−}), c_{+}) ∉ O. Granted the action of G on Δ_{+} and Δ_{−} we may assume that d_{+} = c_{+} in (2), (3) and (4).

(2) Suppose that y_{−}, c_{−}^{1} and c_{−}^{2} are chambers as in (ii) with (c_{+}, y_{−}) ∉ O. Then y_{−} = gB_{−}, c_{−}^{1} = b_{+}^{1}B_{−} and c_{−}^{2} = b_{+}^{2}B_{−} for g ∈ G, b_{+}^{i} ∈ B_{+}. Let d_{−}(c_{−}^{1}, y_{−}) = w_{1} and d_{−}(c_{−}^{2}, y_{−}) = w_{2}. It follows from the assumptions that l(w_{1}) = l(w_{2}). Assume w_{1} ≠ w_{2}. Because we work in a non-spherical Coxeter group two possibilities occur. Namely w_{1}^{2} = w_{2}^{2} = 1 or w_{1}^{2} ≠ 1 and w_{2}^{2} ≠ 1. Expressing that the distances from c_{−}^{1} and c_{−}^{2} to y_{−} are w_{1} and w_{2} gives:

gB_{−} = b_{+}^{1}b_{−}^{1}w_{1}B_{−}
= b_{+}^{2}b_{−}^{2}w_{2}B_{−}

for b_{−}^{i} ∈ B_{−}. Hence

b_{+}^{1}b_{−}^{1}w_{1} = b_{+}^{2}b_{−}^{2}w_{2}b_{−}

for b_{−} ∈ B_{−}. But then

(b_{+}^{2})^{−1}b_{+}^{1} = b_{−}^{2}w_{2}b_{−}w_{1}^{−1}(b_{−}^{1})^{−1}.

If w_{1}^{2} = w_{2}^{2} = 1 then

b_{−}^{2}w_{2}b_{−}w_{1}^{−1}(b_{−}^{1})^{−1} = b′_{−}w_{2}w_{1}b′′_{−}

for b′_{−}, b′′_{−} ∈ B_{−}. If w_{1}^{2} ≠ 1 and w_{2}^{2} ≠ 1 then w_{1}w_{2} = 1 and

b_{−}^{2}w_{2}b_{−}w_{1}^{−1}(b_{−}^{1})^{−1} = b′_{−}w_{2}^{2}b′′_{−}

for b′_{−}, b′′_{−} ∈ B_{−}. In all cases we find that if w_{1} ≠ w_{2} then for a v ≠ 1, b′_{−} and b′′_{−} ∈ B_{−}

b′_{−}vb′′_{−} ∈ B_{+},

with l(v) = 0 mod 2. This means that b′_{−}vb′′_{−} has to fix the chamber c_{+}. Write b′_{−}vb′′_{−} = u′_{−}vu′′_{−}h for h ∈ H. Then u′_{−}vu′′_{−} has to fix c_{+}. Two cases occur:

(a) u′′_{−} = 1. Then we have u′_{−}(v(x_{0}) = x_{0} and u′_{−}(v(x_{1})) = x_{1}. This is only possible if v = 1 and u′_{−} = 1.

(b) The element u′′_{−} ≠ 1. Suppose that W = {s, t}, ∂α_{s} = x_{0}, ∂α_{t} = x_{1}. If u′′_{−} ∈ U_{−α_{s}} we find that u′_{−}vu′′_{−}(x_{0}) = x_{0}. Granted the condition on u′′_{−} this implies that u′_{−}(v(x_{0})) = x_{0}. Again a contradiction. Hence there exists an index j such that x_{j} ∼ y_{1} ∼ y_{2} ∼ … ∼ u′′_{−}(x_{0}) ∼ u′′_{−}(x_{1}) is the gallery in Γ from Σ_{0} to u′′_{−}(x_{1}). Suppose that j < 0 (we already excluded the case where j = 0). Because l(v) = 0 mod 2, v acts as a translation of Σ_{0}, i.e.

v(x_{l}) = x_{l+k_{0}}, ∀l

for a fixed k_{0} ∈ ℤ. Let v(x_{j}) = x_{m}. If m ≤ 0 then d_{+}(x_{j}, u′′_{−}(x_{0})) ≠ d_{+}(x_{m}, x_{0}). One easily checks that there cannot exist a u′_{−} ∈ U_{−} with u′_{−}(vu′′_{−}(x_{0})) = x_{0}. If m ≥ 1 then

d_{+}(x_{m}, vu′′_{−}(x_{0})) < d_{+}(x_{m}, vu′′_{−}(x_{1})).

Using this fact one also checks that for no u′_{−} ∈ U_{−} we can have u′_{−}(vu′′_{−}(x_{0})) = x_{0}. If j > 0 one uses similar arguments to deduce a contradiction.

(3) If (y_{+}^{1}, y_{−}) and (y_{+}^{2}, y_{−}) ∈ O then

y_{−} = g(c_{−})
y_{+}^{1} = g(c_{+})
y_{+}^{2} = gb_{−}(c_{+})

for g ∈ G and b_{−} ∈ B_{−}. A symmetric proof completely analogous to (2) gives d_{+}(y_{+}^{1}, c_{+}) = d_{+}(y_{+}^{2}, c_{+}).

(4) Let y_{−} and c_{−}^{1} be chambers of Δ_{−} with (c_{+}, c_{−}^{1}) ∈ O and d(c_{−}^{1}, y_{−}) being minimal as in (ii). Then we look for a chamber y_{+} in Δ_{+} such that (y_{+}, y_{−}) ∈ O and d_{+}(c_{+}, y_{+}) = d_{−}(c_{−}^{1}, y_{−}).

This will imply (iv). Without loss of generality we can assume that c_{−}^{1} = c_{−}. Let the minimal gallery in Δ between c_{−} and y_{−} be

y_{−}^{0} = c_{−} ∼^{s} y_{−}^{1} ∼^{t} y_{−}^{2} ∼^{s} … ∼^{t} y_{−}^{m} = y_{−}

If y_{−}^{1} = u_{−α_{s}}s(c_{−}) let y_{+}^{1} be u_{−α_{s}}s(c_{+}). If y_{−}^{2} = u_{−α_{t}}tu_{−α_{s}}s(c_{−}) let y_{+}^{2} be u_{−α_{t}}tu_{−α_{s}}s(c_{+}). If we do this for all y_{−}^{i} we get a gallery

y_{+}^{0} = c_{+} ∼^{s} y_{+}^{1} ∼^{t} y_{+}^{2} ∼^{s} … ∼^{t} y_{+}^{m}

from c_{+} to y_{+}^{m}. One shows with a proof similar as in (2) that for no v ∈ W and b_{−}, b′_{−} ∈ B_{−} we can have that b_{−}vb′_{−} ∈ B_{+}. This ensures us that all the y_{+}^{j} are different. The gallery is therefore non-stammering and d_{+}(c_{+}, y_{+}^{m}) = d_{−}(c_{−}^{1}, y_{−}). By construction we have (y_{+}, y_{−}) ∈ O. This completes the proof that (i), (ii), (iii) and (iv) are satisfied for O. Hence O is the opposition relation of a twinning between Δ_{+} and Δ_{−}. □

6. Constructing a 2-twinning

In this paragraph we will show that the building (Δ, W, S, d) is half of a twin building using a result of B. Mühlherr in [2]. We restate the main result of loc. cit.

Theorem 6. Let M be a Coxeter matrix over I, let (Δ_{+}, W, S, d_{+}) and (Δ_{−}, W, S, d_{−}) be two thick buildings of type M and let O ⊆ (Δ_{+} × Δ_{−}) ∪ (Δ_{−} × Δ_{+}) be a non-empty symmetric relation. Then O is the opposition relation of a twinning between (Δ_{+}, W, S, δ_{+}) and (Δ_{−}, W, S, δ_{−}) if and only if the following condition is satisfied: If J ⊆ I is of cardinality at most 2 and if R_{+} ⊆ Δ_{+} and R_{−} ⊆ Δ_{−} are J-residues, then either O ∩ ((R_{+} × R_{−}) ∪ (R_{−} × R_{+})) = ∅ or O ∩ ((R_{+} × R_{−}) ∪ (R_{−} × R_{+})) is the opposition relation of a twinning between R_{+} and R_{−}.

We now have:

Theorem 7. Given a Moufang building (Δ, W, S, d) with root groups (U_{α})_{α∈Φ} then Δ is half of a twin building, i.e. there exists a building (Δ_{−}, W, S, d_{−}) and a codistance function d^{∗} such that ((Δ, W, S, d), (Δ_{−}, W, S, d_{−}), d^{∗}) is a twin building.

Proof. By Theorem 2 and Corollary 3 we know that there are two BN-pairs involved. Namely (G, B_{+}, N, S) and (G, B_{−}, N, S). The building (Δ_{+}, W, S, d_{+}) associated to (G, B_{+}, N, S) is by construction isomorphic to Δ. We define the symmetric relation O between Δ_{+} and Δ_{−} as before. Consider s_{i}, s_{j} ∈ S. Let R_{s_{i}s_{j}}^{+} and R_{s_{i}s_{j}}^{−} be the {s_{i}, s_{j}}-residues in Δ_{+} and Δ_{−} containing c_{+} and c_{−} respectively. Then it follows from Theorem 4 and Theorem 5 that O defines the opposition relation of a twinning between R_{s_{i}s_{j}}^{+} and R_{s_{i}s_{j}}^{−}. By construction this implies that O satisfies the conditions of Theorem 6. Hence O defines a twinning between Δ_{+} and Δ_{−}. This means that Δ ≅ Δ_{+} is half of a twin building. □

References

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[6] Mühlherr, B.; Van Maldeghem, H.: On certain twin buildings over tree diagrams. Bull. Belg. Math. Soc. (1998), 393–402.

[7] Ronan, M.: Lectures on Buildings. Academic Press 1989.

[8] Ronan, M.; Tits, J.: Twin trees I. Invent. Math. 116 (1994), 463–479.

[9] Tits, J.: Buildings of Spherical Type and Finite BN-Pairs. Lecture Notes in Math. 386, Springer 1974.

[10] Tits, J.; Ensembles ordonnés, immeubles et sommes amalgamées. Bull. Soc. Math. Belgique 38 (1986), 367–387.

[11] Tits, J.: Uniqueness and presentation of Kac-Moody groups over fields. J. Algebra 105 (1987), 542–573.

[12] Tits, J.: Twin buildings and groups of Kac-Moody type. London Math. Soc. Lecture Notes 165, (Proceedings of a conference on groups, combinatorics and geometry, Durham 1990), Cambridge University Press 1992, 249–286.

[13] Tits, J.; Weiss, R.: The classification of Moufang polygons. In preparation.

Received January 21, 2000