3-spherical twin buildings

We classify thick irreducible 3-spherical twin buildings of rank at least 3 in which every panel contains at least 6 chambers. Together with the Main result of [11] we obtain a classification of thick irreducible 3-spherical twin buildings.


Introduction
In [20] Tits gave a complete classification of all thick irreducible spherical buildings of rank at least 3.The decisive step in this classification is the extension theorem for isometries (Theorem 4.1.2in loc.cit.).It implies that a thick spherical building is uniquely determined by its local structure.Inspired by the paper [21] on Kac-Moody groups over fields, Ronan and Tits introduced twin buildings.These combinatorial objects appear to be natural generalisations of spherical buildings, as they come equipped with an opposition relation which shares many important properties with the opposition relation in a spherical building.In [15] Mühlherr and Ronan gave a proof of the extension theorem for 2-spherical twin buildings satisfying an additional condition they call (co).Condition (co) turns out to be rather mild (see [15,Introduction]).In order to classify such twin buildings it suffices to determine all possible local structures which appear in twin buildings.
The local structure of a building mentioned before is essentially the union of all rank 2 residues of a chamber.In [18] Ronan and Tits introduced the notion of a foundation which is designed to axiomatize geometric structures that may occur as local structures of a building.Roughly speaking, it is a union of rank 2 buildings which are glued along certain rank 1 residues (for the precise definition we refer to Section 3).A foundation is called integrable, if it is the local structure of a twin building.One can associate with each twin building of 2-spherical type a foundation and all such are Moufang foundation, i.e. the irreducible rank 2 buildings are Moufang and the glueings are compatible with the Moufang structures induced on the rank 1 residues.
Inspired by [18], Tits conjectured that a Moufang foundation is integrable if and only if each of its spherical rank 3 restrictions is integrable (cf.[22,Conjecture 2]).It turned out that this conjecture was too optimistic and he gave a reformulation by omitting the word "spherical" in his earlier conjecture (cf.[23]).A proof of this conjecture would reduce the classification of 2-spherical twin buildings to the rank 3 case.In [12] and [13] a strategy for the classification of 2-spherical twin buildings is outlined that does not make use of this conjecture.Several important steps in this classification programme have been carried out in the meantime (cf.[11], [25], [26]).The results obtained up until now suggest that the conjecture will be a consequence of the classification once it will be accomplished.However, it would be most desirable to have a proof of this conjecture that is independent of the classification.Our main result is a conceptual proof of this conjecture in the 3-spherical case.It turned out that we only need the integrability for the restriction to irreducible types (cf.Corollary (5.3)): Theorem A: Let F be a Moufang foundation of irreducible 3-spherical type and of rank at least 3 such that every panel contains at least 6 chambers.Then the following are equivalent: (i) F is integrable.
(ii) Each irreducible rank 3 restriction of F is integrable.
A consequence of Theorem A together with [11,4] is the classification of thick irreducible 3-spherical twin buildings (cf.Theorem (5.4)): Corollary B: Let ∆ be a thick irreducible 3-spherical twin building of rank at least 3. Then ∆ is known.
Let (W, S) be a Coxeter system and let Φ be the associated set of roots (viewed as half-spaces).An RGD-system of type (W, S) is a pair (G, (U α ) α∈Φ ) consisting of a group G together with a family of subgroups U α indexed by the set of roots satisfying a few axioms (for the precise definition see Section 2).Let F be a Moufang foundation of type (W, S) satisfying a certain Condition (lco) (e.g. this is satisfied if every panel contains at least 5 chambers).In Section 3 we construct for all s = t ∈ S RGD-systems X s,t acting on the corresponding building of the foundation.Moreover, we construct RGD-systems X s and canonical homomorphisms X s → X s,t .Let G be the direct limit of the groups X s and X s,t .Then there is a natural way of defining a family of root groups (U α ) α∈Φ inside G. Let D F := (G, (U α ) α∈Φ ).We prove the following result (cf.Corollary (3.13)): Theorem C: Let F be a Moufang foundation of 2-spherical type satisfying Condition (lco).If the canonical mappings U ±αs → G are injective and if D F satisfies (RGD3), then D F is an RGD-system and F is integrated in the twin building associated to D F .We use Theorem C to prove Theorem A. Our strategy is to let G act on a building and deduce that the hypotheses of Theorem C are satisfied.Thus we have a twin building ∆(D F ) in which F is integrated.In particular, we have the following corollary (cf.Theorem (5.2)): Corollary D: Let F be an irreducible Moufang foundation of 3-spherical type such that every panel contains at least 6 chambers.If each irreducible rank 3 restriction of F is integrable, then D F is an RGD-system.

Remarks:
1.In [11] Mühlherr accomplished the classification of thick locally finite twin buildings of irreducible 2-spherical type and m st < 8 without residues associated with one of the groups B 2 (2), G 2 (2), G 2 (3), 2 F 4 (2).In particular, the thick locally finite twin buildings of irreducible 3-spherical type without residues associated with B 2 (2) are already known.As we will see in the proof of Theorem (5.4), the assumption about the residues can be dropped in the 3-spherical case.

2.
By Corollary B we have a classification of 3-spherical simply-laced (i.e.m st ∈ {2, 3}) twin buildings.We note that in this case the integrability of those Moufang foundations is already established by different methods: In a 3-spherical simply-laced Moufang foundation, the Moufang triangles are parametrised by skew-fields and all such are isomorphic.If there exists s ∈ S with three neighbours in the Coxeter diagram, then there is a D 4 subdiagram (as there is no Ã2 subdiagram) and hence the Moufang triangles are parametrised by a field.The existence of a twin building with the prescribed foundation follows now from [11].If each s ∈ S has at most 2 neighbours, the diagram is either A n or Ãn .In the first case the existence follows from projective geometry and in the second case from Kac-Moody theory.
Acknowledgement I am very grateful to Bernhard Mühlherr for proposing this project to me, as well as for many helpful comments and suggestions.

Direct limits
This subsection is based on [19].
Let I be a set and let (G i ) i∈I be a family of groups.Furthermore, let F i,j be a set of homomorphisms of G i into G j .Then a group G := lim In this paper we only consider the case where X i and X i,j := X i , X j are groups for i = j ∈ I and f : X i → X i,j are the canonical homomorphisms.We call this direct limit the 2-amalgam of the groups X i .

Coxeter systems
Let (W, S) be a Coxeter system and let ℓ denote the corresponding length function.For s, t ∈ S we denote the order of st in W by m st .The rank of a Coxeter system is the cardinality of the set S. The Coxeter diagram corresponding to (W, S) is the labeled graph (S, E(S)), where E(S) = {{s, t} | m st > 2} and where each edge {s, t} is labeled by m st for all s, t ∈ S. We call a Coxeter system irreducible, if the underlying graph is connected; otherwise we call it reducible.It is well-known that the pair ( J , J) is a Coxeter system (cf.[5, Ch.IV, §1 Theorem 2]).A subset J ⊆ S is called spherical if J is finite; for k ∈ N the Coxeter system is called k-spherical, if J is spherical for each subset J of S with |J| ≤ k.Given a spherical subset J of S, there exists a unique element of maximal length in J , which we denote by r J we define for distinct s, t ∈ S with m st < ∞ the element p(s, t) to mean stst . . .with ℓ(p(s, t)) = m st ; e.g. if m st = 3, we have p(s, t) = sts.It is well-known that for w ∈ W, s ∈ S with ℓ(sw) < ℓ(w), there exists w ′ ∈ W such that ℓ(w ′ ) = ℓ(w) − 1 and w = sw ′ (cf.[2, exchange condition on p.79]).
(2.1) Convention.For the rest of this paper we let (W, S) be a Coxeter system.

Chamber systems
Let I be a set.A chamber system over I is a pair (C, (∼ i ) i∈I ) consisting of a non-empty set C whose elements are called chambers and where ∼ i is an equivalence relation on the set of chambers for each i ∈ I.
The chambers c, d are called adjacent, if they are i-adjacent for some i ∈ I.If we restrict ∼ i to ∅ = X ⊆ C for each i ∈ I, then (X , (∼ i ) i∈I ) is a chamber system over I and we call it the induced chamber system.
A gallery in (C, (∼ i ) i∈I ) is a sequence (c 0 , . . ., c k ) such that c µ ∈ C for all 0 ≤ µ ≤ k and c µ−1 , c µ are adjacent for all 1 ≤ µ ≤ k.Given a gallery G = (c 0 , . . ., c k ), then we put β(G) := c 0 and ε(G) := c k .If G is a gallery and if c, d ∈ C are such that β(G) = c and ε(G) = d then we say that G is a gallery from c to d or G joins c and d.The chamber system (C, (∼ i ) i∈I ) is said to be connected, if for any two chambers there exists a gallery joining them.A gallery G will be called closed if β(G) = ε(G).Given two galleries G = (c 0 , . . ., c k ) and H = (d 0 , . . ., d l ) such that ε(G) = β(H), then GH denotes the gallery (c 0 , . . ., c k = d 0 , . . ., d l ).
Let J be a subset of I.A J-gallery is a gallery (c 0 , . . ., c k ) such that for each 1 ≤ µ ≤ k there exists an index j ∈ J with c µ−1 ∼ j c µ .

Homotopy of galleries and simple connectedness
In the context of chamber systems there is the notion of m-homotopy and m-simple connectedness for each m ∈ N. In this paper we are only concerned with the case m = 2. Therefore our definitions are always to be understood as a specialisation of the general theory to the case m = 2.
Let (C, (∼ i ) i∈I ) be a chamber system over a set I. Two galleries G and H are said to be elementary homotopic if there exist two galleries X, Y and two J-galleries G ′ , H ′ for some J ⊆ I of cardinality at most 2 such that G = XG ′ Y, H = XH ′ Y .Two galleries G, H are said to be homotopic if there exists a finite sequence G 0 , . . ., G l of galleries such that If two galleries G, H are homotopic, then it follows by definition that β(G) = β(H) and ε(G) = ε(H).A closed gallery G is said to be null-homotopic if it is homotopic to the gallery (β(G)).The chamber system (C, (∼ i ) i∈I ) is called simply connected if it is connected and if each closed gallery is null-homotopic.
The rank of ∆ is the rank of the underlying Coxeter system.The elements of C are called chambers.Given s ∈ S and x, y ∈ C, then x is called s-adjacent to y, if δ(x, y) = s.The chambers x, y are called adjacent, if they are s-adjacent for some s ∈ S. A gallery from x to y is a sequence (x = x 0 , . . ., x k = y) such that x l−1 and x l are adjacent for every 1 ≤ l ≤ k; the number k is called the length of the gallery.Let (x 0 , . . ., x k ) be a gallery and suppose s i ∈ S such that δ(x i−1 , x i ) = s i .Then (s 1 , . . ., s k ) is called the type of the gallery.A gallery from x to y of length k is called minimal if there is no gallery from x to y of length < k.Given a subset J ⊆ S and x ∈ C, the J-residue of x is the set R J (x) := {y ∈ C | δ(x, y) ∈ J }.A residue is a subset R of C such that there exist J ⊆ S and x ∈ C with R = R J (x).Since the subset J is uniquely determined by R, the set J is called the type of R and the rank of R is defined to be the cardinality of J.Each J-residue is a building of type ( J , J) with the distance function induced by δ (cf.[2, Corollary 5.30]).Given x ∈ C and a J-residue R ⊆ C, then there exists a unique chamber z ∈ R such that ℓ(δ(x, y)) = ℓ(δ(x, z)) + ℓ(δ(z, y)) holds for every y ∈ R (cf.[2, Proposition 5.34]).The chamber z is called the projection of x onto R and is denoted by proj R x.Moreover, if z = proj R x we have δ(x, y) = δ(x, z)δ(z, y) for each y ∈ R. A residue is called spherical if its type is a spherical subset of S. A building is spherical if its type is spherical; for k Let R be a spherical J-residue.Then x, y ∈ R are called opposite in R if δ(x, y) = r J .A panel is a residue of rank 1.An s-panel is a panel of type {s} for s ∈ S. For c ∈ C and s ∈ S we denote the s-panel containing c by P s (c).The building ∆ is called thick, if each panel of ∆ contains at least three chambers.For an s-panel P we will also write P s instead of P to underline the type of P .We denote the set of all panels in a given building ∆ by P ∆ .For c ∈ C and k ∈ N we denote the union of all residues of rank at most k containing c by E k (c).Let ∆ = (C, δ), ∆ ′ = (C ′ , δ ′ ) be two buildings of type (W, S) and let X ⊆ C, X ′ ⊆ C ′ .Then a map ϕ : X → X ′ is called isomorphism if it is bijective and preserves the distance functions.In this case we will write X ∼ = X ′ .We denote the set of all isomorphisms from a building ∆ to itself by Aut(∆).Coxeter buildings ) is a building of type (W, S).Moreover, W acts on Σ(W, S) by left-multiplication.
(2.4) Convention.For the rest of this paper we let (W, S) be a Coxeter system of finite rank and Φ := Φ(W, S) (resp.Φ + , Φ − ) be the set of roots (resp.positive roots, negative roots).
For α ∈ Φ we denote by ∂α (resp.∂ 2 α) the set of all panels (resp.spherical residues of rank 2) stabilized by r α .The set ∂α is called the wall associated to α.For a gallery G = (c 0 , . . ., c k ) we say that G crosses the wall ∂α if {c i−1 , c i } ∈ ∂α for some 1 ≤ i ≤ k.
(2.5) Lemma.Let α ∈ Φ and let P, Q ∈ ∂α.Then there exist a sequences P 0 = P, . . ., P n = Q of panels in ∂α and a sequence R 1 , . . ., R n of spherical rank 2 residues in ∂ 2 α such that P i−1 , P i are distinct and contained in R i .
Proof.This is a consequence of [7,Proposition 2.7].The fact that P 0 , . . ., P n ∈ ∂α follows from the implication (iii)⇒(ii) in loc.cit.Since Two conditions for buildings (2.6) Example.Let ∆ = (C, δ) be a building of type (W, S).We define x ∼ s y if δ(x, y) ∈ s .Then ∼ s is an equivalence relation and (C, (∼ s ) s∈S ) is a chamber system.
We now introduce two conditions for a building: (lco) A building ∆ satisfies Condition (lco) if it is 2-spherical and if R is a rank 2 residue of ∆ containing a chamber c, then the induced chamber system defined by the set of chambers opposite c inside R is connected.
(lsco) A building ∆ satisfies Condition (lsco) if it is 3-spherical and if R is a rank 3 residue of ∆ containing a chamber c, then the induced chamber system defined by the set of chambers opposite c inside R is simply connected.

Spherical Moufang buildings
Let ∆ = (C, δ) be a thick irreducible spherical building of type (W, S) and of rank at least 2. For a root α of ∆ we define the root group U α as the set of all automorphisms fixing α pointwise and fixing every panel P pointwise, where |P ∩ α| = 2.The building ∆ is called Moufang if for every root α of ∆ the root group U α acts simply transitive on the set of apartments containing α.

Twin buildings
and where δ * is a twinning between ∆ + and ∆ − .The twin building is called thick, if ∆ + and ∆ − are thick.The twin building ∆ satisfies Condition (lco) if both buildings ∆ + , ∆ − satisfy Condition (lco).For ε ∈ {+, −} and c ∈ C ε we define preserves the sign, the distance and the codistance.5.153] twice, we obtain that the s-panel containing c ′ contains at least three chambers for each s ∈ S. Thus ∆ ε is a thick building.The thickness of ∆ −ε follows similarly.

RGD-systems
An RGD-system of type (W, S) is a pair G, (U α ) α∈Φ consisting of a group G together with a family of subgroups U α (called root groups) indexed by the set of roots Φ, which satisfies the following axioms, where Let D = (G, (U α ) α∈Φ ) be an RGD-system of type (W, S) and let Theorem 8.80] that there exists an associated twin building satisfies Condition (lco), it follows by the Main result of [3] that G is isomorphic to the direct limit of the inductive system formed by the groups L s := HX s and L s,t := HX s,t for all s = t ∈ S. Furthermore, the direct limit of the inductive system formed by the groups X s and X s,t (s = t ∈ S) can be naturally endowed with an RGD-system and is a central extension of G (cf. [6, Theorem 3.7]).
For s ∈ S we define For the rest of this subsection we fix 1 = e s ∈ U αs and put n s := m(e s ).By [2, Consequence (11) on p.416] there exist for every As in the case of a Coxeter system, we define p(n s , n t ) to mean n s n t n s n t . .., where n s , n t appear m st times, e.g. if m st = 3, we have p(n s , n t ) = n s n t n s .By [16,Lemma 7.3] we have p(n s , n t ) = p(n t , n s ).
(2.9) Theorem.Let (G, (U α ) α∈Φ ) be an RGD-system spherical type (W, S) such that G = U α | α ∈ Φ .Let B := U + , H . Then G has the following presentation: as generators we have s∈S H s ∪ α∈Φ + U α and {n s | s ∈ S} and as relations we have all relations in B and for s, t ∈ S, α s = α ∈ Φ + , h ∈ H t , u s ∈ U s we have the relations where Proof.All these relations are relations in G. Thus it suffices to show that all relations in [20,Corollary (13.4)] can be deduced from the relations in the statement.A case distinction yields the result.
(2.10) Example.Let ∆ be an irreducible spherical Moufang building of rank at least 2 and let Σ be an apartment of ∆.Identifying the set of roots Φ with the set of roots in Σ, we deduce that G = U α | α ∈ Φ is an RGD-system.
(2.12) Remark.Let G, (U α ) α∈Φ , H, (V α ) α∈Φ be two RGD-systems of the same irreducible spherical type (W, S) and of rank at least 2. Assume that G = U α | α ∈ Φ , the root groups U α are nilpotent and that both twin buildings associated with the RGD-systems satisfy Condition (lco).Then G, H are perfect by the previous lemma.Assume that there exists a homomorphism ϕ : (2.13) Lemma.Let D = G, (U α ) α∈Φ be an RGD-system of 3-spherical type (W, S) such that ∆(D) ± satisfy the Conditions (lco) and (lsco) (e.g.every root group contains at least five elements).Then B + is the 2-amalgam of the groups HU αs .
A parameter system will mean a family of disjoint parameter sets (2.15) Example.Let ∆ = (C, δ) be a building of type I 2 (m), m ≥ 2, with a labelling of type , where S = {s, t}, based at some chamber c ∈ C. Given any chamber x ∈ C at distance d from c one has a minimal gallery (c = c 0 , . . ., c d = x).Let u i be the label attached to c i in the unique panel containing c i−1 and c i .The gallery thus determines the sequence (u 1 , . . ., u d ) where the u i lie alternately in U s and U t , and any such sequence obviously determines a unique gallery starting at c, and hence a unique chamber at the end of this gallery.If two sequences determine the same chamber they are called equivalent.
A blueprint of type (W, S) is a tuple (U ′ s ) s∈S , (∆ st ) s =t∈S consisting of a parameter system (U ′ s ) s∈S and buildings ∆ st = (C st , δ st ) of type I 2 (m st ) for each s = t ∈ S, with a given labelling of type (U ′ s , U ′ t ), based at some chamber ∞ st ∈ C st .A building ∆ of type (W, S) will be said to conform to the blueprint (U ′ s ) s∈S , (∆ st ) s =t∈S if there exists a labelling of ∆ of type (U ′ s ) s∈S , based at some chamber c ∈ C, such that for each {s, t}-residue R of ∆ there is an isomorphism ϕ R : ∆ st → R with the property that x and ϕ R (x) have the same s-and t-labels for each chamber x of ∆ st .We call a blueprint B realisable if there exists a building which conforms to it.

A realisable criterion
We want to construct a building which conforms to a given blueprint.Let B = (U ′ s ) s∈S , (∆ st ) s =t∈S be a blueprint of type (W, S).As a first step we construct a chamber system C(B) as follows: The chambers are sequences ū := (u 1 , . . ., u k ), where u i ∈ U s i and s 1 • • • s k is reduced.We call (s 1 , . . ., s k ) the type of ū.We define s-adjacency via where u k+1 , u ′ k+1 ∈ U s ; this is evidently an equivalence relation, so C(B) is a chamber system.For ū = (u 1 , . . ., u k ) having type (s 1 , . . ., s k ) and v = (v 1 , . . ., v n ) having type (t 1 , . . ., t n ) we define the sequence ūv : reduced.We now define an elementary equivalence to be an alteration from a sequence ū1 ūū 2 of type (f 1 , p(s, t), f 2 ) to ū1 ū′ ū2 of type (f 1 , p(t, s), f 2 ) where ū and ū′ are equivalent in ∆ st .Two sequences ū and v are called equivalent, written ū ≃ v, if one can be transformed to the other by a finite sequence of elementary equivalences.We now consider C(B) modulo the equivalence relation.
Notice that [ū] determines a unique element w ∈ W where (s 1 , . . ., s k ) is the type of ū and Proof.This follows from the proof of [16,Theorem (7.1)].
(2.17) Corollary.A blueprint is realisable if and only if for any two sequences ū, v of the same reduced type, ū ≃ v implies ū = v.In particular, a blueprint is realisable if its restriction to each spherical rank 3 subdiagram is realisable.

Blueprints and Moufang buildings
This subsection is based on [16].
Let ∆ = (C, δ) be a spherical Moufang building of type (W, S) and let c ∈ C. For each s ∈ S we fix 1 = e s ∈ U αs and put n s := m(e s ).Then any chamber of ∆ can be written uniquely in the form This yields a natural labelling of the building ∆.More precisely let P be any s-panel of ∆, and let proj P c = d and w = δ(c, d).As cosets of B + the chambers of P may be written We assign them the s-labels ∞ s and v, using U ′ αs = U αs ∪ {∞ s }.If we let R st be the {s, t}-residue containing c, then R st acquires a labelling and we have a blueprint given by the (e s ) s∈S , namely (U ′ αs ) s∈S , (R st ) s =t∈S .(2.18) Proposition.Let ∆ be a spherical Moufang building.Then ∆ conforms to the blueprint given by the restriction to E 2 (c), i.e. (U ′ αs ) s∈S , (R st ) s =t∈S , and the natural labelling of ∆ as above.
(2.19) Corollary.Let ∆ be a spherical Moufang building and B be the blueprint given by E 2 (c).Then (u 1 , . . ., u k ) and (v 1 , . . ., v k ) are equivalent if and only if In particular, the map ϕ : We now extend the concept of a natural labelling to (arbitrary) buildings of type If ∆ is a spherical Moufang building with a natural labelling given by 1 = e s ∈ U αs then any A 1 × A 1 residue acquires a natural labelling in this sense (because the appropriate root groups commute).
(2.20) Lemma.Let (W, S) be a reducible 2-spherical Coxeter system of rank 3. Then a blueprint of type (W, S) is realisable if the labelling of the restriction to any A 1 × A 1 residue is natural.
Proof.Let B = ((U ′ s ) s∈S , (∆ st ) s =t∈S ) be a blueprint of type (W, S).Let S = {r, s, t} and assume m sr = 2 = m tr .By Corollary (2.17) it suffices to show that for any two sequences ū, v of the same reduced type, ū ≃ v implies ū = v.Therefore let ū = (u 1 , . . ., u k ) and v = (v 1 , . . ., v k ) be two sequences of the same reduced type (s 1 , . . ., s k ) such that ū ≃ v (note that k ≤ m st + 1).If (W, S) is of type A 1 × A 1 × A 1 the claim follows, because an elementary equivalence is just a permutation of the sequence and for each s ∈ S there is at most one element of U s in such a sequence.Thus we can assume that (W, S) is not of type A 1 × A 1 × A 1 and hence m st ≥ 3. Since m sr = 2 = m tr we know that r occurs at most once in the reduced type.If u i is in U r , then the sequence with u i , u i−1 (resp.u i , u i+1 ) reversed is equivalent to (u 1 , . . ., u k ), since u i is the only element in U r in the sequence ū.We also note that we can do an elementary equivalence in ∆ st only if u i is at position 1 or k in the sequence.If we do an elementary equivalence in ∆ st we have to do this twice (because of the type) and get the same sequence as we started with.Thus the claim follows.

The action via left multiplication
Let ∆ = (C, δ) be a spherical Moufang building of type (W, S).For each s ∈ S we fix 1 = e s ∈ U αs and put n s := m(e s ).As we have already mentioned, every chamber of ∆ can be written in the form where u i ∈ U αs i and this decomposition is unique if one fixes the type (s 1 , . . ., s k ).Since left multiplication of G is an action on ∆, we want to know how g.u

11) and U ns
−αs = U αs it suffices to consider this action for U αs , H s and n ±1 s for each s ∈ S. For s 1 , . . ., s k ∈ S and u i ∈ U αs i we denote the chamber u 1 n 1 • • • u k n k B + by (u 1 , . . ., u k ).We remark that we do not consider all chambers g.(u 1 , . . ., u k ) for g ∈ G.

Foundations
A foundation of type (W, S) is a triple F := ((∆ J ) J∈E(S) , (c J ) J∈E(S) , (ϕ rst ) {r,s},{s,t}∈E(S) ) such that the following hold: (F1) ∆ J = (C J , δ J ) is a building of type ( J , J) with c J ∈ C J for each J ∈ E(S).
Let F = ((∆ J ) J∈E(S) , (c J ) J∈E(S) , (ϕ rst ) {r,s},{s,t}∈E(S) ) be a foundation of type (W, S).An apartment of F is a tupel (Σ J ) J∈E(S) such that the following hold: (FA1) Σ J is an apartment of ∆ J containing c J for each J ∈ E(S).

Moufang sets
A Moufang set is a pair (X, (U x ) x∈X ), where X is a set with |X| ≥ 3 and for each x ∈ X, U x is a subgroup of Sym(X) (we compose from right to left) such that the following hold: (MS1) For each x ∈ X, U x fixes x and acts simply transitive on X\{x}.
(MS2) For all x, y ∈ X and each g ∈ U x , g • U y • g −1 = U g(y) .
The groups U x for x ∈ X are called the root groups of the Moufang set.Let (X, (U x ) x∈X ) and (X ′ , (U x ′ ) x ′ ∈X ′ ) be two Moufang sets and let ϕ : X → X ′ be a map.Then the Moufang sets are called ϕ-isomorphic, if ϕ is bijective and for all x ∈ X we have ϕ (3.2) Example.Let ∆ be a thick, irreducible, spherical Moufang building of rank at least 2. Let P be a panel, let p ∈ P and let Σ be an apartment in ∆ with p ∈ Σ.Let α denote the unique root in Σ containing p but not P ∩ Σ.Let U p := {g| P | g ∈ U α }.Then the group U p is independent of the choice of the apartment Σ and M(∆, P ) := (P, (U p ) p∈P ) is a Moufang set (cf. [14, Notation 1.19]).
Let F be an integrable Moufang foundation.Then every panel contains at least 3 chambers.Since F is integrable, there exists a twin building ∆ and a chamber c of ∆ such that F ∼ = F(∆, c).By Lemma (2.7) the twin building ∆ is thick.Moreover, every irreducible integrable Moufang foundation satisfying Condition (lco) determines the isomorphism class of the corresponding twin building: be two thick irreducible 2spherical twin buildings of type (W, S) and of rank at least 3 satisfying Condition (lco).Sup-

The Steinberg group associated with an RGD-system
Let D = (G, (U α ) α∈Φ ) be an RGD-system of irreducible spherical type (W, S) and rank at least 2. Following [21], the Steinberg group associated with D is the group Ĝ which is the direct limit of the inductive system formed by the groups U α and U

The direct limit of a foundation
In this subsection we let F = ((∆ J ) J∈E(S) , (c J ) J∈E(S) , (ϕ rst ) {r,s},{s,t}∈E(S) ) be a Moufang foundation of type (W, S) satisfying Condition (lco) and let (Σ J ) J∈E(S) be an apartment of F. As F is a Moufang foundation, the buildings ∆ J are Moufang buildings.We identify the roots in Σ J with Φ J .For α ∈ Φ J we let U J α be the root group associated with α ⊆ Σ J and we let H J = U J α | α ∈ Φ J ≤ Aut(∆ J ).As J ∈ E(S), it follows from [2, Remark 7.107(a)] that the root groups U J α are nilpotent.We note that for each {s, t} ∈ E(S) the restriction U Let J = {s, s 0 } ∈ E(S).For each s ∈ K ∈ E(S) we denote the image of u ∈ U ±s in Aut(∆ K ) by u K .Then for every 1 = u ∈ U s there exist u ′ , u ′′ ∈ U −s such that (u ′ ) J u J (u ′′ ) J stabilises Σ J and acts on Σ J as the reflection r αs .By [2, Consequence (3) on p.415] u ′ , u ′′ are unique.Let s ∈ K ∈ E(S).By construction of U ±s → Aut(∆ K ) we know that (u ′ ) K u K (u ′′ ) K interchanges the elements in Σ K ∩ P s (c K ) and stabilizes every panel P with |P ∩ Σ K | = 2.As K is spherical, this implies that (u ′ ) K u K (u ′′ ) K stabilizes Σ K and acts on Σ K as the reflection r αs (cf.[2, proof of Lemma 7.5]).This implies that for u ∈ U s the elements u ′ , u ′′ ∈ U −s do not depend on the choice of s ∈ K ∈ E(S).We define for every 1 = u ∈ U s the element m(u) := u ′ uu ′′ , where u ′ , u ′′ ∈ U −s are as above.
(3.5) Lemma.Let π s : U s ⋆ U −s → s∈J∈E(S) Aut(∆ J ) be the canonical homomorphism and let K s := ker π s .Let X s := (U s ⋆ U −s ) /K s .Then U ±s → X s is injective and (X s , (U ±s )) is an RGD-system of type A 1 , where we identify U ±s with its image in X s .
Proof.Note that U ±s → Aut(∆ J ) are injective for every s ∈ J ∈ E(S) and U J −αs ≤ U J αs .Thus U ±s → X s are injective and U −s ≤ U s in X s .It suffices to show (RGD2).We show that m(u) = u ′ uu ′′ conjugates U ±s to U ∓s for every Let F = F(∆, c + ) and Σ J := R J (c + ) ∩ Σ.Then (Σ J ) J∈E(S) is an apartment of F. We consider the homomorphism ϕ : U s ⋆ U −s → Aut(∆) be the canonical homomorphism.Let g ∈ ker ϕ.Then g acts trivial on every rank 2 residue and hence g ∈ ker π s .Now let g ∈ ker π s .Then ϕ(g) ∈ U αs ∪ U −αs .As ϕ(g) fixes P s (c + ), we deduce ϕ(g) ∈ H αs and hence ϕ(g) fixes Σ.If m st > 2, then ϕ(g) fixes R {s,t} (c + ) by assumption.If m st = 2, then ϕ(g) fixes P t (c + ), as the corresponding root groups commute.In particular, ϕ(g) fixes a twin apartment and all neighbours of c + .Thus ϕ(g) = 1 by [17, Theorem 1] and we have ker π s = ker ϕ.In particular, X s → Aut(∆) is injective, where X s is as in the previous lemma.
Let J = {s, t} ∈ E(S) and let ĤJ be the Steinberg group associated with (H J , (U J α ) α∈Φ J ).Let π St J : ĤJ → Aut(∆ J ), π s,J : U s ⋆ U −s → Aut(∆ J ), ϕ s : U s ⋆ U −s → ĤJ be the canonical homomorphisms.As π s : U s ⋆ U −s → s∈K∈E(S) Aut(∆ K ), we deduce ker π s ≤ ker π s,J .As ĤJ .Now ĤJ /K st is again an RGD-system by [2, 7.131].Thus we let D J = ( ĤJ /K st , ( Ûα K st /K st ) α∈Φ J ) and write X s,t := ĤJ /K st .Let ψ st : ĤJ → X s,t be the canonical homomorphism.Then ( We define G to be the direct limit of the inductive system formed by X s , X s,t for all s = t ∈ S. Note that X s,t is generated by the canonical image of X s , X t in X s,t and hence G = X s | s ∈ S .We note that it is not clear whether X s → G is injective.If we write 1 = u ∈ U s we simply mean that u = 1 in the group U s . (3.7) Lemma.Let s 1 , . . ., s k , s ∈ S and let Proof.We show the hypothesis by induction on k.For k = 1 we have m(v 1 ) −1 m(u 1 ) ∈ H s 1 ≤ N Xs 1 ,s (U s ).Thus we assume k > 1.Using [2, Consequences (4) and (5) on p.415] we deduce tm(u i )t −1 = m(tu i t −1 ) and tm(u i ) Proof.At first we show that we have an action of W on the conjugacy class of U s , where s ∈ S acts on every conjugacy class by conjugation with m(u) for some 1 = u ∈ U s .By the previous lemma, conjugation with m(u) does not depend on 1 = u ∈ U s and m(u) 2 acts trivial on every conjugacy class.Moreover, (st) mst acts as (m(u s )m(u t )) mst ∈ H s ∪ H t (cf.[6,Lemma 3.3]) and hence trivial.
We are now in the position to prove the claim.If s = t, there exists w ∈ s, t such that wα s = α t and w −1 U s w = U t holds in X s,t .Thus we can assume s = t.It suffices to show that for w ∈ W with wα s = α s we have w −1 U s w = U s .Thus let w ∈ W such that wα s = α s .Then ws / ∈ α s and hence {w, ws} ∈ ∂α s .By Lemma (2.5) there exist a sequence of panels P 0 = {1, s}, . . ., P n = {w, ws} contained in ∂α s and a sequence of spherical rank 2 residues R 1 , . . ., R n contained in ∂ 2 α s such that P i−1 , P i are distinct and contained in R i .We show that claim via induction on n.For n = 0 there is nothing to show.Let n > 0 and let x ∈ P n−1 ∩ α s .Then x −1 w ∈ J for some J ⊆ S with |J| = 2. Using induction, we deduce For s ∈ S we define U αs := U s and for α ∈ Φ we define U α as a conjugate of U s as in (RGD2).This is well-defined by the previous lemma.We put D F := (G, (U α ) α∈Φ ).
Proof.It follows by the definition of G and U α that (RGD2) and (RGD4) are satisfied.Let {α, β} be a prenilpotent pair with o(r α r β ) < ∞.Then there exists (3.10) Lemma.Let (c 0 , . . ., c k ) be a minimal gallery and let (α 1 , . . ., α k ) be the sequence of roots which are crossed by that minimal gallery.Then we have Proof.We can assume α 1 ⊆ α k .In particular, k ≥ 3 and c 0 , . . ., c k are not contained in a rank 2 residue.Let R be the residue containing c k−2 , c k−1 and c k .Then there exists a minimal gallery (d 0 = c 0 , . . ., d k = c k ) with d i = proj R c 0 for some 0 ≤ i ≤ k.We note that the set of roots crossed by (d 0 , . . ., d k ) coincides with the set of roots crossed by (c 0 , . . ., c k ).Let (β 1 , . . ., β k ) be the set of roots crossed by (d 0 , . . ., d k ) and let 1 ≤ j ≤ k be such that β j = α 1 .Since α 1 ⊆ α k , we have j < i.Let γ = β i be the root containing d i but not some neighbour of d i in R. Since {β j+1 , . . ., In particular, the subgroup is nilpotent because of the commutator relations and the fact that the groups U δ are nilpotent.
(3.11) Lemma.Let (c 0 , . . ., c k ) be a minimal gallery and let (α 1 , . . ., α k ) be the sequence of roots which are crossed by that minimal gallery.Then the product map Proof.As in the definition of an RGD-system we let By the previous lemma we have U α 1 , . . ., U α k = U α 1 • • • U α k and hence the product map is surjective.We prove by induction on k that it is injective.If k = 1 there is nothing to show.Thus let k ≥ 2 and let Using (RGD2) we can arrange it that α 1 = −α s ∈ Φ − for some s ∈ S and Using induction the claim follows.
Using Lemma (2.5) there exist a sequence P 0 = P, . . ., P n = Q of panels in ∂β and a sequence R 1 , . . ., R n of spherical rank 2 residues in ∂ 2 β such that P i−1 , P i are distinct and contained in R i .Let (d 0 = c 0 , . . ., d m ) be a minimal gallery such that Q = {d m−1 , d m }.We will show by induction on n that for every i ∈ I the root α i is crossed by the gallery (d 0 , . . ., d m ).For n = 0 there is nothing to show, as two minimal galleries from one chamber to another chamber cross the same roots.Thus we assume n > 0. Note that by [2, Lemma 3.69] every minimal gallery from c 0 to a chamber which is not contained in β has to cross ∂α and ∂β.Let (e 0 = c 0 , . . ., e l ) be a minimal gallery such that P n−1 = {e l−1 , e l } and e z = proj Rn e 0 for some z.Moreover, we let (β 1 , . . ., β l = β) be the sequence of roots which are crossed by (e 0 , . . ., e l ).Using induction α i is crossed by (e 0 , . . ., e l ) for every i ∈ I.
Now assume that for i ∈ I we have o(r Then a minimal gallery from c 0 to q does not cross the root α i .This yields a contradiction and we infer α i β for every i ∈ I. Using similar arguments, we deduce α α i for every i ∈ I. Since γ ∈ (α, β) if and only if α γ β, the claim follows.Proof.By Theorem (3.12) D F is an RGD-system.Thus it suffices to show that F ∼ = F(∆(D F ), B + ).Since U ±s → G are injective, we do not distinguish between them and their images in G.
Each ∆ {s,t} is a spherical Moufang building for s = t with {s, t} ∈ E(S).Thus there exists an isomorphism β {s,t} : X s,t /B {s,t} → ∆ {s,t} , gB {s,t} → g(c {s,t} ) by [2, Lemma 7.28], where . By Remark (2.12) and Lemma (2.14), we know that α {s,t} : X s,t /B {s,t} → R {s,t} (B + ), gB {s,t} → gB + is an isomorphism for every {s, t} ∈ E(S). Then As γ {r,s} (d) = γ {r,s} (g(c {r,s} )) = gB + , we have For J ⊆ S we define G J to be the direct limit of the groups X s and X s,t with the inclusions as above, where s = t ∈ J.It follows directly from the definition of the direct limit that we have a homomorphism G J → G extending X s , X s,t → G.We define (D F ) J = (G J , (U α ) α∈Φ J ) and note that we have D (F J ) = (D F ) J in general, as the group X s depends on F.
(3.14) Lemma.Let F be an irreducible Moufang foundation of type (W, S) satisfying Condition (lco).If J ⊆ S is irreducible such that |J| ≥ 3 and F J is integrable, then the following hold: (a) Let ∆ be a twin building of type (W, S) and let c be a chamber of ∆ such that F ∼ = F(∆, c).Then we have a canonical homomorphism G J → Aut(∆) (b) The homomorphisms U ±s → G J are injective for every s ∈ J and (D F ) J is an RGDsystem.
Proof.Since F J is integrable, there exists a thick twin building ∆ of type (W, S) and a chamber c of ∆ such that F J ∼ = F(∆, c).Using [15,  We conclude that there exists a homomorphism G J → Aut(∆) mapping U ±s onto U ±αs .Note that U α ≤ U αs | s ∈ S fixes c for every α ∈ Φ J + , whereas U −αs does not.Thus (RGD3) holds and the claim follows from Theorem (3.12).
(3.15) Lemma.Let F be a Moufang foundation and let J ⊆ S be reducible such that |J| = Then U ±s → G J are injective and (D F ) J is an RGD-system.
Proof.Let J = {r, s, t} and assume (3.16) Lemma.Let F be an irreducible Moufang foundation of type (W, S).Then F is integrable if and only if D F is an RGD-system and U ±s → G are injective.
Proof.One implication is Corollary (3.13); the other follows from Lemma (3.14) applied to J = S.
(3.17) Lemma.Assume that for every irreducible J ⊆ S with |J| = 3 the J-residue F J is integrable.Then X s → X s,t is injective for all s = t ∈ S.
Proof.Let s = t ∈ S. If m st = 2 there is nothing to show, as X s,t = X s × X t .Thus we assume m st = 2 and let 1 = g ∈ ker(X s → X s,t ).Since ker(X s → s∈J∈E(S) Aut(∆ J )) = {1}, there exists s ∈ K ∈ E(S) such that g / ∈ ker(X s → Aut(∆ K )).As g ∈ ker(X s → X s,t ), we have g ∈ ker(X s → Aut(∆ {s,t} )) and hence K = {s, t}.Let J = K ∪ {t}.Then J is irreducible.As F J is integrable, there exists a twin building ∆ and a chamber c of ∆ such that F J ∼ = F(∆, c).By Lemma (3.14) we obtain that G J acts on ∆.As g is trivial in G J but not in Aut(∆), this yields a contradiction and hence X s → X s,t is injective for all s = t ∈ S.

The 3-spherical case
In this section we let F be a Moufang foundation of irreducible 3-spherical type (W, S) and of rank at least 3 satisfying Condition (lco).Moreover, we assume that for every irreducible J ⊆ S with |J| = 3 the J-residue F J is integrable and satisfies Condition (lsco).Let (Σ J ) J∈E(S) be an apartment of F and let U ±s , X s , X s,t and D F = (G, (U α ) α∈Φ ) be as before.
Our goal is to show that F is integrable.By Corollary (3.13) it suffices to show that the canonical mappings U ±s → G are injective and that D F = (G, (U α ) α∈Φ ) satisfies (RGD3).We will show that G acts non-trivially on a building and deduce both hypotheses from the action.For all s ∈ S we fix 1 = e s ∈ U s and let n s := m(e s ).
For s = t ∈ S with m st = 2 we define ∆ {s,t} to be the spherical building associated with X s × X t (cf.[2, Proposition 7.116]).Using [2, Proposition 7.116, Corollary 7.68 and Remark 7.69]), we get (similar as for Moufang buildings) a natural labelling.In particular, (U s ∪ ∞ s ) s∈S , (∆ {s,t} ) s =t∈S is a blueprint.We will denote it by B F and remark that the restriction of B F to any A 1 × A 1 residue is natural.
(4.1) Lemma.Let J ⊆ S be irreducible and of rank 3. Then the blueprint B F J is realisable.
Proof.Since F J is an integrable Moufang foundation, there exists a thick twin building ∆ = (∆ + , ∆ − , δ * ) and a chamber c of the twin building ∆ such that F J ∼ = F(∆, c).Without loss of generality let c ∈ C + .Let Σ be a twin apartment containing the images of the apartments (Σ K ) K∈E(J) of the foundation.We deduce from [15, Theorem 1.5] and [2, Theorem 8.27 and Proposition 8.21] that ∆ + is a spherical Moufang building.By Proposition (2.18) the building ∆ + conforms to the blueprint given by its restriction to E 2 (c) and the natural labelling of ∆ + , i.e.U αs ∪ {∞ s }) s∈J , (R {s,t} (c)) s =t∈J , where the U ±αs are the root groups corresponding to the roots in Σ ∩ ∆ + .We will show that ∆ + conforms to B F J .As we have a labelling of ∆ of type (U αs ∪ {∞ s }) s∈J and U s → U αs are isomorphisms, we also have a labelling of type (U s ∪ {∞ s }) s∈J .Let R be an {s, t}-residue of ∆ + .Then there exists an isomorphism ϕ R : R {s,t} (c) → R such that x, ϕ R (x) have the same s-and t-labels for each x ∈ R {s,t} (c).As F J ∼ = F(∆, c), there exist isomorphisms α K : ∆ K → R K (c) for each K ∈ E(J).Then x and α {s,t} (x) have the same s-and t-labels and hence ∆ + conforms to the blueprint B F J .In particular, B F J is realisable.Proof.Let J ⊆ S be of rank 3. Then J is spherical by assumption.If J is irreducible, then F J is integrable and hence B F J is realisable by Lemma (4.1).If J is reducible, then B F J is realisable by Lemma (2.20).Thus each restriction to a spherical rank 3 subdiagram is realisable and hence the claim follows from Corollary (2.17).

Recall that H
An action associated with left multiplication (4.3) Theorem.Let s = t ∈ S. Then B {s,t} acts on C(B F ) as follows: Let s 1 , . . ., s k ∈ S be such that s 1 • • • s k is reduced and let u i ∈ U s i .Let r ∈ {s, t} and g ∈ U r ∪ H r .Then ω B (g, ()) := () and for k > 0 we have Proof.We have to show that every relation in B {s,t} acts trivial on (u 1 , . . ., u k ).We prove the hypothesis by induction on k.For k = 0 there is nothing to show.Thus we assume k ≥ 1.We consider a relation in B {s,t} = H s ∪H t ⋉ U s ∪U t (note that H s , H t normalise U s , U t and their intersection is trivial by [2, Lemma 7.62]).Let h 1 , . . ., h m ∈ H s ∪H t be such that h We distinguish the following cases: and U s acts trivial on the t-panel containing a fundamental chamber, the element w 1 • • • w m ∈ U t acts also trivial on this t-panel.Since the action is simply transitive, we deduce is a relation in G {s,t} and hence in B {s,t} .The claim follows by induction.
As (v(v ′ u 1 ) h −1 ) h = v h v ′ u 1 = u 1 in U s and (h −1 ) ns 1 h ns 1 is a relation in G {s} and hence in B {s} , the claim follows by induction.

( 2 . 7 )
Lemma.Let ∆ = (∆ + , ∆ − , δ * ) be a twin building of type (W, S) and let c be a chamber of ∆ such that |P s (c)| ≥ 3 holds for all s ∈ S. Then ∆ is thick.Proof.Let ε ∈ {+, −} and let c ∈ C ε .For each d ∈ c op and each s ∈ S the s-panel containing d contains at least three elements by [2, Corollary 5.153].Let c ′ ∈ C ε and d ∈ c op .If c ′ ∈ d op the s-panel containing c ′ contains at least three chambers by [2, Corollary 5.153].Otherwise, there exists a chamber d

( 3 .
13) Corollary.If the canonical mappings U ±s → G are injective and if D F satisfies (RGD3), then D F is an RGD-system and we have F ∼ = F(∆(D F ), B + ).

( 4 . 5 )
Lemma.The action in Theorem (4.3) maps equivalent sequences to equivalent sequences.In particular, ω B extends to an action of the building C B F .

( 4 . 9 )
Lemma.For s, t ∈ S and 1 = h ∈ H t we have n s hn s = n 2 s h ns in Aut(C B F ).
Theorem 1.5] and [2, Theorem 8.27], ∆ is a so-called Moufang twin building.Let ε ∈ {+, −} be such that c ∈ C ε .By Lemma (2.8) there exists a twin apartment Σ containing the image of the apartments Σ J of the foundation.As we have seen in Example (3.6), K s acts trivial ∆ and we have homomorphism X s → Aut(∆).Clearly, U ±s → Aut(∆) are injective.Let {α, β} ⊆ Φ {s,t} be a prenilpotent pair.Note that restriction of an automorphism of ∆ to R {s,t} (c) is an epimorphism.Using [2, Corollary 7.66] it is an isomorphism from U [α,β] to its image in Aut(R {s,t} (c)).Thus we have a homomorphism Ĥ{s,t} → Aut(∆) for m st > 2. As K s is contained in the kernel of this map, we obtain a homomorphism X s,t = Ĥ{s,t} /K st → Aut(∆) for m st > 2. As U ±αs commutes with U ±αt in Aut(∆) if m st = 2, we also have a homomorphism X s,t = X s × X t → Aut(∆).
,s 1 } .As the product is contained in B {s,t,s 1 } , it is a relation in B {s,t,s 1 } .Note that G {s,t,s 1 } is an RGDsystem by Lemma(3.14)orLemma(3.15).Using Lemma (2.13) we know that B {s,t,s 1 } is the 2-amalgam of the groups B {r} with r ∈ {s, t, s 1 }.By the universal property of direct limits and induction, we deduce that B {s,t,s 1 } acts on sequences of length at most k − 1 and hence we can write for some g i ∈ B {s,t,s 1 } and relations r i ∈ B {s,t} ∪ B {s,s 1 } ∪ B {t,s 1 } .The claim follows now by induction.Let v 1 , . . ., v m ∈ U s and w 1 , . . ., w m ∈ U t be such that m i=1 ,s 1 } and hence in B {s,t,s 1 } .As before, the claim follows by induction.(CaseII) s 1 ∈ {s, t}: Without loss of generality we can assume s 1 = t.Then we compute ω