a diameter bound for extensions of the f4(2)-building

European Journal of Combinatorics 24 (2003) 685–707

www.elsevier.com/locate/ejc

A diameter bound for extensions of theF4(2)-building

A.A. Ivanova, A. Pasinib

aDepartment of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2BZ, UKbDepartment of Mathematics, University of Siena, via del Capitano 15, 53100 Siena, Italy

Received 9 October 2002; received in revised form 24 April 2003; accepted 25 April 2003

Abstract

We consider extensions of theF4(2)-building with the diagram

such that the residue of every element of the rightmost type is a one-point extension of thecorrespondingC3(2)-residue in the building. Four flag-transitive such geometries are known withthe automorphism groups isomorphic to2E6(2) : 2, 3 · 2E6(2) : 2, E6(2) : 2 and 226 : F4(2).The first example is a folding of the second one. We show that the last three examples are simplyconnected. This brings us close to the complete classification of the flag-transitivec-extensions ofthe F4(2)-building with the local one-point extension property.© 2003 Elsevier Ltd. All rights reserved.

1. Introduction

We continue investigatingc · F4(t)-geometries defined as follows.

Definition. A geometryE is said to be ac · F4(t)-geometry if

(i) E belongs to the diagram

wheret = 1,2 or 4.(ii) E satisfies the intersection property (cf. [13]). In particular, the residues of elements

of type 1 are buildings;

(a) any two elements of type 1 are incident to at most one common element oftype 2;

0014-5793/03/$ - see front matter © 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0195-6698(03)00060-X

686 A.A. Ivanov, A. Pasini / European Journal of Combinatorics 24 (2003) 685–707

(ab) if two elements of type 1 are incident to an elementeof type 2 and to an elementu of type 5 thene andu are incident.

(iii) E satisfies

(b) three elements of type 1 are pairwise incident to elements of type 2 if and onlyif they are incident to acommon element of type 5.

If Γ is the collinearity graph of ac · F4(t)-geometryE (the vertices and edges ofΓ arethe elements of type 1 and 2 with the inherited incidence relation), then (a), (ab) and (b)say thatΓ has no multiple edges and locally it is the graph on the point-set of the residualF4(t)-building F in which two points are adjacent if they are in a common symplecton.These properties ofΓ can be taken as an alternative definition ofc · F4(t)-geometries;cf. [10], where the intersection property was assumed but admittedly not stated explicitly.

Since every flag-transitive automorphism group ofF is well known to be point-primitive, (a) is implied by the flag-transitivity. The conditions (ab) and (b) say that theresidue of an element of type 5 is a one-point extension of theC3(t)-residue inF .

In [10] a general theory ofc · F4(t)-geometries was developed, in particular thepossibilities for theµ-subgraphs in the collinearity graphs were determined. By the mainresult of [10] there isauniquec·F4(4)-geometry which is the flag-transitive one associatedwith the Baby Monster sporadic simple group.

In [11] the c · F4(1)-geometries were classified under a combinatorial condition whichholds in every flag-transitive geometry. Thus we know that there are exactly two flag-transitivec · F4(1)-geometries:E(Fi22 : 2) and its 3-fold coverE(3 · Fi22 : 2).

In [9] four examples:

E(2E6(2) : 2), E(3 · 2E6(2) : 2), E(E6(2) : 2) and E(226 : F4(2))

of c · F4(2)-geometries were constructed and the intersection numbers of their collinearitygraphs were computed. The second geometry possesses a folding onto the first one, andthe folding map commutes with the action of the automorphism group. On the otherhand, the last geometry possesses a number of foldings over orbits of subgroups in theelementary Abelian translation group and the resultingc · F4(2)-geometries are no longerflag-transitive. This fact makes the classification project of allc · F4(2)-geometries abit problematic. On the other hand, the flag-transitive case appears reasonable and thecorresponding amalgams have already been looked at in [14]. In order to achieve theclassification in this case the simple connectedness question for the known examples hasto be addressed. We prove

Theorem 1. The geometriesE(3· 2E6(2) : 2), E(E6(2) : 2) andE(226 : F4(2)) are simplyconnected.

In fact our results are stronger, but in order to state them properly we need to reintroducesome terminology and notation.

Let F be the building of type F4(2), F = F4(2) be the automorphism group ofF(which is the only flag-transitive automorphism group ofF ) andΨ be the collinearitygraph ofF . Let ∆ be the graph on the point-set ofF in which two points are adjacent ifthey are incident to a common symplecton. LetE be ac · F4(2)-geometry andΓ be the

A.A. Ivanov, A. Pasini / European Journal of Combinatorics 24 (2003) 685–707 687

collinearity graph ofE . ThenΓ is locally ∆ and every graph which is locally∆ is thecollinearity graph of somec · F4(2)-geometry (cf. Lemmas 3.1 and 3.3 in [10]). There isa unique injection of the point-set ofF into the set of involutions inF which commuteswith the action ofF . It is common to identify the points ofF with involutions inF via thisinjection and in these terms two points are adjacent in∆ if and only if they commute. Fora vertexx of Γ let Γ (x) denote the set of neighbours ofx in Γ and let

ιx : Γ (x) → ∆

be a bijection which establishes the isomorphism of the subgraph inΓ induced byΓ (x)with ∆.

If π = (a, x,b) is a 2-path inΓ (which means thata,b ∈ Γ (x)with a andb distinct andnon-adjacent), then〈ιx(a), ιx(b)〉 is eitherD6 (in which caseπ is said to be of typeD6)or D8 (in which caseπ is of type D8). With π as above the subgraph inΓ induced byΓ (a) ∩ Γ (b) is called aµ-subgraph. By Lemma 6.3 in [10] wheneverx andy are in thesame connected component of theµ-subgraphΓ (a)∩Γ (b), (a, x,b) and(a, y,b) have thesame type. This allows us to define (in the obvious way) the type of a connected componentof aµ-subgraph.

The connected components of typeD8 are all isomorphic (with 144 vertices) asdescribed in Section 6 in [10], see alsoSection 4.1below. The connected components ofD6-type are much more interesting. IfΞ is such a component then the subgraph induced bythe projectionΣ = ιa(Ξ ) is isomorphic to the collinearity graph of an affine polar space(AP for short) obtained by removing a hyperplane from theSp8(2)-polar spaceP . It iswell known [3] thatP admits three kinds of hyperplanes: ofO+, O− and singular type (cf.Section 3.1for details). The graph∆ contains a subgraphΠ isomorphic to the collinearitygraph ofP , whose stabilizer in F induces onΠ the full automorphism groupSp8(2). Wesay that anAPΣ is classicalif it is contained in the image ofΠ under an element ofF . Itis known (Section 7 in [10]) that everyAP of type O+ or O− is classical. We assume thefollowing two conditions:

(c1) if (a, x,b) is a 2-path of typeD8 then theµ-subgraphΓ (a) ∩ Γ (b) is connected;(c2) all connected components of allµ-subgraphs of typeD6 are isomorphic.

By (c1) everyµ-subgraph has a well-defined type, so (c2) makes sense when (c1) holds.In view of (c2) we can say thatE is of O+, O− or singular type depending on the type ofthe connected components of theD6 typeµ-subgraphs.

It was proved in [9], Lemmas 3.2, 3.5, 3.7 and 3.8 that (c1) and (c2) always hold whenE is flag-transitive.

Theorem 2. Suppose thatE satisfies(c1) and(c2) and letδ(Γ ) be the diameter ofΓ . If Eif of O−-type thenδ(Γ ) ≤ 3. If E is of O+-type thenδ(Γ ) ≤ 4.

It is easy to deduce the simple connectedness of the geometriesE(3 · 2E6(2) : 2) andE(E6(2) : 2) from Theorem 2(cf. Corollary 4.2).

Theorem 3. Suppose thatE is flag-transitive of singular type and G is a flag-transitiveautomorphism group ofE . ThenE ∼= E(226 : F4(2)) and G∼= 226 : F4(2).

As a consequence of the proof ofTheorem 3weobtain the following (cf.Lemma 5.10).


Theorem 4. Suppose that(R, ϕ) is a non-trivial F-admissible (possibly non-Abelian)representation ofF such that the images underϕ of the points in a symplecton generate agroup of order26. Then Ris the26-dimensional irreducible GF(2)-module for F.

We follow [7] and [8] for notation for stabilizers; in particular, ifG is an automorphismgroup ofE andX is a set ofpoints ofE , we denote byG[X] its set-wise stabilizer inG andby G(X) its point-wise stabilizer. Also, given a subgroupA ≤ G[X], we denoted byAX

the action induced byA on X.

2. Basics on the F4(2)-building

2.1. Notation and terminology

As aboveF is the building of typeF4(2) andF = F4(2) is its automorphism group. Theresidue of a 4-element ofF will be called asymp(a shortening for ‘symplecton’). Elementsof F of type 1, 2 and 3 are calledpoints, linesandplanes. We will use the symbol⊥ forthe collinearity relation ofF . Also, for two distinct collinear pointsx, y, we denote byxytheunique line through them.

We recall that the collinearity graphΨ of F has diameter 3. As in [10], for a pointxandi ∈ 1,2,3, Ψi (x) denotes the set of points at distancei from x, but wewill alwayswrite x⊥ insteadΨ1(x) ∪ x. We also recall that there are twotypes of pairs of points atdistance 2, namely pairsx, y where|x⊥ ∩ y⊥| = 1 (these are called special pairs) andpairsx, y where|x⊥ ∩ y⊥| > 1, which are called symplectic pairsand are characterizedby being contained in a (unique) symp. For a symplectic pairx, y, we denote by〈x, y〉the symp containing it. We recall that〈x, y〉 is the convex hull of x, y.

As in [10], for y ∈ Ψ2(x) we write y ∈ Ψ22 (x) or y ∈ Ψ4

2 (x) according to whetherthe pair x, y is symplectic or special. However, it will be convenient to have someterminology for the relationsΨ3, Ψ4

2 andΨ22 more concise than ‘being at distance 3’,

or ‘forming a special pair’ or ‘a symplectic pair’. Thus, when two pointsx, y of F are inrelation Ψ3, we saythat they arefar. Givena set ofpointsX, we denote byΨ3(X) the setof points far from all points ofX. If Y ⊆ Ψ3(X) (or y ∈ Ψ3(X)) then we say thatX andYarefar (that the pointy is far from X).

For two points x, y at distance 2, we say that they arealmost closeor almost faraccording to whethery ∈ Ψ2

2(x) or y ∈ Ψ42(x). Givena setX of points, we denote by

Ψ42 (X) (resp.Ψ2

2(X)) the set ofpoints almost far from (almost close to) all points ofX.

2.2. Properties of symps

The following are well known (cf. [2], for instance):

Lemma 2.1. The intersection S∩ S′ of two distinct symps S, S′ is either empty, a singlepoint or a plane ofF . Explicitly, S∩ S′ = ∅ when Sand S′ are either far or almost far aspoints of the dualF∗ ofF , whereas S∩ S′ is apoint or a plane when S and S′ are almostclose or, respectively, collinear as points ofF∗.

Lemma 2.2. For a symp Sand a point p/∈ S,one of the following occurs:


(1) p⊥ ∩ S = ∅. In this case there is a unique point pS ∈ S that is almost close to p,the points of S collinear with pS but different from pS are almost far from p and theremaining points of S are far from p.

(2) p⊥ ∩ S is aline, say l. The points of S∩ l⊥\l are almost close to p and the remainingpoints of S are almost far from p.

In view of Lemma 2.2, givena point q of F far from p and a sympS on p, there is aunique pointqS ∈ S∩ Ψ2

2 (q). The sympψp,q(S) := 〈q,qS〉 is the unique one onq thatmeetsS non-trivially. The functionψp,q sending every sympS ∈ Res(p) to the sympψp,q(S) ∈ Res(q) is a bijection from the set of symps onp to the set of symps onq, withψq,p beingits inverse.

Note also thatψp,q = π−1q,pπp,q whereπp,q is the bijection from the set of symps onp

to Ψ22 (p)∩ Ψ2

2(q) sending a sympS ∈ Res(p) to qS, andπq,p is defined in the same way,but permuting p andq. Clearly,πp,q andπq,p are bijections.

Lemma 2.3. The functionψp,q extends to an isomorphism fromResF (p) to ResF (q).

Proof. For two sympsS1, S2 on p, suppose thatπ := S1 ∩ S2 is a plane. The linesq⊥S1

∩πandq⊥

S2∩ π meet in a points. By Lemma 2.2, l i = s⊥ ∩ ψp,q(Si ) is a line, fori = 1,2.

Also, pi := q⊥ ∩ l i is apoint for i = 1,2. (That intersection cannot be a line, in view ofLemma 2.2, sincep andq are far.) If p1 = p2, thens andq belong to a symp, whereasthey must be almost far byLemma 2.2. So, p1 = p2. Thus,ψp,q(S1) ∩ ψp,q(S2) containstwo distinct points, namelyq andp1 = p2. Hence that intersection must be a plane.

2.3. Points as involutions and the structure of a point-stabilizer

The points ofF bijectively correspond to the elements of a class of central involutionsof F , apoint p corresponding to the unique involution ofC(p) = Z(F(p)). We will freelyregard points as involutions whenever this is convenient, writing for instanceX p for theimage of a setX via the involution corresponding to a pointp. In this style, we recall thefollowing well known result from [5].

Lemma 2.4. Twopoints x, y ofF are collinear or almost close if and only if, regarded asinvolutions of F, they commute and their product xy is a point. The points x, y are almostfar i f and only if the product xy has order4. Finally, x, y are far if and only if xy hasorder3. In particular, the points ofF form a class of3,4-transpositions of F.

Let p be a point ofF , Q(p) = O2(F(p)), C(p) be the commutator subgroup ofQ(p),Z(p) be the centre of Q(p).

Lemma 2.5. The following assertions hold:

(1) F(p) is the centralizer of the involution p;

(2) F(p) = Q(p)K (p), where K(p) ∼= Sp6(2) is a complement to Q(p) in F(p);

(3) C(p) is oforder2 and isthe centre of F(p); thequotient Q(p)/C(p) is an elemen-tary Abelian2-group and as a K(p)-module it is an indecomposable extension ofthe6-dimensional symplectic module by the8-dimensional spin module;


(4) Z(p) is an indecomposable extension of C(p) by the 6-dimensional symplecticmodule (so that Z(p) is the orthogonal module for K(p) ∼= Ω7(2));

(5) F(p) contains two classes of complements to Q(p); the representatives K1 and K2of these classes can be chosen in such a way that K1 = F(p) ∩ F(q) for someq ∈ Ψ3(p) and NF (K1) ∼= S3 × Sp6(2), while K2 acts fixed-point freely onΨ\pand NF (K2) = NF(p)(K2) ∼= 2× Sp6(2) (K1 and its conjugates in F will be calledLevi complements);

(6) F(p) acts onΨ3(p) as on the cosets of K1; in particular the action is faithful, Q(p)acts onΨ3(p) regularly, Z(p) has 28 orbits of length27 each and, if M is oneof these orbits, then the subgraph inΨ induced by M has no edges and there are135 Z(p)-orbits with whom M is joined by matchings;

(7) if L ∼= [220].(S3 × L3(2)) is the stabilizer in F of a line ofF , then the centre of L istrivial;

(8) if t0(p) is the generator of C(p) andψ0 : p → t0(p), then (F, ψ0) is a repre-sentation ofF .

Proof. These are well known facts. We only note that (5) follows from [1, (17.7)], noticingthat the first cohomology group of the symplectic module is 1-dimensional while that ofthe 8-dimensional spin module is trivial [12]; (7) follows, for instance, from the fact thatF4(2) contains no element of order 42.

Remark. Given K1 = F(p) ∩ F(q) and K2 as inLemma 2.5(5), the Levi complementK1 fixes one more pointr besidesp andq. Thepoints p, q, r are mutually far and thefactor S3 of NF (K1) = S3 × K1 acts transitively onp,q, r . The triple p,q, r is ahyperbolic line of theSp8(2)-polar spaceP = P(Π ) (seeSection 3.1for the definitionof P(Π )) and NF (K1) is the set-wise stabilizer ofp,q, r in F[P] ∼= Sp8(2). If Σ istheAP-substructure obtained fromP by removing the perp ofp (seeSection 3.1), thenq,r ∈ Σ andr is the unique point ofΣ at distance 3 fromq.

The complementK2 can also be taken insideF[P], andK2 (but notK1) is contained insubgroupsOε ≤ F[P] isomorphic toΩε

8(2), for ε = + or −. In particular, the conjugatesof K2 in F contained inO− ∼= Ω−

8 (2) are conjugate inO−, whereas those that arecontained inO+ ∼= Ω+

8 (2) form three classes inO+, fused in Aut(O+) ∼= NF (O+).

3. Affine polar spaces involved in FIn this section we consider certain substructures ofF , which arise from connected

components of typeD6 of µ-subgraphs in the collinearity graphs ofc · F4(2)-geometries.First we recall a few basic properties of those substructures, and then prove two theoremson them, to be exploited inSection 4.

3.1. Definition and basic properties

Let Φ be the graph on the set of points ofF where two points are adjacent when theyare almost close. It is well known ([10, Chapter 7]; also [9]) thatΦ admits several inducedsubgraphsΣ with the following properties:


(ap1) if two vertices ofΣ have distance 2 inΣ , then they are far as points ofF ;(ap2) Σ is isomorphic to the collinearity graph of an affine polar space obtained by

removing a hyperplane from theSp8(2)-polar spaceP .

As the elements of an affine polar space can be recovered as distinguished cliques of itscollinearity graph, a graphΣ as above can be regarded as a rank 4 geometry. (Explicitly,the vertices and the edges of the graphΣ are the 1- and 2-elements of the geometry,the 4-elements are the maximal cliques andthe 3-elements are 4-cliques obtained asintersections of two 4-elements.) We callΣ an AP-substructureof F . (Needless to say,the lettersAPare a shortening for ‘affine polar’.)

The threetypes of AP-substructuresIt is well known [3] thatP admits three kinds of hyperplanes, namely:

(1) Singular hyperplanes, formed by the perp of a point ofP .(2) Hyperplanes isomorphic to the elliptic quadric forΩ−

8 (2). Wecall them hyperplanesof O−-type.

(3) Hyperplanes isomorphic to the hyperbolic quadric forΩ+8 (2). We say they are of

O+-type.

If the affine polar spaceA is obtained by removing a hyperplaneH from the polar spaceP , thenA is said to be ofsingular, O−- or O+-typeaccording to whetherH is of singular,O−- or O−-type. If A is of type O− or O+, we also say it is of non-singulartype. Thesame terminology will be used forAP-substructures.

In the next table we recall a few properties ofA, according to its type. Note that Aut(A)is flag-transitive in all cases. Smaller flag-transitive subgroups, if any, are mentioned in thelast row.

Type Singular O− O+

Number ofpoints 128 136 120Diameter 3 2 2Aut(A) 21+6Sp6(2) Ω−

8 (2) : 2 Ω+8 (2) : 2

Smaller flag-tr. subgr. Ω−8 (2) Ω+

8 (2)

Notice that in the singular case, for every pointp ∈ A, there exists a unique point ofAat distance 3 fromp.

As anAP-substructureΣ of non-singular type has diameter 2, condition (ap1) impliesthat no two points of it are collinear or almost far inF . So, inview of Lemma 2.4, thepoints ofΣ , regarded as involutions of F , form a set of 3-transpositions in the subgroupof F generated by them. On the other hand, ifΣ is of singular type then it admits pairs ofpoints at distance 3, and these might possibly be almost far inF (but we are not aware ofany example where that happens).

Lemma 3.1. No two points of an AP-substructure of singular type are collinear inF .

Proof. In view of the previous remarks, we only need to consider the case of two pointsx, y at distance 3 inΣ . By contradiction, supposex ⊥ y. Givena point z ∈ Σ adjacent to


y, consider the sympS := 〈y, z〉. By Lemma 2.2, x cannot be far fromz in F . However,z has distance 2from x in Σ , asy is the unique point ofΣ at distance 3 fromx. Hencezmust be far fromx, contrary to our previous remark.

Classical AP-substructures and the polar subgraphΠRepresentatives of all three types ofAP-substructures can be obtained as induced

subgraphs of a larger induced subgraphΠ of Φ, which forms the collinearity graph of acopyP(Π ) of theSp8(2)-polar space and such thatF[Π ] ∼= Sp8(2) (a maximal subgroupof F). We callΠ an Sp8(2)-polar subgraphof Φ. Properties ofΠ will be discussed inmore detail later (Section 3.4). We only mention here that no two points ofΠ are eithercollinear or almost far inF . So, thepoints ofΠ , regarded as involutions ofF , form a setP (actually, a class) of 3-transpositions ofF[Π ] andΠ is the commuting graph ofP.

If we remove fromΠ a hyperplaneH of the polar spaceP(Π ), then weobtain anAP-substructureΣ , with F[Σ ] = Aut(Σ ) and I (Σ ) := P\H as the point-set. We say thatΣ is classical.

When H is of non-singular typeOε (ε = − or +), then I (Σ ) is a class of 3-trans-positions ofF[Σ ] ∼= Ω ε

8(2) : 2. Notice that,I (Σ ) lies outside the commutator subgroupΩ ε

8(2) of F[Σ ]. On theother hand, letH be of singular type andp be its radical. Then〈I (Σ )〉 = F[Π ] and I (Σ ) ∩ F(Σ ) = ∅. We havepx = p for everyx ∈ I (Σ ). HenceΣ x = Σ for everyx ∈ I (Σ ). (In fact,Σ ∩ Σ x is the perp ofx in P(Π ), asone can see bynoticing thatp, x, px is thehyperbolic line ofP(Π ) spanned byp, x.)Lemma 3.2. An AP-substructureΣ is classical if andonly if one of the two following twoequivalent conditions holds:

(1) F[Σ ] ∼= Aut(Σ ) (the automorphism group ofΣ );(2) for everypoint p ∈ Σ , thestabilizer of p in F[Σ ] acts as Sp6(2) on theΣ -neigh-

bourhood of p.

Proof. Exploiting the information in [4] on F and its subgroups, it is not difficult to seethat (1) is equivalent toΣ being classical. Obviously, (1) implies (2). LetG ≤ Aut(Σ ) besuchthatG(p) is transitive on theΣ -neighbourhood ofp for every pointp ∈ Σ . Then, asevery edge ofΣ is contained in a triangle andΣ is connected,G is point-transitive. Thus,if (2) holds, thenF[Σ ] is point-transitive. Now, again by exploiting [4], one can see thateitherF[Σ ] = 21+6 · Sp6(2) or Ω ε

8(2) ≤ F[Σ ] ≤ Ωε8(2) : 2 for ε = − or +. In anycase,

Σ is classical. HenceF[Σ ] = Ω−8 (2) : 2, Ω+

8 (2) : 2 or 21+6Sp6(2) andF[Σ ] = Aut(Σ ),as claimed in (2). Lemma 3.3. All AP-structures of non-singular type are classical. Furthermore, anAP-structure Σ is of non-singular type if and only if its points, regarded as involutions,stabilizeΣ .

Proof. We only need to prove that, ifI (Σ ) ⊆ F[Σ ], thenΣ is classical of non-singulartype, the rest of the lemma being contained in [10, Chapter 7]. SupposeI (Σ ) ⊆ F[Σ ]and, for a pointp ∈ Σ , let I (p) be the subset ofI (Σ ) corresponding to theΣ -neighboursof p. Then the elements ofI (p) stabilize ResΣ (p) (which is a copy of theSp6(2)-polarspace) and behave on it just as the elements of a class of 3-transpositions ofSp6(2). So,


〈I (p)〉 inducesSp6(2) on ResΣ (p). By Lemma 3.2(2),Σ is classical. Clearly, it cannot beof singular type.

More terminologyGiven an AP-substructureΣ , we call the 3- and 4-elements ofΣ planesand 3-spaces

respectively (as they may be regarded as affine planes and affine 3-spaces of order 2), butwe keep calling the 2-elementsedges, even ifΣ is regarded as a geometry.

3.2. Symps and AP-substructures

In the sequelΣ is a givenAP-substructure.

Lemma 3.4. For any symp S, either S∩ Σ = ∅ or S∩ Σ is an edge ofΣ .

Proof. Given a sympS with S ∩ Σ = ∅, suppose that|S ∩ Σ | > 2 and letp, x, y bethree distinct points ofS ∩ Σ . (Note that S ∩ Σ is a clique, by condition (ap1).) In theSp6(2)-polar space ResΣ (p) we can pick an edgep,q coplanar withp, x in Σ butnotwith p, y. So,S′ := 〈x,q〉 is a symp andq /∈ S. In view of Lemmas 2.1and2.2, q⊥ ∩ Sis a line. Thus,y cannot be far fromq in F . On theother hand,y has distance 2from q inΣ , whence it is far fromq. We avoidthis contradiction only assuming that|S∩ Σ | ≤ 2.

In order to finish, we only need to show thatS∩ Σ = p is impossible. By the aboveand the fact thatΣ admits 63 edges onp, there exist 63 sympsS on p with |S∩ Σ | = 2.However, there are exactly 63 symps inF on p. So,|S∩Σ | = 2 for every sympSon p.

Lemma 3.5. For p ∈ Σ , let Σ (p) be its neighbourhood inΣ . The functionϕp,Σ sendingevery x ∈ Σ (p) to the symp〈p, x〉 induces an isomorphism of geometries fromResΣ (p)to thedualResF (p)∗ of ResF (p).

Proof. By Lemma 3.4, the functionϕp,Σ is a bijection fromΣ (p) to the set ofpoints ofResF (p)∗ (namely, symps ofF on p). We only need to prove thatϕp,Σ is an isomorphismfrom the graph induced onΣ (p) to the collinearity graph of ResF (p)∗.

Supposex, y ∈ Σ (p) are adjacent. Thenp is almost close to two points of 〈x, y〉,namely to x andy. By Lemma 2.2, l := p⊥ ∩ 〈x, y〉 is a line. The planeπ := 〈l , p〉 ofF spanned byp andl is contained in both〈p, x〉 and〈p, y〉. Hence〈p, x〉 and〈p, y〉 arecollinear as points of ResF (p)∗. This shows thatϕp,Σ is a bijective morphism fromΣ (p)to the collinearity graph of ResF (p)∗. However, thesetwo graphs are isomorphic. So, asthey are finite, every bijective morphism between them is also an isomorphism.

By combiningLemma 2.3with Lemma 3.5, weobtain the following:

Corollary 3.6. For p ∈ Σ and q far from p, the compositeψp,qϕp,Σ induces anisomorphism fromResΣ (p) to ResF (q)∗.

3.3. Points far from an AP-substructure of non-singular type

Theorem 3.7. Let Σ0 be an AP-substructure of non-singular type and suppose thatΨ3(Σ0) = ∅. ThenΣ0 is of type O+ and there exist two more AP-substructuresΣ1 andΣ2of type O+ such that:


(1) Σ1 andΣ2 are mutually far and both far fromΣ0;

(2) the commutator subgroup O∼= Ω+8 (2) of F[Σ0] stabilizes each ofΣ0, Σ1 andΣ2

whereas its normalizer in F induces S3 on the tripleΣ0,Σ1,Σ2;(3) Σi ∪ Σ j = Ψ3(Σk) for i , j , k = 0,1,2.

Proof. Firstly, we pick a pointp far from all points ofΣ0 and, regardingp as an involutionof F , we consider the imageΣ1 := Σ p

0 of Σ0 underp. Sincep ∈ Ψ3(Σ0), for anyx ∈ Σ0

we have(px)3 = 1, that is 〈p, x〉 ∼= S3. SinceΣ1 = Σ p = x p|x ∈ Σ0 andp, x, x pare the three involutions of〈x, p〉, one easily obtains thatΣ1 ⊆ Ψ3(p). Furthermore, foranyx, y ∈ Σ0 we have

xyp = xpy = (xy p)y = (zp)y,

wherez = xy ∈ Σ0. Hencexyp is of order 3, that isx andyp are mutually far. ThereforeΣ1 ⊆ Ψ3(Σ0). We now take aq ∈ Σ1 and setΣ2 = Σq

0 . If we replacep by q in the aboveargument, we see thatΣ2 ⊆ Ψ3(Σ0)∩Ψ3(q). On theother hand,q = x p for somex ∈ Σ0.Hence

Σ2 = Σq0 = Σ pxp

0 = Σ xpx0 = Σ px

0 = Σ x1 .

Thus, replacingΣ0 by Σ1 and p by x in the first part of this proof, we see thatΣ2 is alsofar from Σ1. Furthermore,p = pxx = x px ∈ Σ2, that is, p stabilizes Σ2. It follows that〈p, x〉 ∼= S3 acts faithfully on the tripleΣ0,Σ1,Σ2. In particular, everyx ∈ Σ0 permutesΣ1 andΣ2. By replacingx ∈ Σ0 with y = x p ∈ Σ1 andΣ0 with Σ1, we also obtain thateveryy ∈ Σ1 permutesΣ0 andΣ2. Similarly, everyz ∈ Σ2 permutesΣ0 andΣ2.

Therefore, denoting byO and O the setwise stabilizer and, respectively, the elemen-twise stabiliser of the tripleΣ0,Σ1,Σ2 in F , we haveO/O ∼= S3. Moreover,F[Σ0],which is generated by the involutionsx ∈ Σ0, has index 3 inO, containsO as a subgroupof index 2 and we haveF[Σ0] = 〈O, x〉, for any x ∈ Σ0. Comparing [4, p. 170], onecan see thatO ∼= Ω+

8 (2) is the only possibility where the above properties are satisfied.Namely,Σ0 is of typeO+.

Definition. A triple of AP-substructuresΣ0, Σ1 andΣ2 as inTheorem 3.7will be calledanO+-triple.

By Theorem 3.7, if an AP-substructure of typeO+ belongs to anO+-triple, then thattriple is unique. It remains to show that such a triple exists, namely that the situationdescribed inTheorem 3.7actually occurs.

Theorem 3.8. Every AP-substructure of O+-type is in a unique O+-triple.

Proof. With Σ0 of typeO+, let O ∼= Ω+8 (2) be the commutator subgroup ofΩ0 := F[Σ0]

and putΣ1 := Σ t0 andΩ1 := F[Σ1] = Ω t

0, for t an element of order 3 of Out(O) =NF (O)/O. ThenΩ0 ∩ Ω1 = O, of index 2 inΩ0 andΩ1. For i = 0,1, let xi ∈ Ωi \O.Thenx0x1, regarded as an elementof Out(O), has order 3. However, the 120 involutionsrepresenting the points ofΣi belong toΩi \O. So, we may assume thatx0 andx1 are indeedinvolutions and represent points ofΣ0 andΣ1 respectively. By the above,(x0x1)

3 = 1. Thatis, Σ1 ⊆ Ψ3(Σ0).


Corollary 3.9. The set of O+-triples is a partition of the set of AP-substructures ofF ofO+-type.

3.4. More on Sp8(2)-polar subgraphs

The results obtained in this section will be exploited to proveTheorem 3.17, but theyhave some interest by themselves.

Given anSp8(2)-polar subgraphΠ of Φ, let P = P(Π ) be theSp8(2)-polar spaceembodied in it (Section 3.1). We recall thatF[Π ] ∼= Sp8(2) acts as Aut(P) onP .

Lemma 3.10. For every symp S, either S∩ Π = ∅ or |S∩ Π | = 3. Thelines ofP are thenon-empty intersections ofΠ with symps ofF and, for every point p∈ Π , the functionsending every line L ofP through p to the (unique) symp S such that L= S∩ Π , extendsto an isomorphism fromResP(p) to the dual ofResF (p).

Proof. SupposeS∩Π = ∅. As no twopoints ofΠ are collinear inF , S∩Π is a clique ofΠ , hence it spans a singular subspaceLS of P . By Lemma 3.4, for every point p ∈ S∩ Πand every hyperplaneH of P not containingp, S ∩ Π \H contains exactly two points,forming a line of the affine polar spaceP\H . On theother hand,H ∩ LS is a hyperplanein LS. By these remarks and some elementary combinatorics it follows thatLS is a line.Namely,S∩ Π = LS, aline ofP . As F[Π ] acts flag-transitively onP , all lines ofP arisein that way. The very last claim of the lemma is a rephrasing ofCorollary 3.6.

Lemma 3.11. The group F[Π ] has three orbits U1,15, U1,1 and U3 on the set of points ofF exterior to Π , of size2295, 34 425and32 640respectively. Explicitly,

(1) p ∈ U1,15 if and only if p⊥ ∩ Π is a maximal singular subspace ofP ,(2) p ∈ U1,1 if and only if|p⊥ ∩ Π | = 1 and(3) p ∈ U3 if and only if p∈ Ψ3(Σ ) for some AP-substructureΣ of Π of type O+ (cf.

Theorem3.7).

Furthermore,every line ofF meetingΠ in a point contains both a point of U1,15 and apoint of U1,1.

Proof. The above is stated in [9, Lemma 2.5], but for the explicit descriptions ofU1,15,U1,1 andU3. Thedescription ofU1,15 is obvious (compare the list of maximal subgroupsof Sp8(2) in [4]). Points as in (3) exist byTheorem 3.8. Clearly, they form an orbit ofF[Π ]and an easy counting argument shows that their number is indeed 32 640. None of thosepointscan be collinear with any pointΠ . Indeed, letp ∈ Ψ3(Σ ) for Σ = P\H with H ofO+-type, and suppose thatp ⊥ q for some pointq ∈ Π . Clearly,q ∈ H . Also, S∩Π ⊂ Hfor every sympS on p,q. In view of Lemma 3.10, this forcesH to contain the perpq∼of q in P , a contradiction with the fact thatH is non-singular.

The orbitU1,1 remains to be considered. Given a pointp ∈ U1,15, let l be one of the15 lines ofF through p that meetΠ non-trivially andq be the point ofl\p not in Π .Put Xp := p⊥ ∩ Π , a maximal singular subspace ofP . Clearly, Xp contains the pointr = l ∩ Π and, as there are exactly seven symps ofF on l and seven lines ofP in Xp

throughr , those seven lines are the intersections ofΠ with those seven symps. As each ofthose lineshas two points different fromr , theirunion coversXp. However, ifq ∈ U1,15,


then the same can be said for seven lines of Xq = q⊥ ∩Π through r . So,X p = Xq, whichcannot be. Therefore, q⊥ ∩ Π = r and q ∈ U1,1.

Corollary 3.12. For every symp S meeting Π non-trivially, S ∩ Π is a hyperbolic line inthe Sp6(2)-polar space ResF (S).

Proof. For every maximal singular subspace X of P containing the line L = S ∩ Π , letpX ∈ U1,15 be the point with p⊥

X ∩ Π = X. Then pX belongs to S and pX is collinearwith all points of L. As there are 15 maximal subspaces of P on L, we get 15 points of Scollinear with all points ofL. HenceL is a hyperbolic line ofS.

Proposition 3.13. The collinearity graph of F induces on U1,15 a graph isomorphic to thecollinearity graph of the dual ofP .

Proof. For x, y ∈ U1,15, put Xx := x⊥ ∩ Π and Xy := y⊥ ∩ Π . If |Xx ∩ Xy| > 1,thenx and y are in a common symp, sayS0. In particular, if Xx = Xy thenx = y, as|x⊥ ∩ y⊥| ≥ 15. If Xx ∩ Xy is a plane ofP , thenx⊥ ∩ y⊥ (⊂ S0) contains an anti-cliqueof S0 of size 7. In this case,x ⊥ y.

Conversely, suppose thatx ⊥ y. Givena line L of P contained inXx, let Sbe the sympon x containingL and letM be theline y⊥ ∩ S. By Corollary 3.12, L is a hyperbolic lineof S. Furthermore, it is contained inx⊥. Hence M⊥ meetsL in at leastone point, sayu.The sympS′ = 〈y,u〉 is one of the seven symps on the linexu. However, asu belongs toseven lines ofXx, all symps onxu meetΠ in a line of Xx. HenceL ′ = S′ ∩ Π is a line ofXx. On theother hand,L ′ is a hyperbolic line ofS′, hence there is a line ofS′ from y to apointv ∈ L ′. By applying to the lineyv the same argument used forxu, we getthat L ′ isalso a line ofXy. So,L meets a line ofP contained inXx ∩ Xy. As L is an arbitrary lineof Xx, it follows thatXx ∩ Xy is a plane.

Proposition 3.14. Let p,q, p0 be aline ofF with p0 ∈ Π , p ∈ U1,15 and q ∈ U1,1.Put X0 = p⊥ ∩Π and H0 := p∼

0 , where∼ stands for the collinearity relation of the polarspaceP . Then

p ∈ Ψ42 (Π \X0) and q∈ Ψ2

2(X0\p0) ∩ Ψ42(H0\X0) ∩ Ψ3(Π \H0).

Proof. Let K be the set-wise stabilizer of X0 in F[Π ]. By Proposition 3.13, K is also thestabilizer of p in F[Π ]. As X0 is a maximal singular subspace ofP , K acts transitively onΠ \X0. Therefore, p is in the same relation (namely, far, almost far or almost close) withall points ofΠ \X0. As X0 is contained in several hyperplanes ofO+-type,Π \X0 containsa number ofAP-substructures of typeO+. Hence p /∈ Ψ3(Π \X0) (indeed,p /∈ U3).Therefore,p is either almost close or almost far fromΠ \X0. However, ifx is apoint ofΠ far from p0 (hencex /∈ X0), thenx is far from one more point ofp,q, p0 and almostfar from the other one. By the above,x is almostfar from p and far fromq. Thus, we haveproved thatΠ \X0 ⊆ Ψ4

2(p) andΠ \H0 ⊆ Ψ3(q). Clearly, X0\p0 ⊆ Ψ22(q). It remains

to prove thatH0\X0 ⊆ Ψ42 (q).

The stabilizerK (p0) of p0 in K is also the stabilizer ofq in F[Π ]. It is transitive onH0\X0. Henceq is in the same relation with all points ofH0\X0. By an argument similarto that applied top and noticing that, sinceq /∈ U3, q ∈ Ψ3(Π \H0), one can see thatqcannot be far fromH0\X0. Suppose thatq is almost close tox ∈ H0 and putS := 〈x,q〉.


By Corollary 3.12, L = S ∩ Π is a hyperbolic line ofS. So,q⊥ ∩ L = ∅. Therefore,p0 ∈ L. So,S containsp and, sincep is almost far from all points ofΠ \X0, we obtainx ∈ X0.

Proposition 3.15. Let p∈ U3. ThenΠ ⊆ Ψ3(p)∪ Ψ42(p), the set Hp := Π ∩ Ψ4

2 (p) is ahyperplane ofP of type O+, andΣp := Π \Hp is the unique AP-substructure ofΠ that isfar from p.

Proof. By the assumption,p ∈ Ψ3(Σ ) for an AP-substructureΣ = Π \H , with H anO+-hyperplane ofP . Also, p⊥ ∩ Π = ∅. As H is not singular, for every pointx ∈ Hthere are lines ofP that are not contained inH . Given such a lineL, let S be the sympcontaining it. As the points ofL\x are far fromp, p⊥∩S = ∅, henceScontains a uniquepoint y ∈ Ψ2

2(p). By Corollary 3.12, L is a hyperbolic line ofS. Hence y⊥ ∩ L = ∅ and,since L\x ⊂ Ψ3(p), L ∩ y⊥ = x. If y = x, then x ∈ Ψ4

2 (p) and we are done.Supposex = y. Then〈x, p〉 is a symp. It also contains a lineM of P throughx and, byCorollary 3.12, p⊥ ∩ M = ∅. Therefore p⊥ ∩ Π = ∅, contrary to the assumption thatp ∈ U3.

The next corollary immediately follows fromPropositions 3.14and3.15:

Corollary 3.16. Ψ22(Σ ) = ∅ for any classical AP-substructureΣ .

3.5. Points almost far from an AP-substructure of non-singular type

Theorem 3.17. LetΣ be an AP-substructure of non-singular type. IfΣ is of type O−, thenΨ4

2 (Σ ) = ∅. If Σ is of type O+, then|Ψ42 (Σ )| = 270and the collinearity relation induces

on Ψ42 (Σ ) a graph isomorphic to the collinearity graph of the dual of theΩ+

8 (2)-polarspace.

Proof. By the assumptions,Σ = Π \H for an Sp8(2)-polar subgraphΠ and a suitablenon-singular hyperplaneH of P = P(Π ).

Assume there exists a pointp ∈ Ψ42 (Σ ). Thenp /∈ U3, as in theproof ofTheorem 3.7.

Therefore,p ∈ U1,15 ∪ U1,1. We shall firstly prove thatΣ cannot be of typeO−. Supposep ∈ U1,1, to fix ideas. (The argument we will use for this case works for the case ofp ∈ U1,15 as well.) Letq be the unique point ofU1,15 collinear withp, put X0 := q⊥ ∩Π ,l := pq and letS(l ) be the set of symps on the linel . For everyS ∈ S(l ), p is eithercollinear of almost close from the points ofS∩Π . So,S∩Σ = ∅. However,S∩Π S∈S(l)

is just the set of lines ofX0 throughp0 = l ∩ X0. Therefore,X0∩Σ = ∅, namely X0 ⊂ H .This forcesH to beof O+-type.Proposition 3.14now implies thatΨ4

2(Σ ) consists of thepoints ofU1,15 corresponding to the maximal singular subspaces ofH . The conclusionfollows fromProposition 3.13.

Remark. It is not difficult to describe the setsΨ42 (Σ ) andΨ3(Σ ) in the classical singular

case, too. We give that description here for the sake of completeness, but avoiding proofs,as we will not make any use of it in the sequel.

Let Σ be a classicalAP-substructure of singular type, namelyΣ = Π \p∼ for anSp8(2)-polar subgraphΠ and a pointp ∈ Π . Onecan prove thatΨ3(Σ ) ∪ Ψ4

2(Σ ) = p⊥.In particular,Ψ4

2(Σ ) contains exactly 135 points andmeets each of the 135 lines ofF


through p in a point different fromp, whereasΨ3(Σ ) containsp and the remaining 135points of p⊥.

4. Bounds for the diameter of E when E is of type O− or O+

We shall proveTheorem 2in this section. After a few preliminary lemmas (Section 4.1),we shall split the proof in two parts. We firstly consider theO−-case inSection 4.2. TheO+-case will be discussed inSection 4.3.

Before turning to the proof ofTheorem 2, we prove the following corollary of thattheorem.

Corollary 4.1. Suppose thatE satisfies the conditions(c1) and (c2) of the Introduction.If E is of type O− then it is simply connected. IfE is of type O+ and theµ-subgraphs ofΓare connected, thenE is simply connected.

Proof. Let f : E → E be the universal covering ofE and leta, b be two points in the samefibre of f .

Suppose firstly thatE is of O−-type. By Theorem 2, the collinearity graphΓ of Ehas diameterδ(Γ ) ≤ 3. So, if a = b, thena and b have distance 3. However,f mapsevery 3-path from a to b onto a triangle ofΓ and the triangles ofΓ , beingcontained inelements ofE , cannot lift throughf to open paths ofΓ . Hence a = b. Namely, f is anisomorphism.

Suppose now thatE is of typeO+ and that theµ-subgraphs ofΓ are connected. Supposethat a = b. By an argument similar to that applied in theO−-case we see thata and bhave distance at least 4. Hence their distance is 4, sinceδ(Γ ) ≤ 4 by Theorem 2. Letα = (a, x, c, y, b) be a path froma to b. Thepointsa = f (a) = f (b) andc = f (c) havedistance 2 inΓ and, by the assumption, theµ-subgraphΓ (a) ∩ Γ (c) is connected. Hencethe pathf (α) splits in triangles. Therefore,a = b.

Corollary 4.2. The geometriesE(E6(2) : 2) andE(3 · 2E6(2) : 2) are simply connected.

Proof. Trivial, by Corollary 4.1 and since theµ-subgraphs ofE(3 · 2E6(2) : 2) areconnected (see [9]).

Remark. (1) We haveδ(Γ ) = 3 when E is E(E6(2) : 2) and δ(Γ ) = 4 whenE = E(3 · 2E6(2) : 2). Thus, the bound obtained forδ(Γ ) in the considered cases isas sharp as possible.

(2) The proof we will give for theO+-case ofTheorem 2could be repeated with onlyminor changes for the singular case as well, except for one crucial step in the proofof a lemma (Lemma 4.9, discussionof case (3)). Regretfully, we have not been ableto overcome that difficulty.

4.1. Preliminaries

Let Γ (a) ∩ Γ (b) be aµ-subgraph ofΓ type D8. By (c1),Γ (a) ∩ Γ (b) is connected. Itcan be described as follows (see [10, Chapter 6] and [9]). The setΩ := ιa(Γ (a) ∩ Γ (b))contains 144 points of the building ResE (a), contributed by 72 lines of ResE (a) passing


through a given pointe ∈ ResE (a)\Ω and forming the complement of a hyperplaneHof the dual polar space ResE (a,e), isomorphic to theG2(2)-generalized hexagon. We calle the a-pole of Γ (a) ∩ Γ (b). The following is proved in [10, Lemma 6.7] (see also [9,Lemma 3.2]):

Lemma 4.3. Let e be the a-pole of aµ-subgraphΓ (a) ∩ Γ (b) of type D8. A pointx ∈ Γ (a) is adjacent inΓ to somepoints ofΓ (a) ∩ Γ (b) if and only if the pointa, x ofResE (a) is not far from e.

We say that a pathα = (a0,a1, . . . ,an) with n ≥ 2 is ann-path if ai−1 andai+1 havedistance 2 for alli = 1,2, . . . ,n − 1. For(k1, k2, . . . , kn−1) ∈ 6,8n−1, we say that then-pathα is of type(Dk1, Dk2 , . . . , Dkn−1) if theµ-subgraphΓ (ai−1) ∩ Γ (ai+1) is of typeDki for i = 1,2, . . . ,n − 1.

Lemma 4.4. If two points ofΓ are joined by a4-path of type(D8, Dj , D8), then theirdistance inΓ is at most3.

Proof. Given a 4-path (a, x, c, y,b) of type (D8, Dj , D8), let ea, eb be thec-polesof Γ (a) ∩ Γ (c) and Γ (c) ∩ Γ (b) respectively and putΩa := ιc(Γ (a) ∩ Γ (c)) andΩb := ιc(Γ (c) ∩ Γ (b)). As everyline of ResE (c) contains at most two points that arefar from a given point, either at least 72 points ofΩa are non-far fromeb, or ea is almostfar from eb andΩa ⊆ Ψ3(eb).

In the first case,Lemma 4.3forces at least 72 vertices ofΓ (a) ∩ Γ (c) to be adjacentwith some vertices ofΓ (c) ∩ Γ (b), hencea has distance at most 3 fromb. Suppose thatea is almostfar from eb andΩa ⊆ Ψ3(eb). Thenea is collinear with at least 144 points farfrom eb. However, this cannot be, as|e⊥

a ∩ Ψ3(eb)| = 27 = 128 (see Fig. 2 of [10]).

4.2. The O−-case

Henceforth we assume thatE is of O−-type. The adjacency relation ofΓ will be denotedby ∼.

Lemma 4.5. Letα = (a,b, c,d) be a3-path of type(D6, D6). Then either thepoint a hasdistance at most2 from d, or we can replaceα with a3-path(a,b, c′,d) of type(D8, D6).

Proof. Let α = (a,b, c,d) be of type(Di , D6) and letΞ be a connected componentof Γ (b) ∩ Γ (d). In ResE (b) we consider the pointp := b,a and theAP-substructureΣ := ιb(Ξ ). By Theorem 3.7, p cannot be far from all points ofΣ . If p is collinear oralmost close to a pointq of Σ then, denoted byc′ the vertex ofΓ suchthatq = b, c′,we havea ∼ c′ and, asc′ ∼ d, thepointsa andd have distance at most 2. Otherwise,p isalmost far for some pointq ∈ Σ . In thiscase, takenc′ so thatq = b, c′, thepath(a,b, c′)has typeD8 (by (c1)) whereas(b, c′,d), beingcontained inΞ , has still typeD6.

Lemma 4.6. If two points ofΓ are joined by a4-path of type(D6, Dj , D8), then theirdistance inΓ is at most3.

Proof. Given a4-path(a, x, c, y,b) of type (D6, Dj , D8), let e be thec-pole ofΓ (c) ∩Γ (b) andΣ := ιc(Ξ ) for a connected componentΞ of Γ (c) ∩ Γ (a). By Theorem 3.7,


e /∈ Ψ3(Σ ). Hence, byLemma 4.3, at least one vertex ofΞ is adjacent to some vertices ofΣ and, therefore,a andb have distance at most 3.

Lemma 4.7. Two vertices ofΓ at distance4 can only be joined by a4-path of type(D6, D8, D6).

Proof. Supposea, b are vertices at distance 4 and letα = (a, x, y, z,b) be a 4-path froma to b and(Di , Dj , Dk) its type. In view ofLemmas 4.4and4.6, andpossibly permutinga andb, we mayassume that(i , j , k) = (6,6,6) or (6,8,6). Suppose(i , j , k) = (6,6,6).Then we can applyLemma 4.5to replace(x, y, z,b) with a path of type(D6, D8), thusreplacingα with a path of type(Di , D6, D8). (Note that, when doing that, we replaceywith another vertexy′ and the resulting 2-path(a, x, y′) might turn out to be of typeD8.)However,Lemma 4.4wheni = 8 andLemma 4.6wheni = 6 now forcea andb to havedistance 3. Therefore,(i , j , k) = (6,8,6) is the only possibility.

End of the proof in the O−-caseBy contradiction, suppose thatδ(Γ ) ≥ 4, let a,b be vertices ofΓ at distance 4 and

pick a vertexc at distance 2 from botha andb. Given connected componentsΞa andΞb

of Γ (c) ∩ Γ (a) andΓ (c) ∩ Γ (b) respectively, putΣa := ιc(Ξa) andΣb := ιc(Ξb). ByLemma 4.7, the path(a, x, c, y,b) is of type (D6, D8, D6) for everyx ∈ Ξa and everyy ∈ Ξb. This means thatΣa ⊆ Ψ4

2 (Σb), impossible byTheorem 3.17.

4.3. The O+-case

In this subsectionE is assumed to be ofO+-type.

Lemma 4.8. If two vertices a, b ofΓ are joined by a4-path (a, x, c, y,b) of type(D6, D6, D8), then either a and b have distance at most3 or there exists a vertexy′ ∈ Γ (c) ∩ Γ (b) suchthat (a, x, c, y′,b) is a 4-path of type(D6, D8, D8); furthermore,the vertex y′ can be chosen in such a way thatc, y andc, y′ are the two points of a lineof ResE (c) through the c-pole ofΓ (c) ∩ Γ (b).

Proof. Let e be thec-pole ofΓ (c) ∩ Γ (b) andΣ := ιc(Ξ ) for Ξ a connected componentof Γ (a) ∩ Γ (c) containingx. If e /∈ Ψ3(Σ ), thenLemma 4.3forces some vertices ofΞ tobe adjacent to some vertices ofΓ (c) ∩ Γ (b), hencea andb have distance at most 3.

Supposee ∈ Ψ3(Σ ) and letΩ := ιc(Γ (c) ∩ Γ (b)). As remarked at the beginning ofSection 4.1, Ω∪e is the join of 72 lines of ResE (c) througheande /∈ Ω . Let l = e, p,qbe the line throughe that contains the pointp := c, y and puts := c, x. Sincee is farfrom s and the points of ResE (c) non-far froms form a hyperplane in ResE (c), s is almostfar from a point ofl . As the2-path(x, c, y) is of type D6, s is far from p. So,s is almostfar from q. If y′ is the vertex ofΓ suchthatq = c, y′, thepath(a, x, c, y′,b) is of type(D6, D8, D8).

Lemma 4.9. Two vertices ofΓ at distance5 can only be joined by paths of type(D6, D8, D8, D6) or (D6, D6, D6, D6).

Proof. Let a, b be vertices ofΓ at distance 5 andα = (a, x, y, c, z,b) a 5-path froma to b. Let (Di , Dj , Dk, Dh) be its type. In view of Lemma 4.4, neither i = k = 8 nor


j = h = 8 are possible. Thus, modulo replacingα with its inverse, the following are thecases to consider:

(1) (i , j , k,h) = (6,6,6,6),(2) (i , j , k,h) = (6,8,8,6),(3) (i , j , k,h) = (6,6,8,8),(4) (i , j , k,h) = (6,6,6,8),(5) (i , j , k,h) = (6,6,8,6),(6) (i , j , k,h) = (8,6,6,8).

We shall prove thatcases (3)–(6) are impossible. In case (6) we can applyLemma 4.8to the subpath (x, y, c, z,b) of α, thus obtaining a path(a, x, y, c, z′,b) of type(D8, D6, D8, D8). By Lemma 4.4applied to(a, x, y, c, z′), thepointsa andz′ have dis-tance at most 3. Hencea andb have distance at most 4, contrary to our assumptions.

In case (5) we applyLemma 4.8to the subpath(a, x, y, c, z), thus obtaining a path(a, x, y, c′, z,b) of type (D6, D8, D8, Di ). We havei = 6, by Lemma 4.4applied to(x, y, c′, z,b). So, z,b is far from z, c′ in ResE (z). However,z,b is also far fromz, c (as the 2-path(c, z,b) is of type D6 by assumption) and far from thez-pole e ofΓ (z)∩ Γ (y) by Lemma 4.3. Also, according toLemma 4.8, the vertexc′ can be chosen insuch a way that the quadruplel = z, c, c′,e is a plane ofE . Hence, regardingl as a lineof ResE (z), z,b is almost farfrom a point ofl . We have reached a contradiction. Case (5)is ruled out.

Case (4)can be reduced to case (3) by applyingLemma 4.8to the subpath(x, y, c, z,b).Case (3) remains to examine. Denoting bye the y-pole ofΓ (y) ∩ Γ (z), let l be thelineof ResE (y) through e and y, c. Let Ξ be the connected component ofΓ (a) ∩ Γ (y)containingx andΣ := i y(Ξ ). By Theorem 3.7, at mostone point ofl belongs toΨ3(Σ ).That point must bee, otherwiseLemma 4.3forcesx andz to have distance less than 3.Therefore, we can pickx′ ∈ Ξ in such a way thaty, x′ andy, c are almost far. Thus,the path(a, x′, y, c, z,b) has type(D6, D8, D8, D8). Consequently,a andb have distanceat most 4, byLemma 4.4.

Lemma 4.10. Two vertices ofΓ at distance5 can always be joined by a path of type(D6, D8, D8, D6).

Proof. Givena, b at distance 5 inΓ and a 5-pathα = (a, x, y, c, z,b), Lemma 4.9forcesthe type ofα to be either(D6, D8, D8, D6) or (D6, D6, D6, D6). Assume the latter andputΣ := ιc(Ξ ) andΣx := ιc(Ξx), for Ξ a connected component ofΓ (c) ∩ Γ (b) andΞx

a connected component ofΓ (x) ∩ Γ (c). Clearly,Σ ∩ Σx = ∅ (otherwisea andb havedistance less than 5).

If Σx Ψ3(Σ ), then some points ofΣ are non-far from some points ofΣx and wecan replaceα with a pathβ of type (Di , D6, D8, D6), as in theproof of Lemma 4.8.By Lemma 4.4, i = 8 (otherwisea andb are forced to have distance less than 5). So,i = 6 and, by applyingLemma 4.8to the subpath ofβ of type (D6, D6, D8), we canreplaceβ with a pathγ of type(D6, D8, D8, Dj ). We havej = 6, otherwiseLemma 4.4applied to the(D8, D8, D8)-subpath ofγ would forcea andb to have distance less than 5.In this case, we are done.


SupposeΣx ⊆ Ψ3(Σ ). By Theorem 3.7, Σx is a member of theO+-triple Σ0,Σ1,Σ2of ResE (c) containingΣ . We mayassume thatΣ = Σ0 andΣx = Σ1. Denoting by Θthe connected component ofΓ (a) ∩ Γ (y) containingx, let t be any of the 119 vertices ofΘ\x. If the path(t, y, c) is of type D8, then wecan replaceα with β = (a, t, y, c, z,b),which is of type(D6, D8, D6, D6). By Lemma 4.8on (b, z, c, y, t) we can in turn replaceβ with a path of type(Di , D8, D8, D6). As above, if i = 8 thepointsa,b cannot be atdistance 5. In this case too, we are done.

Finally, suppose that(t, y, c) is of type D6 for everyt ∈ Θ . If Σt /∈ Σ1,Σ2 for somet ∈ Θ then we are in the same situation as whenΣx Ψ3(Σ ) and we are done. So,we may assume thatΣt ∈ Σ1,Σ2 for all t ∈ Θ . However,y ∈ Σt for any sucht . So,Σt = Σx = Σ1 for all t ∈ Θ . We now consider this situation as it looks in ResE (y). Theset X of verticesr ∈ Γ (y) with c, r ∈ Σ1 has size 63, the pointp = i y(c) is far fromi y(Θ) but almost closeto all points ofX, which inturn are almost closeto all points of theAP-substructurei y(Θ). However, this is a contradiction to Corollary 3.16.

End of the proof ofTheorem2Suppose thatδ(Γ ) ≥ 5, leta,b be vertices ofΓ at distance 5 andα = (a, x, c, y, z,b)

be a 5-path froma to b of type (D6, D8, D8, D6), which exists byLemma 4.10. Given acomponentΞ of Γ (a) ∩ Γ (c), putΣ := ιc(Ξ ).

Suppose thatΩ := ιc(Γ (c) ∩ Γ (z)) is contained inΨ42(Σ ). By Theorem 3.17, Ω is

a subgraph of the collinearity graph of the dual of theΩ+8 (2)-polar space. However, the

latter contains no triangles whereasΩ contains many of them. So, this case is impossible.ThereforeΩ Ψ4

2 (Σ ), namely at least one pointc, y′ ∈ Ω is far from a pointc, x′of Σ . We can replaceα with β = (a, x′, c, y′, z,b), which has type(D6, D6, D8, Di ).Lemma 4.9now forcesa,b to have distance less than 5.

5. The singular case

In this section we proveTheorem 3. Thus, assuming thatE is a c · F4(2)-geometryof singular type andG is a flag-transitive automorphism group ofE , we show thatE ∼= E(226 : F4(2)) and G ∼= 226 : F4(2). The proof will be achieved in a series oflemmas. Firstly notice thatE satisfies the conditions (c1) and (c2) ofSection 1(because ofthe flag-transitivity) and recall the following result proved in [9, Lemma 3.7(iii)]:

Lemma 5.1. If Ξ is a connected component of aµ-subgraphΓ (a) ∩ Γ (b) of Γ of typeD6, then the AP-substructureΣ = ιa(Ξ ) is classical.

Lemma 5.2. All µ-subgraphs ofΓ are connected.

Proof. In view of (c1), we only need to prove the connectedness ofµ-subgraphs of typeD6. By contradiction, suppose thatΓ admits a disconnectedµ-subgraphΓ (a) ∩ Γ (b)of type D6. The stabilizer G(a,b) of a and b in Aut(E) is transitive onΓ (a) ∩ Γ (b).Put Σ := ιa(Γ (a) ∩ Γ (b)) and regard it as an induced subgraph of the graph∆ withResE (a) as the set of vertices and ‘being collinear or almost close’ as the adjacencyrelation. Let A be the group induced byG(a,b) on Σ . Also, if Ξ1,Ξ2, . . . ,Ξn are the


connected components ofΓ (a) ∩ Γ (b), let Ai be the group induced onΣi := ιa(Ξi )

by the stabilizer of Ξi in G(a,b). By Lemma 5.1, all AP-structuresΣ1,Σ2, . . . ,Σn areclassical andAi ∼= 21+6 : Sp6(2) for i = 1,2, . . . ,n. Furthermore, by [9, Lemma 3.8(iii)],either A = F[Π ] ∼= Sp8(2) for an Sp8(2)-polar subgraphΠ or A = F(p) for a pointp of ResE (a). However,Σ1, Σ2, . . . ,Σn are the images ofΣ1 underA and, as they areιa-projections of the connected components ofΓ (a) ∩ Γ (b), they form apartition Σ in nconnected components. AsA is transitive onΣ , n is equal to the index ofAi in A.

Suppose first thatA = Sp8(2). Thenn = 255 and|Σ | = 32 640. So,Σ can only be thesubgraph of∆ induced on the orbitU3 of Lemma 3.11. Theorbit U3 contains subgraphsisomorphic to theAP-substructure of typeO+ and stabilized byΩ+

8 (2) < Sp8(2) (cf.Theorem 3.8). However, noAP-substructure ofO+-type is a subgeometry of anAP-substructure of singular type. Therefore,A = F(p) is the only possibility. Hencen = 28

and|Σ | = 215. Accordingly,Σ is the graphinduced by∆ onΨ3(p) andΣi is an orbit Mas inLemma 2.5(6). HoweverM is not a connected component ofΨ3(p). We have reacheda final contradiction.

In view of (c1), the following convention (borrowed from [9]) is consistent: given twopointsx, y of E at distance 2, we writey ∈ Γ 3

2 (x) or y ∈ Γ 42 (x) according to whether

Γ (x) ∩ Γ (y) is of type D6 or D8.Let y ∈ Γ 3

2 (x). By Lemma 5.2, theιx-imageΣ = ιx(Γ (x)∩Γ (y)) in Γ (x) is a classicalAP-substructure of ResE (x). So,denoted byC the set-wise stabilizer ofΓ (x) ∩ Γ (y) inthe permutation groupG(x, y)Γ (x) induced byG(x, y) on Γ (x), C ∼= Aut(Σ ) ∼= 21+6 :Sp6(2). This isomorphism implies thefollowing (see [9, proof of Lemma 3.5]):

Lemma 5.3. If z ∈ Γ 32 (x) is such thatΓ (x) ∩ Γ (z) = Γ (x) ∩ Γ (y), then z= y.

The above in its turn implies the next (cf. [9, Lemma 3.7(iii) and remarks beforeLemma 3.6]):

Lemma 5.4. G(x) actsfaithfully onΓ (x), i.e. G(x) = F ∼= F4(2).

In view of this, we take the liberty of identifying Γ (x) and the point-set ofF , thusgetting rid of the ‘embedding’ιx : Γ (x) → ResE (x). Accordingly, we may assume thatG(x, y) ≤ G(x, p) = F(p) for a given pointp ∈ Γ (x) and thatΓ (x) ∩ Γ (y) and theorbit M of Lemma 2.5(6) are the same thing. So,y is theunique vertex ofΓ 3

2 (x) suchthatΓ (x) ∩ Γ (y) = M (seeLemma 5.3).

Let q ∈ M, K1 = F(p) ∩ F(q) ∼= Sp6(2) (a Levi complement inG(x) ∼= F4(2))andr be the second vertex inM fixed by K1 (see Remark at the end ofSection 2.3). LetS ∼= Sp8(2) be such that its orbitΘ of length 255 onΨ containsp and M (notice thatS is uniquely determined). Notice also thatNF (K1) = NS(K1) ∼= S3 × Sp6(2). ThenM(p) := M = Θ ∩ Ψ3(p), we can defineM(q) and M(r ) similarly and letu andv bethe vertices inΓ 3

2 (x) havingM(q) andM(r ) asµ-subgraphs. Theny,u, v are fixed byK1and we have the following.

Lemma 5.5. The following assertions hold:

(1) the subgroup K1 is a Levi complement in G(p);


(2) an element t∈ Gx, p\G(x, p) can be chosen to normalize (and even centra-lize) K1;

(3) the connected componentΛ containing x in the subgraph ofΓ induced by the verticesfixed by K1 is the 3-dimensional cube and the stabilizer X ofΛ in NG(K1) is23 : S3 × Sp6(2);

(4) the element t in(2) can be chosen to be an involution.

Proof. SinceK1 fixes (at least) three vertices, namely,x, u andv, adjacent top, (1) fol-lows from Lemma 2.5(5). ThusGx, p preserves the classes ofSp6(2)-complements ofG(x, p) = F(p) and (2) follows (sinceSp6(2) is complete, normalization implies central-ization). By (2), X acts vertex-transitively onΛ and by the paragraph before the lemma wehaveΛ(x) = p.q, r , Λ2(x) = u, v, y andNG(x)(K1) ∼= S3 × Sp6(2) induces the fullsymmetric group of bothΛ(x) andΛ2(x). By noticing that any two vertices at distance 2in Λ have exactly two common neighbours, we easily identifyΛ with the 3-dimensionalcube and the action ofX on Λ with the full automorphism group 23 : S3 of the cube.If we chooset ∈ X ∩ CG(K1), we obtain (4), sinceCG(K1) ∩ X also induces the fullautomorphism group ofΛ. Lemma 5.6. The element t inLemma5.5(4) commutes with G(x, p), so that Gx, p ∼=2 × 21+6+8 : Sp6(2), in particular the centre of Gx, p is elementary Abelian of order4.

Proof. Suppose to the contrary thatt induces a non-trivial automorphism ofF(p) =G(x, p) and consider the semi-direct productR of F(p) and〈t〉 with respect to the naturalaction. Further consider the action ofR on the cosets ofK1. This would give an extensionof the action of F(p) onΨ3(p) by an automorphism of order 2 commuting withK1. It isclear thatt acts trivially onC(p), Z(p)/C(p) andQ(p)/Z(p) since each is an absolutelyirreducibleK1-module. Next it is easy to check thatt must act trivially on Z(p), since itcentralizes a complement. Nowt fixes the orbitM of Z(p) onΨ3(p) point-wise and eachother orbit (they are indexed by the elements ofQ(p)/Z(p)) as a whole. Since there arematchings connecting some pairs of the orbits (seeLemma 2.5(6)), and the correspondinggraph on the orbits is connected,t acts trivially on the cosets ofZ(p) in Q(p), hence theresult.

Let t0(p), t1(p), t2(p) be the non-identity elements of the centre ofG(x, p). Heret0(p)is the non-trivial element ofC(p) (cf. Lemma 2.5(8)),

ti (p) = t3−i (p)t0(p) for i = 1,2

andt in Lemma 5.6is eithert1(p) or t2(p). For any pointq of F defineti (q) = f −1ti (p) f ,where f ∈ F mapsp ontoq. By Lemma 5.6, ti (q) is well defined. Letψi : p → ti (p) bea mapping of the point set ofF into G, i = 0,1,2.

Lemma 5.7. ψi defines a representation ofF for i = 0,1,2.

Proof. For i = 0 the claim is byLemma 2.5(8). Fori = 1 or 2 all we need is to check theline-relations. Letl = p,q, r be aline in F , we need to check thatsi := ti (p)ti (q)ti (r )is the identity element inG.

We claim that for everyα ∈ l the elementti (α) induces an even permutation on theextended lineE := x, p,q, r . First assume the claim is true. Thensi acts trivially onE,


hence it is contained in the stabilizerL of l in F = G(x) and furthermore it is in the centreof L by the choice ofti (α) (it commutes withG(x, α) and it is uniquely specified by theedgex, α). By Lemma 2.5(7) the centre ofL is trivial and hencesi = 1 as desired.

Now it remains to prove the claim. Consider a planeπ = p1, . . . , p7 in F (contain-ing l ), let π = x ∪ π be the extended plane. LetD be the action induced onπ by itsstabilizer in G. ThenD ∼= 23 : L3(2) and the image ofti (p1) in D must commute withD(x, p1) ∼= S4. This shows that the image ofti (p1) belongs to the translation subgroup ofD, henceti (p1) acts onπ fixed-point freely, which gives the claim.Lemma 5.8. For exactly one i∈ 1,2 the representationψi is such that the images of thepoints incident to a symp S generate an elementary Abelian group of order26.

Proof. Notice first that in any non-trivialF-admissible representation ofF the imagesof the points in a symp generate an elementary Abelian 2-group of order 26 or 27.Furthermore it is easy to check (cf.Lemma 5.10below) that with respect toψ0 theimages generatea group of order 27 which is the centre ofO2(F[S]) (in fact, it is theimage of Z(p) under an outer automorphism ofF4(2)). The extensionS of the sympS is a complete graph on 64 vertices. The stabilizer ofS in G induces onS the group26 : Sp6(2). Therefore Y := 〈ψ0(p), ψ1(p), ψ2(p)〉p∈S is elementary Abelian of order213. Furthermore,R = F[S]/O2(F[S]) ∼= Sp6(2) acting onY centralizes a unique non-identity element (which generates the centre ofF[S]) and has two non-trivial chief factorseach isomorphic to the 6-dimensional symplectic module. HenceY/Z(F[S]), as amodulefor R, contains exactly three irreducible 6-dimensional submodules. It is easy to see thatthe preimages of two are indecomposable (orthogonal modules) while exactly one is thedirect sum of 1- and 6-dimensional modules. Hence the result follows.

Notice that ifE = E(226 : F4(2)) andG = 226 : F4(2) then the image ofψ0 generates acomplement toO2(G) in G. If i is as inLemma 5.8then the image ofψi generatesO2(G)and that ofψ3−i generates the wholeG (which is also a representation group ofF ).

To complete the job we need the following preliminary lemma.

Lemma 5.9. Let (P, χ) be the Sp6(2)-admissible8-dimensional representation of thedual polar spaceD of Sp6(2) in the spin module andΩ be the collinearity graph ofD(of valency14 in 135vertices). Then

(1) χ is the only Sp6(2)-admissible representation ofD in which the images of the (dual)points in a quad generate an elementary Abelian group of order24;

(2) if ω ∈ Ω then thepoints at distance1 fromω generate a group X of order24, whilemodulo X the points at distance2 fromω generate a group of order23.

Proof. Given anSp6(2)-admissible representationρ : D → R as in (1), letAfρ(D) bethe (right) affine extension ofD by ρ, namely the rank 4 geometry with the elements ofthe groupR as 1-elements and the (right) cosets of the subgroups〈ρ(Y)〉 of R as elementsof type 2, 3 and 4, forY respectively a point, a line or a quad ofD. Then Afρ(D) isan extended near hexagon as considered by [6]. The assumption that for a quadQ thesubgroup〈ρ(Q)〉 is elementary Abelian of order 24 implies hypothesis(∗) of [6]. Thus,[6, Theorem 1.3] can be applied toA fρ(D) and the equalityρ = χ follows. In its turn,claim (2) follows from (1).


Lemma 5.10. Let (R, ϕ) be an F-admissible representation ofF such that the imagesunderϕ of the points in a symp generate a group of order26. Then R isthe irreducible26-dimensional F4(2)-module.

Proof. SinceE(226 : F4(2)) is ac·F4(2)-geometry, we see that the 26-dimensional modulesatisfies the hypothesis of the lemma. Thus it is sufficient to show thatR is of order atmost 226.

For a subsetΛ of the point-set ofF put RΛ := 〈ϕ(q)〉q∈Λ. Let p be a point ofF andS be a symp containingp. By the hypothesis of the lemmaϕ(q)q∈S together with theidentity is the element-set of the groupRS, hence it is closed under multiplication.

Consider anSp8(2)-polar subgraphΠ , put D := F[Π ] ∼= Sp8(2), P := P(Π ) andassume thatp ∈ Π .

Claim A. The restriction ofϕ to Π defines a representation ofP .

We need to check the line-relations. Letl = p,a,b be aline of P (so thata,b ∈Ψ2

2 (p)). Thenl is the intersection ofΠ with a symp and we can assume that this symp isS.By the aboveparagraphϕ(p)ϕ(a) = ϕ(c) for somec ∈ S. On theother hand,D containsthe stabilizerF([S], p,a) of p anda in F[S]. Henceb is the only point in S\p,a fixedby F([S], p,a). Thereforec = b, as claimed.

By Claim A and [8, 3.5.1 and 3.5.5] we immediately obtain the following:

Claim B. The group RΠ is elementary Abelian of order at most29.

In order to finish the proof ofLemma 5.10we only need to prove the following

Claim C. |R| ≤ 226.

We shall proveClaim Cin a few steps. We firstly state the following definitions:

R1 = Rp,R2 = Rp ∪ Ψ1(p),R3 = Rp ∪ Ψ1(p) ∪ Ψ2

2 (p),R4 = Rp ∪ Ψ1(p) ∪ Ψ2

2 (p) ∪ Ψ3(p).Step 1. [R4, ϕ(p)] = 1.

This follows fromClaim B, noticing that, ifq ∈ Ψ3(p), thenp andq are contained in acommonSp8(2)-polar subgraph ofF .

Step 2. R4 = R.

This is immediate from the factthat every line intersectingΨ3(p) intersectsΨ3(p) intwo points and the third point is inΨ4

2(p). Thenext step is obvious.

Step 3. |R1| = 2.

Step 4. |R2/R1| ≤ 28.

This follows fromLemma 5.9(1), sinceRS∩ Ψ1(p)/R1 is of order 24.

Step 5. |R3/R2| ≤ 27.


Sinceboth RSR2/R2 andRS∩ Π R2/R2 are of order 2, the equalityR3 = R2RΠ ∩Ψ2

2 (p) holds, while byClaim B, RΠ ∩ Ψ22 (p)R1/R1 is of order at most 27.

Step 6. |R4/R3| ≤ 210.

By Claim B, RΠ R3/R3 has order (at most) 2. On the other handΠ ∩ Ψ3(p) is a Z(p)-orbit. HenceR4/R3 is generated by 28 involutions corresponding to theZ(p)-orbits onΨ3(p). Let ∆ be the graph on these orbits in which two orbits are adjacent if they containcollinear points. Then∆ is of valency 135 (cf.Lemma 2.5(6)) and it is the Cayley graphassociated with the representation of theSp6(2)-dual polar space in the spin module. Thenthe result follows from Step 4 and the fact that the∆-vertices equal to or adjacent to agiven vertex form a hyperplane with connected complement.

The bound|R| ≤ 226 immediately follows from the above.

Proof ofTheorem3

By Lemmas 5.8and5.10.

Acknowledgements

We thank both referees for their useful remarks and suggestions. In particular, one ofthem has suggested a proof ofTheorem 3.7far shorter and more perspicuous than ouroriginal proof, which was admittedly rather involved. We have been glad to replace ourproof with that suggested by the referee. A part of the research was conducted whilst thefirst author was visiting the Department of Mathematics, University of Siena under anINDAM grant.

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a diameter bound for extensions of the f4(2)-building

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