A R J E H M. C O H E N

AN AXIOM

SYSTEM

FOR METASYMPLECTIC

SPACES

ABSTRACT. Metasymplectic spaces are geometries representing Tits' weak buildings of Type F 4. For metasymplectic and polar spaces an axiom system is given that characterizes them in terms of points and lines.

Key words and phrases: Metasymplectic spaces, buildings of type F4, polar spaces.

1. T E R M I N O L O G Y

AND NOTATIONS

An incidence system (P, S ) is a nonempty set P of points together with a collection ~o of subsets of cardinality > 1, called lines. If (P, ~ ) is an incidence system then the point graph or colinearity graph of (P, ~ ) is the graph (P, F) whose vertex set is P and whose edges consist of the pairs of colinear points. The incidence system is called connected whenever its colinearity graph is connected. Likewise terms such as (co)cliques, paths will be applied freely to (P, 2 ~) when in fact they are meant for (P, F). (Recall that a set of vertices of a graph is called a clique whenever each pair of vertices of this set is an edge, and a coclique whenever no pair of vertices of the set is an edge.) We let d(x, y) for x, y e P denote the ordinary distance in (P, F) and write Fi(x) = { y e P [ d ( x , y ) = i}. Also F ( x ) = F l(x)

and

x±={x}wF(x).

For a subset X of P we write X ± -- (~x~Xx±. Ifz e P and X, Y are subsets of P, then d(z, X) = inf d(z, x) xeX

and

d(X, Y) = inf (y, X). yeY

A subset X of P is called a subspace of (P, co) whenever each point of P on a line bearing two distinct points of X is itself in X. A subspace x is said to be nondegenerate if none of its points is colinear with all points of X. A subspace X is called singular whenever it induces a clique in (P, U). The length i of a longest chain X~u c~ X,I c ". ~ Xi = X of nonempty subspaces Xj of a singular subspace X is called ~ e rank of X. For a subset X of P, the subspace generated by X is denoted ( X ) . Note that it is well defined as P is a subspace containing X. Furthermore, L(X) denotes the set of lines contained in X. The incidence system (P, ~ ) is called linear if any two distinct points are on, at most, one line. If x, y are colinear points of a linear incidence system, then xy denotes the unique Geometriae Dedicata 12 (1982) 417-433. 0046-5755/82/0124-0417502.55. Copyright (~ 1982 by D. Reidel Publishing Co., Dordreeht, Holland, and Boston, U.S.A.

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line through them. Thus x y = ( x , y ), where ( x , y ) If ~,~ is a family of subsets of P and x e P, then ~ x of ~- of members containing x. Also, L ( ~ ) denotes for X e J~. Finally, a line is called thick if there are at and thin if it has exactly two points. 2.

INTRODUCTION

is short for ( { x, y} ). denotes the subfamily the collection of L(X) least three points on it

A N D S T A T E M E N T OF T H E T H E O R E M

Metasymplectic spaces were introduced by H. Freudenthal [8] as part of the study of geometries of type F 4 . A set of axioms together with a characterization in terms of buildings of type F 4 under mild additional conditions, can be found in Tits' work [10, pp. 215-217]. Our starting point will be the definition given there. In order to state the definition, however, we need the notion of polar spaces. For a definition of polar spaces of rank n we could cite [1] or [7]. Instead, we shall formulate the celebrated BuekenhoutShult theorem from [2] for two reasons: (i) it can be used as an alternative definition, and (ii) it suffices for our purposes. 2.1. T H E O R E M (Buekenhout-Shult). Let (p,~qo) be an incidence system and let n be a natural number. Then P together with its singular subspaces is a polar space of rank <<.n if and only if the following holds." (P1) (P2) (P3)

No point of P is colinear with all of P. For each point x of P and line I of Z~ with x ~ l, x is colinear with either one or all points of I. The singular subspaces of P are of rank <. n - 1.

Buekenhout [-1] observed that the extra hypothesis in [2] that all lines be thick is superfluous. Therefore, it is omitted here. We note that in contrast to other places in the literature, here polar spaces are nondegenerate (i.e., axiom (P1)holds). The theorem can be used as a definition of a polar space of rank <~n. Of course, a polar space of rank n is a polar space of rank ~< n which is not a polar space of rank ~< n - 1. Polar spaces of rank 2 are called generalized quadrangles (this means that all generalized quadrangles are nondegenerate). We shall often abuse language and call incidence systems polar spaces when in fact the associated geometry of its points and all its singular subspaces is a polar space. Similarly for projective spaces. The singular subspaces of polar spaces have the structure of projective spaces. Here a projective space is what is usually called a generalized projective space (thus thin lines may occur in projective spaces). A projective plane is a projective space of rank 2. Finally, (P, L) is said to be a dual polar space of rank n( >>-2) if P can be identified with the maximal singular subspaces of a polar space of rank n in such a way that any line is the set of such subspaces containing

AN AXIOM SYSTEM FOR METASYMPLECTIC

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419

a singular subspace of rank n - 2 of the polar space (and conversely) (see

[3]). 2.2. DEFINITION(Tits [-10], 10.13, p. 215). A metasymplectic space is a set P in which some subsets, called lines, planes and symplecta are distinguished, and satisfy the following axioms: (M1) (M2) (M3)

The intersection of two distinct symplecta is empty, or is a point, a line or a plane. A symplecton, together with the 'linear subspaces', empty set, points, lines and planes contained in it, is a polar space of rank 3. Considering the set x* of all symplecta containing a given point x of P, and calling lines (resp. planes) of x* the subsets of x* consistin 9 of all symplecta containing a plane (resp. a line) through x, one also obtains a polar space of rank 3.

A classical result of Veblen and Yong characterizes projective spaces in terms of points and lines. The Buekenhout-Shult Theorem is a characterization of polar spaces in terms of points and lines. More recently, Cameron [-3] and Shult and Yanushka [9] obtained a characterization of dual polar spaces in terms of incidence systems. Also Cooperstein's work [-7] provides characterizations of this kind of certain geometries associated with buildings of Lie type. We shall make use of some of these results (see Section 3) to prove the following characterization of metasymplectic spaces in terms of points and lines. 2.3. THEOREM. Let (P, Aa) be a connected incidence system. Then P and ~,¢ can be identified with the points and lines of a connected metasymplectie space or a polar space if and only if(P, ~ ) satisfies the following axioms."

(F 1) (F2) (F3)

(F4)

For each x ~ P and each l ~ ~ either 0 or 1 or all points of I are colinear with x. For each pair x, y ~ P with x ~ y±, the graph on x ' n y i is not a clique. For each pair x , y ~ P with xq~y ±, such that x ' ~ y ± contains at least two points, x" ~ y" together with the lines whose points are all in x" n y± is a polar space of rank >>-2. There are no minimal 5-circuits, i.e., given xl , x2, x 3, x4, x 5 e P with xi ~ F~(xi+ 1)\x;L+2 for each i (indices taken rood 5), there is an i for which x i is colinear with a point on a line through xi+ 2 and Xi+ 3 •

(F5) To

I f x, y, z E P are such that x i n y . has at least two points and y ~ z ±, then x ± c~ z ± :/: ~ .

facilitate notations,

a connected

incidence

system satisfying

420

ARJEH

M. C O H E N

(F1) .... , (F5) is called a metapolar space. The theorem states that the metapolar spaces of diameter 3 are precisely the connected metasymplectic spaces (whose planes and symplecta are discarded). The original theorem found by the author and announced in [5] is stated for geometries with thick lines only and characterizes metasymplectic spaces under a stronger version of (F3) (namely that for any two points x, y of P with d(x, y ) = 2 and Ix ± c~y±l > 1 the subspace x ± c~yI is a polar space of rank 2). The removal of both the 'thick line' and the 'rank 2' condition were suggested by Buekenhout. Professor Shult proved to the author that the rank 2 condition may indeed be dropped. His result is incorporated in this paper via Lemmas 4.2 and 4.3. Lines of an arbitrary length ( ~< 2) could be allowed, thanks to an extension of Cooperstein's Corollary to 3.10 in [7], given in the proof of 3.8. 2.4. Remark. An incidence system satisfying (F1),(F2), (F3) is linear and completely determined by its colinearity graph. Moreover l±± = l for any For, if (P, ~ ) is such a system and x e P , y~Fl(x) then there are u, v E P with u 6 v ± such that x, y e u ± c~ v ± by (F2), so that u ± c~ v± is a polar space by (F3). If I is a line on x, y, then all its points belong to u ± c~ v± according to (F1), so that l ~_ u±c~v ± and I±±___{u, v}±±z= u±c~v 1. Thus 1z± is a clique containing l and contained in all maximal singular subspaces of {u, v}l on l and therefore by (F3) equal to l. But l±__ {x, y}±___ l± (again, using (F1)), so l = Iaz = {x, y}a±. This settles the remark. We shall now briefly discuss the individual axioms. Axiom (F1) states that (P, ~ ) is a Gamma space in D. G. Higman's terminology. It is known [6], that the points and lines in a metasymplectic space arising from a building of type F 4 satisfy this property. Axiom (F2) is a nondegeneracy condition. Together with (F1),(F3) it ensures that the incidence system is, in fact, determined by its colinearity graph (as we have just seen) and that it contains polar spaces as subspaces. Generalized hexagons, but also near octagons without 4-circuits (see [4] ), are examples of incidence systems satisfying all axioms (F1) . . . . . (F5) except for (F2). Axiom (F3) provides the building blocks for simplecta. It may be regarded as a slight extension of Cooperstein's axiom 1.4 of [7]. Axiom (F4) is a weakened version of the near n-gon property (of [9] ) that for any x e P and leSe with d(x, l) = 2 there is a unique point on I nearest x. The geometries described by Cooperstein in Theorem B of [7] are examples of incidence systems in which all other axioms hold and which are not metasymplectic spaces.

AN A X I O M SYSTEM FOR M E T A S Y M P L E C T I C S P A C E S

421

Axiom (F5) may be viewed as another nondegeneracy condition. For it rules out bouquets of polar spaces, i.e., geometries (P, ~ ) containing a point x such that any connected component of P\{x} joined with {x} forms a subspace which is a polar space. Clearly, these bouquets satisfy all other axioms. The proof of the theorem consists of Sections 4 and 5. To conclude this introduction we give three more axioms which hold in metapolar spaces. It might be of interest to know to what extent they could replace axioms in the theorem. (F6) (F7) (F8)

For x e P and l ~ Y either l or allpoints yoflsatisfyd(x, y)= d(x,l). For l, m s S f either 1 or all points x of l satisfy d(x, m) = d(1, m). For x e P , 1~5~ with d(x, l) = 2, there is at most one point y in Ifor which x ±c~y± contains at least two points.

The proofs of (F6), (F7) for metapolar spaces are omitted as they are immediate consequences of the results in Section 4. Axiom (F8) is shown to hold in 4.6. 3. P R E L I M I N A R Y

RESULTS

Most of the material in this section is taken from the sources [3], [7] and [9]. The first lemma is a slightly altered version of Proposition 2.5 in [9], to which the reader is referred for a proof. 3.1. L E M M A (Shult-Yanushka). Let (P, ~ ) be an incidence system with thick lines such that for x~P, l~Lf with d(x, l)<. 2, there is a unique point of l nearest x. Then (P, Lf) is linear. Moreover, if al, a 2 , a s , a4 is a minimal 4circuit in P(i.e., a i6 F(a i + 1)\ai+ 2 for all i, indices taken mod 4), then Q = {xEP[ d(x, aiai+ 1) <" 1 for each i} is subspace of P which, together with the lines it contains, is a generalized quadrangle such that for any x e P\ Q, the intersection x± c~ Q contains at most one point. 3.2. DEFINITIONS. Let (P, ~ ) be an incidence system such that for x s P , l ~ with d(x, l) ~< 2, there is a unique point of I nearest x. A quad in in (P, ~o) is a subspace Q of P which is a generalized quadrangle such that any point of P collinear with two distinct points of Q is inside Q. Two points x, y of P with d(x, y) = 2 are contained in at most one quad, denoted by Q(x, y) whenever it exists. By the above lemma Q(x, y) exists if [x-Lc~y±l > 1 and the lines of (P, L~°) are thick. Moreover, if Q(x, y) exists, then for any two mutually non-colinear ul, u 2 ~ {x, y}Z we have Q(x, y) = {ze P I d(z, xui), d(z, YUi) -%<1 for i = 1, 2}. The intersection of two quads is either empty or a singular subspace.

422

ARJEH

M. C O H E N

If (P, £,e) is a connected incidence system, it is called a near hexagon (cf. [9] ) whenever it satisfies the following two axioms: (N 1) (N2)

For each x • P and l• ~ there is a unique point of l nearest x; The diameter of (P, 5#) is at most 3.

The near hexagon (P, 5¢) is said to be classical (cf. [3] ) if it also satisfies: (N3) (N4)

For any x, y • P with d(x, y) = 2, there is a quad on x, y; For each quad Q and x ~ P\Q, the intersection x I c~Q is nonempty.

3.3. LEMMA. Let (P, 2P) be a connected incidence system such that for x • P , l • ~ with d(x, l)<<,2, there is a unique point of l nearest x, and suppose (N3) holds for (P, ~ ) . Then (P, ~ ) is a classical near hexagon if and only if the intersection of no two quads is a point. Moreover, if this is the case, then: (N5) (N6)

For each quad Q and x • P\Q the intersection x ±~ Q is a singleton. Given a quad Q and distinct colinear x, y e P for which there are noncolinear u, v • Q satisfyin9 ve y ± and u • y ±, then xy ~_ Q.

Proof. We shall leave the proof of (N5) and (N6) for a classical nearhexagon to the reader. If (P, 5 °) is a classical near-hexagon and Q, R are distinct quads with x • Q n R for some x • P, we need to verify that Q c~R contains another point. Take zerz(x)c~R. Then z~Q as Q ~ R is a clique. In view of (N4) there is y e P with y e z ~ ~ Q. Note that x # y. If y • x ±, then y • x ± n z ± ~ R, so Q c~R contains x, y as wanted. On the other hand, y ~ x ± implies d ( x , y ) = 2 = d(x,z), so that there is w e y z c ~ x ± by (N1). Now w e ( x ± ~ z ± ) c ~ ( x ± ~ y l) c_ Q ~ R and we are done since w :p x. To prove the converse, let (P, ~ ) be an incidence system satisfying the hypothesis and assume that the intersection of no two quads is a point. Let Q be a quad and consider x • P \ Q for which there is y • x I ~ Q. Take z e x ± \ x y and consider the quad Q(y, z) whose existence is guaranteed by (N3). Since y e Q n Q(y, z), there must be a line m such that Q c~Q(y, z)= m and there is w e m c~ z I ~ Q c~ z ' . By induction with respect to the length of a path from an arbitrary point z of P\Q to a point of Q, we get Q c~z ± ~= ~ for any z • P\Q, proving (N4). It remains to show that for z e P and l•~e we have d(z, l)~< 2. We may assume that the diameter of (P, ~ ) is ~> 3. Thus there exists a quad R containing I. Note that for z • R , we have d(z, l) ~< 1. Hence for arbitrary z e P , axiom (N4) yields d(z, l) <<.d(z, R) + 1 ~ 2. This finishes the proof. []

The significance of classical near hexagons will become clear from the following result, which is a consequence of Theorem 2.3 in [3]. 3.4. T H E O R E M (Cameron, Shult-Yanushka). Let (P, 5¢) be a connected

AN A X I O M S Y S T E M F O R M E T A S Y M P L E C T I C

SPACES

423

incidence system. Then (P, 5¢) is a generalized quadrangle if and only if it is a classical near hexagon of diameter 2. Moreover, (P, ~ ) is a dual polar space of rank 3 if and only if it is a classical near hexagon of diameter 3. We are now in a position to reformulate axiom (M3) in terms of near hexagons. Recall that if ~ is a collection of subsets of P, then L(~- )~ for x e P denotes the collection of subsets of lines through x contained in a member of ~ on x. 3.5. COROLLARY. Let (P, ~a) be an incidence system in which collections

7Y, 5 ¢ of subsets of P are distinguished. Then (P, 5q) together with ~//-for its collection of planes and 5¢ for its collection of symplecta, is a metasymplectic space if and only if (M1), (M2) and (M3) 3 hold, where (M3) 3 is the following axiom." (M3)3

For any x e P , ( S x, L(~/fx)x) is a classical near hexagon of diameter 3 whose quads are the members of L(SCx)~.

In the remainder of this section we recall some properties of incidence systems satisfying (F1), (F2) and (F3). They can be found in Section 3 of [7], where, instead of (F3), a slightly stronger statement holds. However, except for the last statement of Proposition 3.8 the proofs are still valid in our case, so we shall not bother to copy them. 3.6 LEMMA. Let (P, f ) be an incidence system for which (F1) holds. Then (i) maximal cliques are singular subspaces; (ii) for any clique X of P, the subspace ( x ) is singular; (iii) if X is a subset of P, then X l is a subspaee. 3.7. LEMMA. Suppose (P, S ) is an incidence system for which (F1), (F2), (F3) hold. Then (i) if l, m e S with d(l, m) = 1, then d(y, m) = 1 for exactly 1 or all points y ofl, (ii) /f d(x, y ) = 2 and x±m y ± contains at least two points, then {x, y}±± is a coclique. 3.8 P R O P O S I T I O N (Cooperstein). Let (P, 5~) be an incidence system satisfying (FI), (F2), (F3). I f x, y e P with d(x, y) = 2 have at least two common

neighbors, the subset S(x, y) of all points z e P satisfying zX ~ l ¢ ;Z5 for each line l of x x ~ y± is a subspace which is a polar space. Moreover, S(x, y) = S(a, b) for any two a, b e S(x, y) with d(a, b) = 2. Proof. For all but the last statement, see [7]. Let S = S(x, y). Let a, be S(x, y) with d(a, b) = 2. In view of Lemma 3.10 of [7], we may assume a, b~x x my ± (if, e.g., a(~x ±, then S = S(x, a) = S(b, a) = S(a, b)). If [xa[ i> 3, take weax\{a, x} ; then S = S(w, y) = S(w, b) = S(a, b), so we may assume that [xa I = 2 and, because of similar reasons, ]xb I = !ay[ =

424

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M. C O H E N

Iby[ = 2. We shall show S(a, b) ~_ S. As x, y e a ± ~ b ± ~_ S(a, b), we then get S c_ S(a, b) by symmetry. Take z e S and let 1 be a line in {a, b} ±. We want to show that z I c~ l # ~ . Note that we may assume z ¢ a ± u b ± for otherwise z, I are in the polar space a ± c~ b ±, so that z I c~ l ~: ~ . Moreover, z, bx, ax, by, ay are all in S, so z e b - w x ~, a I u x ±, a ± u y±, b I w y± in view of cardinal±ties of the lines involved. Thus z ~ x ± or zCy ± would imply z ~ a i c~b ±. It follows that z ~ x i ~ y ±. As x , y , l are all in {a,b} ±, there are u, vEl with {u} =u~c~l and {v} = y±c~l. Suppose u = v . Then u E { x , y } ~ _ S , so z±c~ua-~ ~ and z ± c ~ u b ~ ~ . If u ~ z ±, we have u~z±c~l and we are done. Therefore assume u C z £ Then there are u 1 ~z ~ c~ ua\{u, a} and u 2 ~z ± c~ ub\{u, b} with ul Cu~. By the same argument as before, we get z ~ S ( u l , u2) = S(ul, b) = S(a, b) as wanted. Suppose u @ v. Now x, vy are both contained in {a, b} ±, so there is w e x ± c~ vy\{v,y}. It follows that w~S, so v ~ w y ~_ S. Similarly, u e S , whence l _ S. Now z ± c~ l # ~ as both z, I are in S. This finishes the proof. [] 3.9. D E F I N I T I O N S . Let (P, ~ ) be as in the preceding proposition. Any pair x, y ~ P with d(x, y) = 2 such that x ± c~y± contains at least two points, will be called a sympIectic pair. The subspace S(x, y) from above for such x, y will be called the symplecton on x, y and will be denoted as in the proposition. The following corollary shows that the symplecton on x, y for x ¢y" is the unique symplecton containing x, y. 3.10. COROLLARY. If(P, 5~) is an incidence system in which (F1), (F2), (F3) hold, and if S, T are distinct symplecta, in P, then S c~ T is a singular subspace. 3.11. LEMMA. Let (P, 5~) be as in Proposition 3.8. (i) Each singular subspace of rank <, 2 is contained in a symplecton. Hence it is a point, a (projective) line or a projective plane," (ii) if M is a singular subspace and M properly contains a line, then M is a projective space of rank >- 2.

4. M E T A P O L A R

SPACES ARE EITHER

POLAR

OR METASYMPLECTIC

Throughout this section (P, 5¢) is a metapolar space which is not a polar space. Thus P is not a symplecton. Let # be the collection of maximal singular subspaces. Their members are called planes. This name will be justified by Lemma 4.4(i). Furthermore, write 5~ for the collection of symplecta. We shall show that (P, 5~) with points, lines, planes and symplecta satisfies axioms (M1), (M2) and (M3)a (cfi 3.5). 4.1. LEMMA. Let x l , x2, x 3 , x 4, x s be a 5-circuit in P with x ~ F l ( x i + 1)\x/X+2 (indices mod 5)for all i . Then there is a unique symplecton S containing all x i.

AN A X I O M S Y S T E M F O R M E T A S Y M P L E C T I C

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425

Proof. By (F4), there is i, say i = 1, and w e P with {w} = F(xl) n x3x 4. N o t e that w 4= x 3 , x 4 . If w e x~, then x 2 e (x 3w) ± by (F 1), and x 2 e x~, contradiction. Hence wCx~;similarly wCx~. But x l , x 3 e x 2 n w ±, so x 2, w is a symplectic pair and x g e wx 3 ~ S(xz , w) in view of Proposition 3.8. Thus x~ , x 4 e S(x2 , w) and S(xl,x4)= S(x2, w) as d(xt,x4)= 2 by the same proposition. The conclusion is that all x~ are in S(xz, w). The uniqueness of S is immediate as {xi[i = 1,... ,5} is not a clique. [] 4.2. L E M M A (Shult). Suppose Seso and zeP\S with z± n S ~ ;3. Then (i) For any x e S \ ( z l & S) ± the subspace {x, z} ± consists of a single point in

S," (ii) z± c~ S is a line.

Proof. (i) By Proposition 3.8, z±c~ S is a singular subspace of S. Let x e S and y e z ± n S with xCy ±. Thus x ¢ z ± n S and S = S(x,y). By (F5), we get x±c~z±5~ £j. N o w suppose there is u e x ± n z ± \ S . Let v be any point of x±c~y ±. If ve{x,y}±\(uiwz±), then v,y,z,u,x is a 5-circuit as in 4.1, so ueS(x, y)= S by the same lemma, which contradicts the assumption on u. Thus {x, y}± = {x, y, u} ± u {x, y, z} ±. Since {x, y}± is nondegenerate and {x, y, u} ±, {x, y, z} ± are both singular subspaces of {x, y}±, we must have {x, y, z, u} ± = £5. By connectivity of {x, y}±, there are ae{x, y, z}±\u± and be {x, y, u, a}±\z± with b # a. N o w x, b, z, e a ± c~ u I so S(a, u) exists and contains x, b, z, and also y (as y e b ± n z±). Thus S(a, u)= S(x, y) = S contradicting z¢S. Consequently x ± n z ± ___S. Finally, if Ix ± n z'] > 1, then {x, z} ± would not be a clique in view of (F3). This conflicts that the subspace z -Ln S containing x ± n z ± is a clique. Hence {x, z} ± is a singleton contained in S. (ii) Let y e z ± n S . T a k e x e S \ y ±. T h e n x e S \ ( z ± n S ) ±, so (i) applies. It follows that z ± n S is a singular subspace containing the line (y, x±c~ z ± ) . O n the other hand, x ± n (z ± n S) = {x, z} ± has rank 0, so z ± c~ S has rank 1, by a well-known property of polar spaces applied to S. This establishes that z± n S e Se.

[]

4.3. L E M M A (Shult). Symplecta are polar spaces of rank 3; in particular axiom (M2) holds. Proof. Let S e s o . Since for x e S and y e S \ x ± the subspace {x, y}± is a polar space (of rank ~> 2), the polar space S has rank ~> 3. Let z e P \ S (its existence is guaranteed by P ¢ 5O). Since (P, L) is connected, we m a y assume z ± c~ S @ ~ . Then l = z ± n S e S a by 4.2. Let b e I± n S\l. Then 1 ___b ± n z ' , so S(b, z) exists. Set V = ( b , l ) . If a e V ~, then aZc~SnS(b,z)~_ V, so that aeSc~S(b,z) by 4.2 (ii). Thus V ± ~ S c~ S(b, z). Since S c~ S(b, z) is a clique, this implies that V ± = V is a maximal singular subspace in 5 °. As Ve ~ , the lemma results. []

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4.4. LEMMA. (i) Each maximal singular subspace of(P, ~ ) is a plane contained in at least one symplecton ; (ii) Axiom (M1) holds. Proof. (i) Let M be a maximal singular subspace of P. In view of (F2), M has rank ~> 2. Take x, y, z E M not all three on a line. By Remark 2.4 there is ue(xy) ± with uCz ±. Thus u, z is a symplectic pair and x, yeS(u, z). Now < x, y, z ) ± c_ S(u, z) in view of Lemma 4.2 (ii), and M ~ ( x, y, z ) ~, so M e ~U(S(u, z)) owing to Lemma 4.3. (ii) Use Lemma 4.3 and Corollary 3.10. [] 4.5. COROLLARY. Let x ~ P and I E ~ satisfy d ( x , l ) = 2 . Then x a n l A contains at most one point. Proof. Suppose y, z are distinct points of x Ac~ l±. Axiom (F3) applied to x, l in y± c~ z ± yields that y, z are coil±near. By the above lemma, yz c~ l ¢ ;g, say wElc~yz. Now xe(yz) A, so that x E w ± and d(x, l)~< 1. This settles the corollary. [] 4.6. LEMMA. Let xEP, I~5¢ with d(x, l) = 2 and suppose x± c~l ± @ £j. Then there is exactly one point y in l such that x, y is a symplectic pair. Thus, (FS) holds. Proof. We shall first establish existence. By Corollary 4.5 there is c ~ P with {c} = xAn I± Thanks to Remark 2.4 we get bEl±\c ±. Now [bAn cA[/> 2 and c ~ x ±, hence according to axiom (F5), d(x, b) ~< 2. Ifb = x, then d(x, l) = 1 contradiction. So b ¢ x. But b 6 x A by Corollary 4.5. Hence d ( x , b ) = 2. Let w e b I ~ x ± . Suppose there is no yEl such that x, y is a symplectic pair. Then x, c, y, b, w is a 5-circuit which is not contained in a symplecton for any yEI. As d(x, b) = d(b, c) = d(x, y) = 2, Lemma 4.1 implies that wEc A u y ± . But if wEc ±, then { w } , l ~ b ± c ~ c a, so there is uel, with w, c E { u , x } A and u, x is a symplectic pair, as desired. Assume w 6 c A. Then w e y ± for each yEl, so wEx± c~ 1±, conflicting with Corollary 4.5. This shows that there is at least one yEl for which x, y is a symplectic pair. Let us now suppose that y, z are distinct points of I such that both x, y and x, z are symplectic pairs. Let c be as before and choose aEx ± c~ yA, d e x A c~ z ± with a, d ¢ c. Note that a -~ d, d 6 y ±, a 6 z ± by Corollary 4.5. Now a, y, z, d, x is a 5-circuit not contained in a symplecton, for d(x, yz) = 2. Thus Lemma 4.1 yields a e d ±. Take v e x ± c~z±\d ±, v ¢ c. Then aEv ± by the same reasoning as above (with v instead of d), so that S(x, z) = S(d, v) = S(a, z) = S(d, y) by repeated application of 3.8. It follows that x, y, z are in one symplecton, hence d(x, yz) = 1, which is absurd. This concludes the proof. [] 4.7. LEMMA. The intersection of two distinct symplecta is empty, a point, or a plane.

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Proof. Let S, T be two distinct symplecta. In view of(M1) (see Lemma 4.4 (ii)) it suffices to show that if S c~ T contains a line l, then S c~ T is a plane. Choose a, b e l with a ~ b and xeFz(b)~Fl(a)c-~S, y e F z ( a ) ~ F ~ ( b ) ~ T . Note that x ¢ T, y ¢ S and x 4: y. According to (F5), d(x, y)~< 2. If d(w, y ) = 1 for a point w e x a , then x e w a ~_ S(a, y ) = T and Sc~ T contains x, contradiction. Thus d(x, y) = 2. Let z e x ± c ~ y ± and consider the 5-circuit x,a, b,y, z. Clearly z¢ab. As x, a, y are not in a symplecton. Lemma 4.1 implies that z e a ± u b ±. Without harming generality we may assume z e a ±. But then ze{a, y}± ~_ S(a, y) = T. Notice that a e b ± c~ (xz) ± and x, beS, while z, be T. It follows from Lemma 4.6 applied to b and xz that d(b, xz) = 1. Let weFl(b) c~xz. Ifw @ z, then x e w z ~_ S(b, z) = T, contradiction. The conclusion is that z e b ±, so that zeb± c~ x ± ~_ S(b, x) = S and z e S c~ T, which shows S c~ T to be a plane. [] Recall that for w e P the collection of lines (planes, symplecta, respectively) containing w is denoted by 5¢ w, (Vw, 5~w, respectively) and that L ( V ,)w (and L(Sew)~) denotes {L(M) w I M e ~w} (and {L(S)~ [ SeSe }, respectively). 4.8. LEMMA. For each weP,(L~w,L(~IF~)w) is a classical near hexagon whose quads are the members of L( Sew)w. Proof. Owing to 4.4 (i), we have L w (= ~ and each line in L is in at least one member of L(~Fw)w. We verify the hypotheses of Lemma 3.3 Denote by d w the distance function in the colinearity graph of (£aw, L(Uw)~). To begin, let M e Yfw and l e ~ w. Then there is m e ~ \ ~ w with M = ( m, w ) and y e F(w) such that I = yw. Ifd~(l, L(M)w) = O, then 1 ___M and I is the unique line in L(M)w nearest I. If dw(l, L(M)~) = 1, then any line n in L(M)~ nearest l spans a plane with I. So if nx, n 2 are two such lines, l _ n ± 1 ~ nl2 = M by Lemma 4.4, contradicting d j l , L(M)w ) = 1. Furthermore, d ( l , L ( M ) w ) > 1 implies d(y,m)= 2 and by Lemma 4.6 there is a unique x e m such that dw(l, xw) = 2. So far, we have seen that (Sew, L(~w)~) satisfies (N1) and is connected with diameter 2 or 3. We next claim that the quads of (~w, L(Uw)~) are the members of L(SPw)~. Let 11,12, ls, 14 be a minimal circuit of ( ~ w , L('ffw)w)" Choose aieli\{w } for each i (1 4 i ~< 4). The quad containing ll, 13 is

Q(ll , 13 ) = { I e 5f I dw (l, L( ( Ii + 1 ' li ) )w) ~ 1 for each i } = {zw ] z e F(w) and d(z, aiai+ 1) ~< 1 for each i} = {zw [ z e F (w) ~ S (a i , a 3 ) } = L(S(a i , a 3 ))w" Conversely, if S is a symplecton on w, there is u e F 2 (w) such that u ± c~ w± is a generalized quadrangle. Take a minimal 4-circuit al, a2, a3, a 4 in u j- c~ w ± and define li = al w. Then

L(Se)w = L(S(aa, a3))w = Q(ll, 13)

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by the same argument as above. This establishes the claim that L(5~ w)w is the collection of quads of (~w, L( ~//~w)w)" Let l, m ~ ~¢w be two lines of distance 2 with respect to dw, and let n ~ ~ x satisfy dw(n , l) = d~(n, m) = I. Taking a 1 ~ l, a z ~ n, a 3 E m, all distinct from w, we get a t Ca 3 and n ~ aic c~ a 3 so that a t , a 3 is a symplectic pair. It follows that l, m are in the quad L(S(a~, a3)w) ~. Thus all hypotheses of L e m m a 3.3 are satisfied. By L e m m a 4.7 two quads never intersect in a point of (~w, L ( ~ )~) so that this incidence system is a classical near hexagon indeed. [] 4.9. LEMMA. Let x e P . There are at least two symplecta containin9 x. Proof. By 4.4(i), x is in at least one symplecton S. Suppose S is the only symplecton on x. We shall derive the contradiction P = S. If I s 2a x, then l _c S, for any line is contained in at least one symplecton. N o w let y e F (x). Then y e S, as y x _ S. We claim that S is the only symplecton on y. If the claim does not hold, there is a symplecton T containing y distinct from S. As x ± n T :p ~ and x ¢ T, L e m m a 4.2(ii) yields that x ± c~ T is a line, say n, through y. For any w ~ n, we have w e x ±, so w e S, whence n __ S ~ T. According to L e m m a 4.7, this implies that M -- S n T for some plane M. Take v e T \ M with v e n ±. Then n ___x I n v±, so x, v is a symplectic pair and y e S ( x , v ) = S, contradicting v ~ T\(Sc~ T). This settles the claim that each neighbor y of x is contained in a unique symplecton namely S. To end the proof of this lemma, use induction on the length of a path from an arbitrary point z to x to establish that z~S. [] 4.10 C O R O L L A R Y . (P, ~ ) satisfies axiom (M3)3. Proof. In view of L e m m a 4.8, we only need to verify that the diameter of (&ax, L(~x)x) is 3 (independent of x ~ P). But this is a direct consequence of the above lemma. [] We have shown that if (P, ~ ) is a metapolar space, then it is either a polar space or the collection of points and lines of a connected metasymplectic space by 3.5, 4.3, 4.4(ii) and 4.10. 5. C O N N E C T E D

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In this section, P is a set of points in which three families of subsets ~ , U , 5e are distinguished whose members are called lines, planes and symplecta, such that P, 5a, "U, 5 ~ form a metasymplectic space whose incidence.system (p, £,e) is connected. According to Corollary 3.5, also (M3)a is satisfied. We shall verify axioms (F 1). . . . . (F5) in order to finish the proof of Theorem 2.3. First of all, (F2) is immediate: since any line is in a symplecton by (M3) and since (F2) holds for polar spaces, it holds for (P, ~ ) by (M2).

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5.1. LEMMA. (i) Any plane together with the lines it contains is a projective plane. (ii) Given M, N e ~ such that M n N is a line, then there is S e 5: with M,N~_S. (iii) The intersection of two symplecta is either empty, a point, or a plane. (iv) I f S e S : and x, y e P with x ¢ S and y e x ± c ~ S , then there is a unique line I in S on y such that (1, x ) is a plane. Proof. (i) By (M3), any plane is contained in a symplecton, so is a projective plane due to (M2). (ii) Take x e M n N and set 1 = M n N. Then I is a point of ( Y , L ( ~ )2) on the two lines L(M)~, L(N)x, so by (M3)a there is a quad L(S)x for S e 5 : containing both L(M)x and L(N) x. Now S is a symplecton as wanted. (iii) Suppose S, T are distinct symplecta and I is a line in Sm T. By (M1), it suffices to prove that S n T \ l ~ ~3. For x e l, the intersection of the quads L(S)x and L(T)~ contains the point l of 5 e and therefore another point, say m e L . So m\ {x} _~ S n T\I and we are done. (iv) Consider (~y,L(Uy)y). It is a classical near hexagon by (M3)3. By (N5) applied to the point, quad pair x y, L(S)y there is a unique line 1 on y in S such that ( x, l ) is a plane. This ends the proof of the lemma. [] 5.2. LEMMA. I f x, y, z e P satisfy z e x i n y ± and if there is no symplecton containing all three of x, y, z, then there are S, T .~ 5: with x E S, y e T such that S n T is a plane containing z. Proof. In the near hexagon ( ~ z , L(~Uz)z), the lines x z and y z are points of distance 3. Let u, v e F~(z) be such that xz, uz, vz, y z is a minimal path in ~ . Then by Lemma 5.1(ii) there is a symplecton S on x, u, z, v and a symplecton T on y, v, z, u. Note that ( u , v , z ) is a plane as uz, vz are colinear in £ 0 . Since S n T contains u, v, z, we are done. [] 5.3. LEMMA. Let a,b, c e P be distinct points forming a clique. Then ( a, b, c ) is a line or a plane. Proof. In view of axiom (M2), it suffices to show that all three points are in a symplecton. Assume there is no such symplecton. Then Lemma 5.2 yields a symplecton S containing b and a symplecton T containing c such that M = S n T is a plane containing a. Notice that b , c ¢ M . Lemma 5.1(iv) implies that there is a line l through b in S such that (l, c ) is a plane. As l, M are both in S, axiom (M2) leads to a point z in M such that ( l, z ) is a plane. Now consider a symplecton U on z, c and l, whose existence is guaranteed by Lemma 5.1(ii). It intersects T in z and c, so z e c ~ by axiom (M1). If z = a then a, b, c ~ U, contradiction. But if z @ a, then (b, z a ) and ( c, za ) are projective planes as they are contained in S and T, respectively, so that b, c, za are in a symplecton (cf. Lemma 5.1 (ii). This establishes the lemma. []

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5.4. C O R O L L A R Y . (P, Y ) satisfies (F1). 5.5. LEMMA. I f S ~ 5¢ and x, y ~ S are of mutual distance 2, then x ± n y± c_ S. Proof. Suppose z ~ x ± n y ± is not in S. Then by L e m m a 5.1(iv), there is a line l in z ± n S on y such that (l, z ) is a plane. Take T s 5 P containing this plane (such a T exists by M3). As S n T contains l, it is a plane, say M, according to L e m m a 5.1(iii). Of course, x C M as d(x, y) = 2. As a consequence of axiom M2 applied to S, the intersection x ± n M is a line, say m. N o w l, m are both in the projective plane M, so they intersect, say, in u. As ( x , m ) and ( x , u, z ) are both planes (use L e m m a 5.3) there is a quad L(U)x for U ~ ~ containing both L( ( x , m ) )x and L ( ( x, u z ))x. Thus U is a symplecton containing x, m, z. It results that U n T contains m and z. Moreover, U = T would imply x, y e T so that S = T by (M1), and z e S which is excluded. Thus U and T are distinct and intersect in a plane. Consequently, (m, z ) is a plane contained in U, and x E ( m, z )± c~ U. This yields x ~ (m, z ) _ T and T contains x, y. Again we are led to S = T and z ~ S. This ends the proof. [] 5.6. C O R O L L A R Y . I f S ~ 5¢ and x ~ P \ S , then xZ c~ S is either empty or a line. Proof. If x ± n S is nonempty, then it is a clique by the above lemma, hence a singular subspace. In view of L e m m a 5. l(iv), it cannot be a point or a plane, hence it must be a line. [] 5.7. C O R O L L A R Y . I f x, y, z e P form a clique and if on a line, then {x, y, z} z = { x, y, z ). Proof. Let S e 5 ~ contain x , y , z (use L e m m a 5.3). u ± n S contains the plane {x, y, z ) , so that u ~ S by the implies u ~ ( x , y, z ) according to (M2). We have shown The other inclusion is trivial.

they are not all three If u @ { x , y , z } z, then above corollary. This {x, y, z} ± _c {x, y, z ) . []

5.8. L E M M A . I f a ~ P and c ~ F 2 (a) are such that a Z n c ± has at least two points, then there is a unique symplecton containing both a, c. Proof. The uniqueness results from (M1), so we need only prove the existence of a symplecton on a, c. If a ± n c ± contains two distinct col±near points, this follows from Lemmas 5.1(ii) and 5.3. So we may assume that a ± n c z is a coclique. Let b, d be distinct points of a ± c~ c ±. In view of L e m m a 5.5., it suffices to prove the existence of a symplecton on some three points of a, b, c, d. Using L e m m a 5.2, we get symplecta A and C containing a and c respectively such that M = A n C is a plane on d. By the above remark, we need only consider the case where bq~AwC. Then l = b Z n A and m = b ± n C are lines on a and c respectively by Corollary 5.6. Applying (M2) to A, C, we obtain points z 1, z e s M with z l e l ± and Zz~m ±. Moreover, by (P2) for C and A, there are Yt e m n z~ and y z ~ I n z~. Notice that zl eb ± implies the existence of a quad in ~ a on ab, az I , ad so that we are done. Similarly

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for z2~b z. Thus we may restrict attention to the case where d(b, z 2 ) = d(b, zl) = 2. N o w let S 1 be the symplecton containing b , z 1, 1 and let S 2 be the symplecton containing b, z z , m . Using L e m m a 5.5, we get y l s S ~ and y2~$2 . But y ~ 6 m ~ S 2 and similarly yzES1, SO that y~, Y2~S1 ( ) S 2. Hence y~ e y e . Since y~ = Yz would lead to the existence of a symplecton on a, by 1 and c, we may assume Yl ~: Y2. Also, we may assume z z ¢ S ~, for else dez~c~a ± C _ S 1 . N o w z~c~S~ contains y l , Yz, zl so z l E y l y 2 by Corollary 5.6. But then, again, z~ Eb I and the L e m m a is proved. [] 5.9. C O R O L L A R Y . Axioms (F3) and (F5) hold for (P, 5~). Proof Suppose x ~ P and y e F 2 (x) are such that x ± n y± contains at least two points. Then they are contained in a symplecton S as we have just seen, so x ± n y ± is a generalized quadrangle with thick lines. This proves (F3). N o w let z ~ y±. If z e S, then clearly d(x, z)~< 2. Suppose therefore z~S. Since y s z±c~ S, there is a line l in z Z n S by Corollary 5.6. N o w there is w e x ± n I by axiom (P2) for S, so that w e x ± n z ±. This shows that d(x, z) <<,2, whence (F5). [] According to L e m m a 5.5, a point is contained in a symplecton if and only if it has two mutually noncolinear neighbors in the symplecton. The following lemma provides the line analogue of this criterion of containment in a symplecton. 5.10. L E M M A . Given a symplecton S and two distinct colinear points x , y for which there are distinct mutually noncolinear u, v ~ S with u ~ x ± n S \ y l and v ~ y± n S \ x ±, then x, y ~ S. Proof If x y n S-~ ~ , then L e m m a 5.5 suffices for the proof. Assume x y c_ P \ S . Set l -- x -~c~ S and m = y± c~ S. By Corollary 5.6, l, m are lines on u, v respectively. Thus l:~ m. If z e I n m, then L(S) z is a quad in L , and L(M)z where M is the plane ( x y , z ) , is a line of ( S e , L(~Uz)z) such that L(M)~ has x z and y z colinear with the noncolinear points u z and vz of the quad. As a consequence of (N6), L(M)~ is in the quad L(S)z, so x, y e S as wanted. Therefore, we m a y assume I c~ m = ~ for the rest of the proof. Let a ~ u ± c~m. N o w ua is a point in L a of distance 2 from both va and y a (use L e m m a 5.8 to derive that u, a, x, y are in a symplecton and recall that u, v are noncolinear). Therefore, there is a line n through a in (m, y ) n u ±. Observe that y ¢ n because y ~ u ±. Due to L e m m a 5.1 (ii), there is a symplecton, say T, on u, n, y. But x-tc~ T contains u and y, so x E T. Furthermore, v ~ ( m , y ) _ _ T and for any w e v ± n l we have x, v ~ w X ~ T, so that w and t h e r e f o r e / i s in T. We conclude that S = T and that x, y are in S. [] 5.11. C O R O L L A R Y . There is no minimal 5-circuit in P three of whose points are in a sympIecton.

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Proof. Let S be a symplecton and let a , , a2, a 3, a 4, a 5 be a minimal 5circuit with three points in S. If three consecutive points of the circuit are in S, the above lemma shows that there is a minimal 5-circuit in the polar space S, which is absurd. So the proof is reduced to the case where (up to a shift of indices modulo 5) a l , a2, a 4 are members of S. But then a4l ~ a 1a 2 ¢ by (M2) contradicting the rain±reality of the circuit. This settles the corollary. [] 5.12. L E M M A . (P, 5~) satisfies (F4) Proof Assume a l , a2, a3, a4, a 5 is a minimal 5-circuit. By Corollary 5.11 there is no symplecton on a 1 , a z , a 3. According to L e m m a 5.2 there are symplecta S, T such that a~ e S and a 3 e T while M = Sc~ T is a plane containing a 2. Again by Corollary 5.11, a 4 , a s q ~ S w T . Set l = a ~ c ~ S and ± nl = a 4 ~ T. Clearly, a I e I and a 3 em. Since M, l are in the polar space S, there is a point z ~ l± c~ M. As z, m are in T, we have w e z ± c~ m. Of course, z ¢ a l , a 3 as a 1, a3c~M. Notice that z ~ l (and z ¢ m ) for else there would be a symplecton on a 2 , l , a 5 (or a2,m, a4, respectively) by L e m m a 5.1(ii), contradicting Corollary 5.11. Thus z ¢ a~ u a~. Denote by U the symplecton on z, l, a 5 . N o w z e w ± c~ U \ a 2 and a s ea~4 c~ U \ z ±, so w a 4 is contained in U by L e m m a 5.10. This, however, leads to the final contradiction (Corollary 5.11) that a 4 , a 5 , a~ are in the symplecton U. [] Having verified axiom (F1) in Corollary 5.4, (F2) in the beginning of this section, (F3) and (F5) in Corollary 5.9, and (F4) in L e m m a 5.12, we conclude that the points and lines of any connected metasymplectic space form a metapolar space. As the same is obviously true for a polar space, we have established the converse of the result in Section 4, and hence finished the proof of Theorem 2.3. ACKNOWLEDGEMENT

The author is grateful to Professor Buekenhout for a very useful discussion on the topic of this paper and to Professor Shult for the extension of the main theorem incorporated in 2.3. REFERENCES i. Buekenhout, F.: Les quadriques et leurs gkn~ralisations. Lecture Notes Universit+ Libre de Bruxelles, 1978. 2. Buekenhout, F. and Shult, E. E. : 'On the Foundations of Polar Geometry.' Geom. Dedicata 3 (1974), 155-170. 3. Cameron, P. J. : 'Flat Embeddings of Near 2n-gons', in Cameron, Hirschfeld and Hughes (eds.), Finite Geometries and Designs. LMS Lecture Note 49 (Proc. Second Isle of Thorns Conf. 1980), London, 1981.

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4. Cohen, A. M. : 'Geometries Originating from Certain Distance-Regular Graphs'. in: Cameron, Hirschfeld and Hughes (eds.), Finite Geometries and Designs. LMS Lecture Note, 49, London, 1981. 5. Cohen, A. M. : 'On the Points and Lines of Metasymplectic Spaces' to appear in Proc. Convegno Internazionale Geometrie Combinatorie e Ioro applicazioni, Rome, 1981. 6. Cooperstein, B. N. : 'Some Geometries Associated with Parabolic Representations of Groups of Lie Type'. Can. J. Math. 28 (1976), 1021-1031. 7. Cooperstein, B. N. : 'A Characterization of Some Lie Incidence Structures'. Geom. Dedieata 6 (1977), 205-258. 8. Freudenthal, H. : 'Beziehungen der E v und E 8 zur Oktavenebene', I-XI, Proe. Kon. Ned. Akad. Wet. A57 (1954), 218-230, 363-368; A58 (1955), 151-157, 277-285; A62 (1959), 165-201,447-474; A66 (1963), 457-487 ( = Indag. Math., 16, 17, 21, 25). 9. Shult, E. E. and Yanushka, A. : 'Near n-gons and Line Systems', Geom. Dedicata 9 (1980), 1-72.

10. Tits, J. : Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Mathematics 386, Springer, 1974.

Author's address:

Arjeh M. Cohen, Mathematical Centre, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands (Received November 16, 1981)