The completion of the classification of the regular near octagons with thick quads

DOI 10.1007/s10801-006-9099-2

The completion of the classification of the regular near octagons with thick quads

Bart De Bruyn^∗

Received: 25 February 2005 / Accepted: 16 November 2005

CSpringer Science+Business Media, LLC 2006

Abstract Brouwer and Wilbrink [3] showed the nonexistence of regular near octagons whose parameters s, t2, t3 and t satisfy s ≥2, t2≥2 and t3=t2(t2+1). Later an arithmetical error was discovered in the proof. Because of this error, the existence problem was still open for the near octagons corresponding with certain values of s, t2and t3. In the present paper, we will also show the nonexistence of these remaining regular near octagons.

MSC2000 05B25, 05E30, 51E12 Keywords (regular) near polygon

1. Introduction

A near polygon ([9]) is a partial linear space S =(P,L,I), I⊆P×L, with the property that for every point x ∈Pand every line L ∈L, there exists a unique point on L nearest to x. Here distances are measured in the point graph or collinearity graph of S. If d is the diameter ofS, then the near polygon is called a near 2d-gon. The unique near 0-gon consists of one point (no lines). The near 2-gons are precisely the lines. Near quadrangles are usually called generalized quadrangles (GQ’s, [7]). We call a generalized quadrangle thick if every line is incident with at least three points and if every point is incident with at least three lines.

If A and B are two nonempty sets of points, then d( A,B) denotes the minimal distance between a point of A and a point of B. If A is a singleton{x}, then we will

∗Postdoctoral Fellow of the Research Foundation - Flanders B. D. Bruyn ()

Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, B-9000 Gent, Belgium

E-mail: [email protected]

also write d(x,B) instead of d({x},B). If A is a nonempty set of points and if i∈N, then we denote byi( A) the set of points y for which d(y,A)=i. If A is a singleton {x}, then we also writei(x) instead ofi({x}).

A near 2n-gon, n≥1, is said to have order (s,t) if every line is incident with precisely s+1 points and if every point is incident with precisely t+1 lines. A near 2n-gon, n ≥1, is called regular if its point graph is a so-called distance-regular graph ([2]), or equivalently, if it has an order (s,t) and if there exists constants ti, i ∈ {0, . . . ,n}, such that for any two points x and y at distance i from each other, there are precisely ti+1 lines through y containing a point at distance i−1 from x.

Obviously, t0= −1, t1=0 and tn =t.

A sub near polygonSof a near polygonSis called geodetically closed if it satisfies the following properties:

(i) the points ofSdetermine a subspace ofS;

(ii) every point of on a shortest path (inS) between two points ofSis again a point ofS.

A near polygon is called dense if every line is incident with at least three points and if every two points at distance 2 have at least two common neighbours. By [3, Theorem 4], every two points of a dense near polygon at distanceδfrom each other are contained in a unique geodetically closed sub near 2δ-gon. These sub near polygons are called quads ifδ=2 and hexes ifδ=3.

In [3, Theorem 7], Brouwer and Wilbrink showed the nonexistence of regular near octagons whose parameters s, t₂, t₃and t satisfy the following conditions: s≥2, t₂≥2 and t₃=t2(t₂+1). Later an arithmetical error was discovered in the proof (lines 5 and 6 of page 172), causing a gap in the proof. The authors of [2] ask in their book to fill this gap (see page 206) and in the present paper we will do that.

The regular near octagons which still need to be ruled out all have parameters of the following form: t2=q, s=q² and q⁴+q³+2q≤t3 ≤q⁴+q³+q²+2q+18.

The lower bound for t3arises from the divisibility condition t2|t3and the inequality t3+1>(t2+1)(st2+1) which are know to hold for any nonclassical dense regular near hexagon which is not isomorphic to the M24 near hexagon. We will rule out the remaining regular near octagons by improving this lower bound for t3+1 and subsequently dealing with the remaining cases. As a consequence, we have

Theorem 1 ([3, Theorem 7] + Section 4). There exist no regular near octagons whose parameters s, t2, t3and t satisfy s≥2, t2≥2 and t3=t2(t2+1).

2. Some properties of generalized quadrangles

As we mentioned before the generalized quadrangles are precisely the near quadran- gles. Any generalized quadrangle which is not degenerate, not a grid and not a dual grid must have a certain order (s,t2). The aim of this section is to collect some known and easy properties of generalized quadrangles.

Lemma 1 ([7, 1.2.2 and 1.2.3]). If Q is a generalized quadrangle of order (s,t2), then

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r_{s+t2|st2(s+1)(t2+1);}

r_s≤t²

2 if t2=1, or dually, t₂≤s²if s=1 (Higman’s inequality).

Lemma 2 ([7, 2.2.1]). If Q is a generalized quadrangle of order (s,t2) and if Qis a proper subquadrangle of order (s,t₂) of Q, then t₂ ≤ ^t_s². Equality holds if and only if every line of Q meets Q.

Lemma 3 ([7, 2.3.1]). Let Q be a generalized quadrangle of order (s,t2), s=1, and let X be a nonempty set of points of Q. If every line of Q which has at least two points in common with X is completely contained in X , then X is one of the following sets:

(a) a set of mutually noncollinear points;

(b) the set of points on a pencil of lines (i.e. a set of lines through a distinguished point);

(c) the set of points of a subquadrangle of order (s,t₂).

3. Restrictions on the parameters of regular near hexagons

LetS be a regular near hexagon with parameters s, t₂ and t₃ and let A denote the collinearity matrix ofS. There exist well-known techniques for calculating the eigen- values and corresponding multiplicities of A, see e.g. [2] or Section 7 of [8]. The graph A has four distinct eigenvaluesλi, i ∈ {0,1,2,3}, with−(t₃+1)=λ0< λ1 < λ2<

λ3=s(t3+1). Hereλ1andλ2are the roots of the quadratic polynomial

X²−(s−1)(t2+2)X+(s²−s+1)t2−st3+(s−1)².

The multiplicity f3of the eigenvalue s(t3+1) is equal to 1. The multiplicity f0of the eigenvalue−(t3+1) is equal to

s³(t2+1)+st3(t2+1)+s²t3(t3−t2) s²(t2+1)+st3(t2+1)+t3(t3−t2), and the multiplicity f₃₋i of the eigenvalueλ3−i, i∈ {1,2}, is equal to

λi(m−1)+s(t3+1)−(λi+t3+1) f0

λi−λ3−i ,

where

m=1+s(t3+1)+s²(t3+1)t3

t2+1 +s³t3(t3−t2) t2+1 .

The fact that all these multiplicities are integers gives already severe restrictions on the parameters. Another restriction is the so-called Mathon bound ([3, 5, 6]) which holds if s=1:

t3≤s³+t2(s²−s+1).

Another inequality which holds if s is different from 1 is the following:

t₃²−(s²t2+s²+t2)t₃+s⁴(t₂+1)≥0.

This inequality follows from one of the Krein conditions, see Section (i) of [3] or Remark 2.4 of [6].

Suppose now thatSis dense, so suppose that s≥2 and t2≥1. Then every two points at distance 2 are contained in a unique quad of order (s,t2). So, Lemma 1 provides additional parameter restrictions. The number of quads through a point, respectively line, is equal to^t_t^3(t3+1)

2(t2+1), respectively^t_t³

2. Hence, we also obtain the following divisibility conditions:

t2|t3; t2(t₂+1)|t3(t₃+1).

Now, let Q denote an arbitrary quad ofS. If x is a point of1(Q), then a unique line through x meets Q and t2(t2+1) lines through x are completely contained in1(Q).

As a consequence, t3≥t2(t2+1). Moreover, t3=t2(t2+1) if and only if2(R)= ∅ for every quad R, or equivalently, if and only ifSis a so-called classical near hexagon (i.e. a dual polar space of rank 3 ([4])). If t3=t2(t2+1), then there exists a point x∈2(Q). The set2(x)∩Q is an ovoid of Q and the set of st2+1 quads through x which meet Q determine (t2+1)(st₂+1) distinct lines through x (see Lemma 25 of [3]). So,

(t3+1)≥(t2+1)(st2+1).

If t3+1=(t2+1)(st2+1), then Theorem 5 of [3] shows that s=2, t2 =2 and t =14. By [1], there is a unique near hexagon with these parameters, namely the near hexagon which is obtained in the following way from the unique Steiner system S(5,8,24) (=the threefold extension of PG(2,4)):

rthe points of the near hexagon are the blocks of S(5,8,24);

rthe lines are the triples of mutually disjoint blocks;

rincidence is containment.

We will refer to this near hexagon as the M24 near hexagon since its automorphism group is isomorphic to the Mathieu group M24.

Theorem 2. LetS be a nonclassical regular near hexagon with parameters s≥2, t2 ≥1 and t3which is not isomorphic to the M24near hexagon. Let t₂ ∈Nsuch that no quad ofShas subquadrangles of order (s, α) with t₂< α ≤ ^t2−1_s . Then t3+1= (st2+1)(t2+1)+η·t2withη≥min

t2,^(t2+_st^{1)(st 2+1)}

2+1 −(st2+s+1) .

Proof: Since t3+1≥(t₂+1)(st₂+1) and t₂|t3, there exists anη∈N such that t3+1=(st₂+1)(t₂+1)+η·t2. Let R denote an arbitrary quad ofS. SinceSis not classical, t₃=t2(t₂+1) and there exists a point x∈2(R). SinceSis not isomorphic to the M₂₄ near hexagon, t₃+1>(t₂+1)(st₂+1) and hence there exists a line L

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through x which is completely contained in2(R). Let V denote the set of lines ofS which meet L and1(R). For every quad Q through L, let VQdenote the set of lines of V which are contained in Q. Since|V| =(s+1)(st₂+1)(t₂+1) and since there are^t_t³ quads through L, there exists a quad Q^∗through L for which|VQ^∗| ≥^(s+1)t2(st_t²₃^+1)(t2+1)2.

Put X :=R∪1(R). By Lemma 8 of [3], every line which has at least two points in common with X is completely contained in X . Hence, if Q is a quad through L, then either Q∩X is empty or satisfies the conditions of Lemma 3. We distinguish the following possibilities:

(i) Q∩X is a (possibly empty) set of mutually noncollinear points. In this case,

|VQ| = |Q∩X| ≤st2.

(ii) Q∩X is the set of points on k≥1 lines through a distinguished point. Then

|VQ| =1+sk≤1+st2.

(iii) Q∩X determines a subquadrangle of order (s, α). Since Q contains a line which is disjoint from the subquadrangle,α≤ ^t2−1_s by Lemma 2. By our assumptions, α≤t₂. Hence,|VQ| =(s+1)(sα+1)≤(s+1)(st₂+1).

Taking Q=Q^∗, we see that^(s+1)t2(st_t^2+1)(t2+1)

3 ≤st2+1 or^(s+1)t2(st_t^2+1)(t2+1)

3 ≤(s+

1)(st₂+1). The first inequality is equivalent with η≥t2 and the latter with η≥

(t2+1)(st2+1)

st₂+1 −(st₂+s+1). The theorem now immediately follows.

Corollary 1. LetSbe a nonclassical regular near hexagon with parameters s ≥2, t2 ≥1 and t₃ which is not isomorphic to the M24near hexagon. Suppose no quad of S has subquadrangles of order (s, α) withα≤ ^t2−_s¹. Then t3≥(s+1)t₂(t₂+1). In particular, this inequality holds if t2≤s.

Proof: By the proof of Theorem 2,η≥t2and hence t₃≥(s+1)t₂(t₂+1).

Remark. In Theorem 2 we can take for t₂ the biggest integer smaller than or equal to ^t2−1_s for which the divisibility condition s+t₂|s(s+1)t₂(t₂+1) is satisfied. If t₂ =^t2−1_s (which is certainly the case if^t2−1_s ∈N), then we get an improvement of the lower bound (t2+1)(st2+1) for t3.

4. The nonexistence of the regular near octagons 4.1. Description of the gap

Suppose thatS is a regular near octagon with parameters s, t₂, t₃and t, with s ≥2, t2 ≥2 and t₃=t2(t₂+1). The case s=t₂² has been ruled out in [3]. So, suppose that there exists a q≥2 such that t₂ =q and s =q². For q=2, we have s=4, t2 =2, t₃≥(t₂+1)(st₂+1)=27 and t₃≤90 by Mathon’s bound. Each value of t3 ∈ {27, . . . ,90}violates however at least one of the following conditions: (i) t₂|t3, (ii) t2(t2+1)|t3(t3+1), (iii) all multiplicities fi are integral. So, we may suppose that q≥3. Put

fq(x) :=x²−(q⁵+q⁴+q)x+(q⁹+q⁸).

For q≥3, this polynomial has two roots r₁(q) and r₂(q) with 0<r1(q)<r2(q). By Section 3, we have fq(t₃)≥0 and hence either t₃≤r1(q) or t₃≥r2(q). The case t₃≥ r2(q) has been ruled out in [3]. The case t₃≤r1(q) has not yet been ruled out because of an arithmetical error. Since fq(q⁴+q³+q²+2q+19)= −16q⁵+23q⁴+41q³+ 40q²+57q+361<0 for every q ≥3, t₃≤q⁴+q³+q²+2q+18. Because t₃>

(t2+1)(st2+1) and since t2is a divisor of t3, q⁴+q³+2q≤t3. As a consequence, q⁴+q³+2q≤t3≤q⁴+q³+q²+2q+18.

This is precisely the description of the gap as given on page 206 of [2].

4.2. Filling of the gap

By Corollary 1, we have t₃≥(s+1)t₂(t₂+1)≥q⁴+q³+q²+q. Since q|t3and t3 ≤q⁴+q³+q²+2q+18, either t3=q⁴+q³+q²+q, t3=q⁴+q³+q²+ 2q or q⁴+q³+q²+3q≤t3≤q⁴+q³+q²+2q+18. (So, q ≤18 in the latter case.) We will kill each of these cases in the following lemmas.

Lemma 4. The case t3=q⁴+q³+q²+q cannot occur.

Proof: In this case the multiplicity of the eigenvalue−(t₃+1) is equal to q¹⁵+2q¹⁴+3q¹³+3q¹²+2q¹¹+2q¹⁰+2q⁹+2q⁸+2q⁷+q⁶+q⁴+q³

q⁵+3q⁴+5q³+6q²+5q+2

=q¹⁰−q⁹+q⁸−q⁷+q⁶+q⁵−3q⁴+3q³−q²− 2q⁴−2q²

q⁵+3q⁴+5q³+6q²+5q+2. Since 0<2q⁴−2q²<q⁵+3q⁴+5q³+6q²+5q+2, this multiplicity would not

be integral, a contradiction.

Lemma 5. The case t3=q⁴+q³+q²+2q cannot occur.

Proof: In this case the multiplicity of the eigenvalue−(t3+1) is equal to q¹⁶+2q¹⁵+3q¹⁴+5q¹³+4q¹²+4q¹¹+4q¹⁰+2q⁹+3q⁸+2q⁷+q⁵+q⁴

q⁶+3q⁵+5q⁴+8q³+8q²+5q+2 , or to

q¹⁰−q⁹+q⁸−q⁷+2q⁶−2q⁵+3q⁴−8q³+15q²−19q+28

−55q⁵+73q⁴+131q³+159q²+102q+56 q⁶+3q⁵+5q⁴+8q³+8q²+5q+2 .

If q ≥54, then 0<55q⁵+73q⁴+131q³+159q²+102q+56<q⁶+3q⁵+ 5q⁴+8q³+8q²+5q+2, contradicting the fact that the multiplicity is integral. So, 3≤q ≤54. Also for the remaining possibilities of q one can verify (individually) that

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the multiplicity of the eigenvalue−(t₃+1) is not integral. So, also this case cannot

occur.

Lemma 6. The case q⁴+q³+q²+3q≤t3≤q⁴+q³+q²+2q+18 cannot oc- cur.

Proof: If q≥6, then t3≥q⁴+q³+q²+2q+6. On the other hand, since fq(q⁴+ q³+q²+2q+6)= −3q⁵+10q⁴+15q³+14q²+18q+36<0 if q ≥6, t3<

q⁴+q³+q²+2q+6. So, we have a contradiction.

If q =5, then q⁴+q³+q²+3q≤t3≤r1(q) implies that t3 =790. But for this value of t3, the divisibility condition t2(t2+1)|t3(t3+1) is not satisfied.

If q=4, then q⁴+q³+q²+3q≤t3≤r1(q) implies that 348≤t3≤351. No possible value of t3survives the conditions t2|t3and t2(t2+1)|t3(t3+1).

If q =3 then q⁴+q³+q²+3q≤t3≤r1(q) implies that 126≤t3 ≤141. From t2|t3and t₂(t₂+1)|t3(t₃+1), t₃∈ {132,135}. None of the possible values of t₃gives rise a integral multiplicity for the eigenvalue−(t₃+1).

Acknowledgements The author wants to thank both referees for their suggested improvements regarding the proof of Theorem 2.

References

1. A. E. Brouwer, “The uniqueness of the near hexagon on 759 points,” in N. L. Johnson, M. J. Kallahar, and C. T. Long, (eds.), Finite Geometries, volume 82 of Lecture Notes in Pure and Appl. Math., Marcel Dekker, New York, Basel, 1982, pp. 47–60.

2. A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs. Springer-Verlag, Berlin, 1989.

3. A. E. Brouwer and H. A. Wilbrink, “The structure of near polygons with quads,” Geom. Ded., 14 (1983), 145–176.

4. P. J. Cameron, “Dual polar spaces,” Geom. Dedicata, 12 (1982), 75–86.

5. R. Mathon, “On primitive association schemes with three classes,” preprint.

6. A. Neumaier, “Krein conditions and regular near polygons,” J. Combin. Theory Ser. A, 54 (1990), 201–

209.

7. S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, volume 110 of Research Notes in Mathe- matics. Pitman, Boston, 1984.

8. S. Shad and E. E. Shult, “The near n-gon geometries,” preprint.

9. E. E. Shult and A. Yanushka, “Near n-gons and line systems,” Geom. Dedicata, 9 (1980), 1–72.