Abstracts

A near polygon is a partial linear space with the property that for every point x and every line L there exists a unique point on L nearest to x (nearest with respect to the distance in the collinearity graph \Gamma of the partial linear space). If d is the diameter of \Gamma, then the near polygon is called a near 2d-gon. A near 0-gon consists of one point, a near 2-gon is a line with a number of points on it, and the class of the near quadrangles coincides with the class of the generalized quadrangles. Near polygons were introduced by Shult and Yanushka while studying tetrahedrally closed system of lines in Euclidean spaces. If a near polygon satisfies some mild conditions then the existence of nice substructures can be proved. Examples are quads which are nondegenerate generalized quadrangles. Brouwer et al. determined all near hexagons with three points on every line and with a quad through every two points at distance 2. This talk treats the case where all lines have four points. We give the known examples and summarize the open cases of the incomplete classification.

Dual polar spaces are near polygons, a concept introduced by Ernie Shult. An embedding of a point-line geometry is an injective map which take the points of the geometry to a subset of the point of some projective space and the lines to a collection of full projective lines. Embeddings of Lie Incidence geometries have been extensively investigated by Ernie Shult. We now know the absolutely universal embeddings of all dual polar spaces and this talk will report on that classification.

This talk is based on joint work with R. Guralnick and K. Magaard.

A permutation group G has genus g if it acts as the mondromy group of a covering of the punctured projective complex line by a surface of genus g. Let E(g) be the set of all simple groups S such that S is a composition factor of a group of genus g and S is neither cyclic nor alternating. It is now known that, as conjectured by Guralnick and Thompson in 1990, E(g) is finite for every non-negative integer g. The set E(0) is of particular interest. The majority of simple groups in E(0) are of low Lie rank, and in the majority of cases the related action is on points of the natural module. The goal is to establish this precisely and to do so in a way that sheds light on why it turns out to be true. This work should be seen in the context of furthering that goal.

Let n >= 11, let F be a skewfield and let (P,L, 'perpendicular' ) be a partial linear space endowed with a symmetric relation 'perpendicular' on the point set P such that for x a point and l a line x 'perpendicular' p \in l and x 'perpendicular' q \in l \{ p } implies x 'perpendicular' y for all y \in l. If for any line k \in L the space k ^{'perpendicular'} is isomorphic to the hyperbolic root group geometry of PE_n+1(F) with l 'perpendicular' m if and only if [ l, m ] = 1 for lines l, m inside k ^{'perpendicular'} , and the graph (L, 'perpendicular' ) is connected, then (P,L) is isomorphic to the hyperbolic root group geometry of PE_n+3(F).

Posing additional conditions, I can decrease n. The ideas used in the proof can be adjusted to characterize the hyperbolic long root group geometries of any classical group, for high enough dimension. There might even be hope for a characterization of the geometries of some exceptional groups.

Nearly thirty years ago Ernie Shult classified all finite graphs with the cotriangle property. The first step reduces the problem to the classification of certain finite partial linear spaces of order 2. In particular the class of all cotriangular spaces (reduced or not, finite or not) is seen to be the class of all partial linear spaces in which each plane is a Latin square design. The classification up to isomorphism of all cotriangular spaces follows Shult's model. Every cotriangular space has a canonical reduced image, and all such have been classified. In the non-reduced case the extension problem can be difficult. A definitive solution is only known when each planar Latin square design comes from the multiplication table of a group of order 2, a step in the classification of 3-transposition groups. We discuss the more general situation where planes are Latin square designs coming from cyclic groups of arbitrary order.

A conic blocking set (CBS) is a set of lines in PG(2,q) with the property that every conic of the plane intersects at least one of the lines. A CBS is irreducible provided it contains no smaller CBS. The study of these sets is in its infancy, and we report our preliminary findings.

In this talk, the special case where the lines of the CBS are concurrent is examined. We provide constructions for CBSs in both odd and even characteristic; and in the particular case where q is even, a construction for an irreducible conic blocking set is provided. Some of these constructions rely on Thas' results related to the theory of flocks of quadratic cones.

Finally, we report on the progress made in obtaining bounds for the sizes of minimal conic blocking sets of this type. An efficient algorithm is developed by dualizing the problem and incorporating optimization techniques. The results of these computers searches are presented for planes of order less than 200 for odd characteristic and in planes of order less than 1024 for even characteristic.

For connected and locally path-connected (pointed) topological spaces, there is a well-developed theory relating covering spaces to subgroups of the fundamental group. The latter group acts as the group of deck transformations of a universal covering space, whose existence is assured if certain conditions are met. Paths seem to play an essential role in this well-known theory that relates covers and subgroups.

We shall instead develop, using ideas that mix incidence geometry and topology, an alternate and completely path-free approach to the classification of covers of general topological spaces (without going beyond that category, as others such as Grothendieck do). It gives particularly good results for locally connected spaces. In place of the fundamental group, which can fail to give the desired results unless the space is at least locally connected, we define another group which works in general.

Let S(k) be the sphere with k handles. The graph theoretic genus of a group G is the smallest k such that a Cayley graph of G can be drawn on S(k). In 1992, the symmetric genus of 23 of the sporadic simple groups was determined. In 1996, it was determnined that every finite simple group can be generated by an element of order 2 and an element of order m. With the above as two of our starting points, we will give bounds for the graph theoretic genus of the finite simple groups. This is a prelininary report.

Extending the work of Thompson, Klinger and Mason in 1975 initiated the study of the groups of characteristic {2,p}-type. More recently Gorenstein and Lyons weakened the notion of characteristic 2-type to even type. In this more general context we prove the following theorem of Klinger-Mason type:

Theorem Let G be a finite simple K-group of even type which satisfies the following conditions:

1. e(G) = 3;
2. If p is an odd prime such that m_2,p(G) = 3 and B \cong E₂₇ is contained in a 2-local subgroup of G, then for every nontrivial subgroup A of B, F^*(C_G(A)) = O_p(C_G(A)).

In 1979 Andriamanalimanana conjectured a formula for p-rank r of these unitals when p | q+1. This formula has attracted substantial interest because it would settle one of the last open questions concerning the modular irreducible representations of finite groups of Lie type. I will show how to ``encrypt'' the Hermitian unital as a 2 by 2 matrix over a certain group ring, to eventually produce a formula for r. This formula requires the computation of the p-rank, a, of a certain q by q matrix and has allowed verification of the conjecture for q <= 2<sup>6</sup>. In case both p and q are odd, I will show how to ``decrypt'' the Hermitian unital as a configuration of planar ovals in PG(3,q). The collinearity graph of the generalized quadrangle W(q) leads to a new lower bound on a and therefore r.

In [2], m-systems of polar spaces were introduced by Shult and Thas and a description of several classes of examples is given. However, few other examples are known until now. First, we briefly discuss the construction of a new class of 1-systems of Q(6,q), q odd, starting from a generalization of a flock of a quadratic cone in PG(3,q). Secondly, we mention a uniqueness result about 1-systems of the quadric Q^-(7,q).

An i-flock of a quadratic cone LQ(2,q) with line vertex in PG(4,q), q odd, is defined as a partition of LQ(2,q) \L in q² mutually disjoint conics such that the planes of the elements of the i-flock pairwise intersect in internal points of LQ(2,q). It can be shown that to every i-flock of LQ(2,q) a locally hermitian 1-system of Q(6,q) is associated and conversely.

Applying the theory of the i-flocks to the semi-classical non-hermitian spread S_[9] of the hexagon H(q), q odd and q \equiv 1 mod 3 (see [1]), which is locally hermitian at some line L, we find an interesting geometric construction of S_[9] starting from a rational normal cubic scroll R³ having L as directrix line. Apparently the conics on R³ determine the q² conic planes of the i-flock associated with S_[9] and hence the i-flock can be reconstructed from the cubic scroll. Surprisingly this geometric construction not only yields the 1-system S_[9]; it turns out that different cubic scrolls may give rise to non-isomorphic locally hermitian 1-systems of Q(6,q). In particular, there are (q-3)/2 orbits in the set of all non-hermitian locally hermitian 1-systems of Q(6,q) constructed from a cubic scroll, under the subgroup of PGL(7,q) fixing Q(6,q).

It can be shown that a locally hermitian non-hermitian 1-system of Q(6,q), q odd, is semi-classical if and only if it arises from a rational normal cubic scroll R³ with directrix line L 'subset or equal' Q(6,q) and with the property that all points of R³\L are internal points of Q(6,q). As it is possible to determine all such cubic scrolls, this yields a complete characterization and determination of the locally hermitian semi-classical 1-systems of Q(6,q) for q odd.

Now, let M be a 1-system of Q^-(7,q). Then it can be shown that every line of Q^-(7,q) contains 0, 1, 2 or q+1 points on lines of M and that the latter holds if and only if the line itself belongs to M. For q odd, this implies that every plane of Q^-(7,q) either contains a line of M or an irreducible conic of points on lines of M. This observation enabled us to prove that Q^-(7,q) has a unique 1-system for q odd, up to a projectivity. This is an important result because 1-systems of Q^-(7,q) were candidates to yield new generalized quadrangles, as described in [3].

References

[1] ¯ I. Bloemen, J. A. Thas and H. Van Maldeghem. Translation ovoids of generalized quadrangles and hexagons.
 Geom. Dedicata, 72(1):19-62, 1998.
[2]  E. E. Shult and J. A. Thas. m-systems of polar spaces.
 J. Combin. Theory Ser. A, 68(1):184-204, 1994.
[3]  E. E. Shult and J. A. Thas. Constructions of polygons from buildings.
 Proc. London Math. Soc. (3), 71(2):397-440, 1995

Homepage of D. Luyckx

Let G be a quasisimple finite group, K an algebraically closed field and M an irreducible KG-module. We say that M is imprimitive with block stabilizer H 'subset' G if there exists some KH-module M₀ such that M = Ind_H^G(M₀). If no such H exists we call M a primitive KG-module. Seitz proved that if G is of Lie type of characteristic p and if char(k) = p, then with four exceptions every irreducible KG-module is primitive. Djorkovic and Malzan proved a similar result for characteristic zero modules of alternating and symmetric groups. I will present joint work with Gerhard Hiss and William Husen that shows that the situation is very different when G is of Lie type of characteristic p and if char(K) =/= p. In fact in our case most irreducible KG-modules are imprimitive. I will also discuss how our results fit into the program of classifying maximal subgroups of classical groups.

Recently Paul Li settled a conjecture by Andries Brouwer regarding the universal embedding dimension of DSp(2n,2), the dual polar space of type Sp(2n,2). Later, Andries Brouwer and Art Blokhuis developed another solution by reformulating the question in terms of DO+(2n,2) < DO(2n+1,2). We apply this reformulation to the problem of finding geometric spanning sets of minimal cardinality. For n < 6 we have found such sets in DO+(2n,2) giving another proof of the result originally established by Bruce Cooperstein. We hope to discuss some of the computer programs were are currently using to handle the case n = 6. This case has been crucial in solving the general case of Brouwer's conjecture on the embedding dimension.

Generalized quadrangles (GQ) of order (s,t) with |s-t| <= 2 have been studied extensively. In the 1980s and 1990s progress was made in the area of characterizing these GQ by, among others, De Soete, Kantor, Miller, Payne, Thas, van Maldeghem, and most recently in a paper by De Bruyn and Payne (to appear in Bull. ICA). During this same period results regarding extended generalized quadrangles whose residual GQ had these parameters were studied in part by Del Fra, Kasikova, Pasechnik, Pasini, Shult, Thas, and Yoshiara.

In this note we consider the affine planes associated with normal ovoids in GQ(s,s-2). Every known GQ with these parameters arises from a q-arc in a plane embedded in PG(3,q) where q = s-1. When the q-arc extends to a translation hyperoval the points of the GQ can be partitioned into a fan of normal ovoids, and the affine planes associated with these ovoids are all desarguesian. We provide sufficient conditions for having isomorphic affine planes arise from normal ovoids in fans of arbitrary GQ(s,s-2).

A Coxeter system is called rigid if the corresponding abstract Coxeter group determines the diagram up to isomorphism. There is a natural way of producing non-rigid Coxeter systems by a process which is called diagram twisting (joint work with N. Brady, J. McCammond and W. Neumann) and there is now the natural conjecture that Coxeter systems a rigid up to diagram twisting. I will describe the diagram twisting procedure and report about recent results, which can be considered as steps towards a proof of this conjecture.

Let \Delta = (P,L) be a parapolar space which is locally A_n,3, n = 6,7. There are two classes of maximal singular subspaces [`A] (which are PG(4)'s) and [`B] (which are PG(n-1)'s). It is proved that there exists a class of 2-convex subspaces D, each subspace isomorphic to D_5,5. Every symplecton of \Delta is contained in a unique element of D. Let \Gamma be a locally D-truncated geometry over K = {P, L, [`A], [`B], D }, where D is a diagram over I of exceptional type E_m, m = 7,8. A residually connected sheaf defined over all nonempty flags of \Gamma is constructed. Then \Delta is the homomorphic image of a building geometry [`(\Delta)] over I, belonging to the diagram D, and the truncation of \Delta to K is isomorphic to \Gamma.

Let G be a classical group acting on the set of singular one-dimensional subspaces of its natural module. We study the structure of the permutation module over a field of positive characteristic different from that of the group. As special cases, we determine submodule structures of mod 2 codes related to some classical generalized quadrangles, whose structure had been conjectured by Bagchi, Brouwer and others.

Since 1996, Michael Aschbacher and I have been working on a new theorem classifying quasithin groups (a result announced by Mason around 1981 but never published). Our proof was written down some time ago, but the process of producing a coherent writeup has been time-consuming. As of late February 2001, we are revising the last few sections of the proof of the Main Theorem; these will soon be available on the web page indicated below.

The talk will survey some of the configurations arising in the last few chapters of our proof: for example, a new version of the theory of large extraspecial subgroups to handle certain 2-locals involving linear groups over F₂; and certain solvable subconfigurations arising in the final sections, roughly corresponding to N-groups.

http://www.math.uic.edu/~smiths/papers/quasithin/quasithin.dvi

An outline and status report will be presented on the Gorenstein-Lyons-Solomon project for the Classification of the Finite Simple Groups. Connections with other projects (Bender-Glauberman-Peterfalvi, Aschbacher-Smith, Meierfrankenfeld-Chermak-Parker-Parmeggiani-Rowley-Stellmacher-Stroth) will be discussed.

Coxeter-Petrie Complexes naturally arise as thin diagram geometries whose rank three residues contain all of the dual forms of an algebraic map (= combinatorial Riemann surface). Corresponding to an algebraic map is its classical dual, which is obtained simply by interchanging the vertices and faces, as well as its Petrie dual, which comes about by replacing the faces by the so-called Petrie polygons. Jones and Thornton have shown that these involutory duality operations generate the symmetric group S₃, giving in all six dual forms, and whose source is as the outer automorphism group of the infinite triangle group generated by involutions s₁,s₂,s₃, subject to the additional relation s₁s₃ = s₃s₁. This outer automorphism group acts transitively on the involutions s₁,s₃,s₁s₃ and fixes s₂. If we allow these four involutions to define the nodes of a Coxeter diagram of type D₄ (but with edges marked with \infty), then there is a natural extension from the original algebraic map to a thin Coxeter complex of rank 4. These are fully explicated in case the original algebraic map is a Platonic map.

Let P be a finite polar space of rank r >= 2. An ovoid O of P is a pointset of P, which has exactly one point in common with each generator of P, that is, with each maximal totally singular subspace of P. A spread S of P is a set of generators, which constitutes a partition of the pointset. It appears that |O| = |S| for any ovoid O and any spread S of any given polar space P; this common number will be denoted by \mu_P and will be called the ovoid number of the polar space P. Ovoids and spreads have many connections with and applications to projective planes, circle geometries, generalized polygons, strongly regular graphs, partial geometries, semipartial geometries, codes, designs. Existence and nonexistence results on ovoids and spreads, and also constructions, are due to Bader, Brouwer, Brown, Conway, Dye, Kantor, Kleidman, Lunardon, Lüneburg, Moorhouse, O'Keefe, Payne, Penttila, Royle, Shult, Thas, Tits, Wilson, ... . Whether or not a particular polar space contains an ovoid or spread can be a very hard problem, and many cases are still open.

In ``m-Systems of polar spaces'', J. Combin. Theory Ser. A, 68:184-204, 1994, E. E. Shult and J. A. Thas introduced partial m-systems and m-systems of polar spaces. A partial m-system of the finite polar space P of rank r, with r >= 2 and 0 <= m <= r-1, is any set { \pi₁, \pi₂, ... , \pi_k } of k ( =/= 0) totally singular m-spaces of P such that no generator containing \pi_i has a point in common with (\pi₁ \cup \pi₂ \cup ... \cup \pi_k)-\pi_i, with i = 1, 2, ... , k. A partial 0-system of size k is also called a partial ovoid, or a cap or a k-cap; a partial (r-1)-system is also called a partial spread. For any partial m-system M of P we have |M| <= \mu_P. If |M| = \mu_P, then the partial m-system M of P is called an m-system of P. For m = 0, the m-system is an ovoid of P; for m = r-1, with r the rank of P, the m-system is a spread of P. The fact that |M|, with M any m-system of the polar space P, is independent of m gives us an explanation why an ovoid and a spread of a polar space P have equal size. Partial m-systems and m-systems have many connections with and applications to codes, graphs and several interesting classes of incidence geometries. Existence and nonexistence theorems, and constructions, for 0 < m < r-1, are due to Hamilton, Mathon, Quinn, Shult and Thas.

We say that a partial m-system M of the polar space P has the BLT property if and only if there is no line of P meeting three distinct members of M non-trivially. In the paper ``Constructions of polygons from buildings'', Proc. London Math. Soc., 71:397-440, 1995, by E. E. Shult and J. A. Thas, these particular partial m-systems play a crucial role. Whether or not a particular (partial) m-system possessing the BLT property exists is still open in many cases.

Here a survey of the known existence and nonexistence theorems will be given, also in the case of the nonclassical finite polar spaces of rank 2, that is, the nonclassical finite generalized quadrangles; also many interesting constructions will be given.

Suppose S is a finite generalized quadrangle (GQ) of order (s,t), s,t > 1, and suppose that L is a line of S. A symmetry about L is an automorphism of the GQ which fixes every line of S meeting L (including L). A line is called an axis of symmetry if there is a full group of symmetries of size s about this line.
Suppose L and M are non-concurrent axes of symmetry of the GQ S; then S is called a span-symmetric generalized quadrangle (SPGQ) with base-span sp(L,M). It was a longstanding conjecture that every SPGQ of order s > 1 is classical, i.e. isomorphic to the GQ Q(4,s) which arises from a nonsingular parabolic quadric in PG(4,s), a result we proved recently using the classification of the finite split BN-pairs of rank 1 (by E. E. Shult and C. Hering, W. M. Kantor and G. M. Seitz) and universal central extensions of groups.
The general problem of classifying SPGQ's of order (s,t), s,t > 1 and s =/= t, seems hopeless at present, although it is worthwhile mentioning that W. M. Kantor was able to show that in such a case we necessarily have that t = s².
As a first step in the aforementioned classification, we focus on a special class of SPGQ's of order (s,s²), s > 1. Let us first recall that a point of a generalized quadrangle is a translation point if every line incident with it is an axis of symmetry, and a GQ with a translation point is often called a translation generalized quadrangle.
In the present talk, we will discuss the classification of generalized quadrangles which have at least two distinct translation points.