Ree geometries

We introduce a rank 3 geometry for any Ree group over a not necessarily perfect field and show that its full collineation group is the automorphism group of the corresponding Ree group. A similar result holds for two rank 2 geometries obtained as a truncation of this rank 3 geometry. As an application, we show that a polarity in any Moufang generalized hexagon is unambiguously determined by its set of absolute points, or equivalently, its set of absolute lines

On collineations and dualities of finite generalized polygons

In this paper we generalize a result of Benson to all finite generalized polygons. In particular, given a collineation theta of a finite generalized polygon S, we obtain a relation between the parameters of S and, for various natural numbers i, the number of points x which are mapped to a point at distance i from x by theta. As a special case we consider generalized 2n-gons of order (1,t) and determine, in the generic case, the exact number of absolute points of a given duality of the underlying generalized n-gon of order t

On the intersection of distance-jj-ovoids and subpolygons in generalized polygons

De Wispelaere and Van Maldeghem gave a technique for calculating the intersection sizes of combinatorial substructures associated with regular partitions of distance-regular graphs. This technique was based on the orthogonality of the eigenvectors which correspond to distinct eigenvalues of the (symmetric) adjacency matrix. In the present paper, we give a more general method for calculating intersection sizes of combinatorial structures. The proof of this method is based on the solution of a linear system of equations which is obtained by means of double countings. We also give a new class of regular partitions of generalized hexagons and determine under which conditions ovoids and subhexagons of order $(s',t')$ of a generalized hexagon of order $intersect in a constant number of points. If the automorphism group of the generalized hexagon is sufficiently large, then this is the case if and only if =s't'$. We derive a similar result for the intersection of distance-2-ovoids and suboctagons of generalized octagons

An infinite family of mm-ovoids of the hyperbolic quadrics Q+(7,q)\mathcal{Q}^+(7,q)

An infinite family of $(q^2+q+1)$-ovoids of $\mathcal{Q}^+(7,q)$, $q\equiv 1\pmod{3}$, admitting the group $\mathrm{PGL}(3,q)$, is constructed. The main tool is the general theory of generalized hexagons.Comment: 9 page

The hyperplanes of the U (4)(3) near hexagon

Combining theoretical arguments with calculations in the computer algebra package GAP, we are able to show that there are 27 isomorphism classes of hyperplanes in the near hexagon for the group U (4)(3). We give an explicit construction of a representative of each class and we list several combinatorial properties of such a representative

Ovoids and spreads of finite classical generalized hexagons and applications

One intuitively describes a generalized hexagon as a point-line geometry full of ordinary hexagons, but containing no ordinary n-gons for n<6. A generalized hexagon has order (s,t) if every point is on t+1 lines and every line contains s+1 points. The main result of my PhD Thesis is the construction of three new examples of distance-2 ovoids (a set of non-collinear points that is uniquely intersected by any chosen line) in H(3) and H(4), where H(q) belongs to a special class of order (q,q) generalized hexagons. One of these examples has lead to the construction of a new infinite class of two-character sets. These in turn give rise to new strongly regular graphs and new two-weight codes, which is why I dedicate a whole chapter on codes arising from small generalized hexagons. By considering the (0,1)-vector space of characteristic functions within H(q), one obtains a one-to-one correspondence between such a code and some substructure of the hexagon. A regular substructure can be viewed as the eigenvector of a certain (0,1)-matrix and the fact that eigenvectors of distinct eigenvalues have to be orthogonal often yields exact values for the intersection number of the according substructures. In my thesis I reveal some unexpected results to this particular technique. Furthermore I classify all distance-2 and -3 ovoids (a maximal set of points mutually at maximal distance) within H(3). As such we obtain a geometrical interpretation of all maximal subgroups of G2(3), a geometric construction of a GAB, the first sporadic examples of ovoid-spread pairings and a transitive 1-system of Q(6,3). Research on derivations of this 1-system was followed by an investigation of common point reguli of different hexagons on the same Q(6,q), with nice applications as a result. Of these, the most important is the alternative construction of the Hölz design and a subdesign. Furthermore we theoretically prove that the Hölz design on 28 points only contains Hermitian and Ree unitals (previously shown by Tonchev by computer). As these Hölz designs are one-point extensions of generalized quadrangles, we dedicate a final chapter to the characterization of the affine extension of H(2) using a combinatorial property