Applications of the amalgam method to the study of locally projective graphs

Since its birth in 1980 with the seminal paper [Gol80] by Goldschmidt, the amalgam method has proved to be one of the most powerful tools in the modern study of groups, with interesting applications to graphs. Consider a connected graph Γ with a family L of complete subgraphs (called lines) with α ∈ {2,3} vertices each, and possessing a vertex- and edge-transitive group G of automorphisms preserving L. It is assumed that for every vertex x of Γ, there is a bijection between the set of lines containing x and the point-set of a projective GF(2)-space. There is a number of important examples of such locally projective graphs, studied and partly classified by Trofimov, Ivanov and Shpectorov, where both classical and sporadic simple groups appear among the automorphism groups. To a locally projective graph one can associate the corresponding locally projective amalgam A = {G(x),G{l}} comprised of the stabilisers in G of a vertex x and of a line l containing it. The renowned Goldschmidt amalgams turn out to belong to this family (α = 3), as well as their densely embedded Djokovic-Miller subamalgams (α = 2). We first determine all the embeddings of the Djokovic-Miller amalgams in the Goldschmidt amalgams, by designing and implementing an algorithm in GAP and MAGMA. This gives, as a by-product, a list of some finite completions for both the Goldschmidt and the Djokovic-Miller amalgams. Next, we consider two examples of locally projective graphs, special for being devoid of densely embedded subgraphs, and we extend their corresponding locally projective amalgams through the notion of a geometric subgraph. In both cases we find a geometric presentation of the amalgams, which we use to prove the simple connectedness of the corresponding geometry. Finally, we use the Goldschmidt’s lemma to classify, up to isomorphism, certain amalgams related to the Mathieu group M24 and the Held group He, as outlined in [Iva18], and we give an explicit construction of the cocycle whose existence and uniqueness is asserted in [Iva18, Lemma 8.5].Open Acces

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