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

Abstract

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].