A characterization of the Mathieu–Conway–Monster series of locally projective graphs

Abstract

Let I=(V,L) be a vertex-line incidence system which is said to be thin or thick if every line contains two or three vertices, respectively. In either case any two lines can intersect in at most one vertex. Let Γ be the collinearity graph of I and let G be a group of automorphisms of I. Then Γ is said to be locally projective in dimension n with respect to G if (a) G is flag-transitive on I; (b) the stabilizer G(x) of a vertex x∈V in G induces on the set of lines containing x, the group Ln(2) for some n≥2 in its natural doubly transitive action of degree 2n−1. Thus a locally projective graph in dimension n has valency 2n−1 or 2(2n−1) depending on whether it is thin or thick. The G(x)-invariant structure of a projective GF(2)-space of linear dimension n on the set of lines containing x, will be denoted by Πx. In the thin case we further assume that (c) the action of G on Γ is 2- but not 3-arc transitive and (d) the action is of collineation type, meaning that for an edge {x,y} of Γ an element in G which swaps x and y can be chosen to centralize G(x)∩G(y) modulo O2(G(x)∩G(y)). The first examples of locally projective graphs come from affine (thin) and projective (thick) GF(2)-spaces. Less trivial classical examples are the dual polar GF(2)-spaces of orthogonal (thin) and symplectic (thick) types. Further examples coming from Petersen and Tilde geometries are associated with some sporadic simple groups. The fundamental problem is to establish an upper bound on |G(x)| in terms of the dimension n, by finding an absolute bound d such that the vertex-wise stabilizer in G of the ball of radius d in Γ centred at a vertex x is always trivial (independently of the graph): Gd(x)=1. In the classical examples G2(x)=1. The first examples of Petersen and Tilde geometries were constructed in the early 1980s to disprove the belief that the classical bound holds in general. In fact, for the vertex-transitive graph of the Petersen geometry of the Baby Monster group the smallest d with Gd(x)=1 is 5. In 1990s V.I. Trofimov in a brilliant series of papers proved that G6(x)=1 in the thin case without assuming (c) and (d). In 2002 it has been proved by S.V. Shpectorov and the author that in the thin case under consideration all the examples with the critical parameter d above the classical bound indeed come from Petersen geometries (one way or another). An ongoing project is to extend the classification of locally projective amalgams to the thick case, which is considerably more complicated. Until recently only a characterization of an important class of 3-dimensional locally projective graphs [3] was available. In a recent paper [6] the author has shown that if the standard geometrization of a thick locally projective graph contains planes locally isomorphic to the generalized quadrangle of order (2,2), then Γ contains a family of thin locally projective subgraphs which are densely embedded in Γ in the sense that the stabilizer in G(x) of such a subgraph containing x has index 2n. This result places in a general setting the famous embeddingsG(M22)↪G(M24),G(Co2)↪G(Co1),G(BM)↪G(M) among Petersen and Tilde geometries. In the present paper we characterize the graphs of the geometries on the right hand side of these embeddings in the class of thick locally projective graphs with planes of the above specified type and with critical parameter d above the classical bound.