A CHARACTERISATION OF WEAKLY LOCALLY PROJECTIVE AMALGAMS RELATED TO A16 AND THE SPORADIC SIMPLE GROUPS M24 AND He

A simple undirected graph is weakly G-locally projective, for a group of automorphisms G, if for each vertex x, the stabiliser G(x) induces on the set of vertices adjacent to x a doubly transitive action with socle the projective group Lnx(qx) for an integer nx and a prime power qx. It is G-locally projective if in addition G is vertex transitive. A theorem of Trofimov reduces the classification of the G-locally projective graphs to the case where the distance factors are as in one of the known examples. Although an analogue of Trofimov’s result is not yet available for weakly locally projective graphs, we would like to begin a program of characterising some of the remarkable examples. We show that if a graph is weakly locally projective with each qx = 2 and nx = 2 or 3, and if the distance factors are as in the examples arising from the rank 3 tilde geometries of the groups M24 and He, then up to isomorphism there are exactly two possible amalgams. Moreover, we consider an infinite family of amalgams of type Un (where each qx = 2 and n = nx + 1 ≥ 4) and prove that if n ≥ 5 there is a unique amalgam of type Un and it is unfaithful, whereas if n = 4 then there are exactly four amalgams of type U4, precisely two of which are faithful, namely the ones related to M24 and He, and one other which has faithful completion A16.