Amalgams determined by locally projective actions

A locally projective amalgam is formed by the stabilizer G(x) of a vertex x and the global stabilizer G{x, y} of an edge (containing x) in a group G, acting faithfully and locally finitely on a connected graph γ of valency 2ⁿ-1 so that (i) the action is 2-arc-transitive; (ii) the subconstituent G(x)^γ(x) is the linear group SL_n(2) ≅ L_n(2) in its natural doubly transitive action and (iii) {t, G{x, y}] ≤ O₂(G(x) γ G{x, y}) for some t ε G{x, y} \ G(x). D. Ž. Djoković and G. L. Miller [DM80], used the classical Tutte's theorem [Tu47], to show that there are seven locally projective amalgams for n = 2. Here we use the most difficult and interesting case of Trofimov's theorem [Tr01] to extend the classification to the case n ≥ 3. We show that besides two infinite series of locally projective amalgams (embedded into the groups AGL_n(2) and O_2n⁺(2)) there are exactly twelve exceptional ones. Some of the exceptional amalgams are embedded into sporadic simple groups M₂₂, M₂₃, Co₂, J ₄ and BM. For each of the exceptional amalgam n = 3, 4 or 5.