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 Gamma of valency 2(n) - 1 so that (i) the action is 2-arc-transitive; (ii) the subconstituent G(x)(Gamma(x)) is the linear group SLn(2) congruent to L-n (2) in its natural doubly transitive action and (iii) [t, G{x, y}] less than or equal to O-2(G(x) boolean AND G {x, y}) for some t is an element of G{x, y} \ G(x). D. Z. Djokovic 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 greater than or equal to 3. We show that besides two infinite series of locally projective amalgams (embedded into the groups AGLn(2) and O-2n(+)(2)) there are exactly twelve exceptional ones. Some of the exceptional amalgams are embedded into sporadic simple groups M-22, M-23, C-O2, J(4) and BM. For each of the exceptional amalgam n = 3, 4 or 5.