Minimal representations of locally projective amalgams

Abstract

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 SLn(2) ≅ Ln(2) in its natural doubly transitive action, and (iii) [t,G{x, y}] ≤ O2(G(x)∩G{x, y}) for some t ∈ G{x, y} \ G(x). Djoković and Miller used the classical Tutte theorem to show that there are seven locally projective amalgams for n = 2. Trofimov's theorem was used by the first author and Shpectorov to extend the classification to the case n ≥ 3. It turned out that for n ≥ 3, besides two infinite series of locally projective amalgams (embedded into the groups AGLn(2) and O2n+(2)), there are exactly twelve exceptional ones. Some of the exceptional amalgams are embedded into sporadic simple groups M 22, M23, Co2, J4 and BM. For a locally projective amalgam A, the minimal degree m = m(A) of its complex representation (which is a faithful completion into GLm(ℂ)) is calculated. The minimal representations are analysed and three open questions on exceptional locally projective amalgams are answered. It is shown that (a) A4(1) possesses SL20(13) as a faithful completion in which the third geometric subgroup is improper; (b) A 4(2) possesses the alternating group Alt64 as a completion constrained at levels 2 and 3; (c) A4(5) possesses Alt256 as a completion which is constrained at level 2 but not at level 3.