Minimal Representations of Locally Projective Amalgams

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 2 n −1 so that (i) the action is 2‐arc‐transitive, (ii) the sub‐constituent 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 }] ⩽ O 2 ( 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 AGL n (2) and O 2 n + (2)), there are exactly twelve exceptional ones. Some of the exceptional amalgams are embedded into sporadic simple groups M 22 , M 23 , Co 2 , J 4 and BM . For a locally projective amalgam A , the minimal degree m = m ( A ) of its complex representation (which is a faithful completion into GL m (C)) is calculated. The minimal representations are analysed and three open questions on exceptional locally projective amalgams are answered. It is shown that A 4 (1) possesses SL 20 (13) as a faithful completion in which the third geometric subgroup is improper; A 4 (2) possesses the alternating group Alt 64 as a completion constrained at levels 2 and 3; A 4 (5) possesses Alt 256 as a completion which is constrained at level 2 but not at level 3.