MINIMAL REPRESENTATIONS OF LOCALLY PROJECTIVE AMALGAMS

Abstract

A locally projective amalgam is formed by the stabilizer G ( x )o f av ertex 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 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 ). Djokovic 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 O + 2 n (2)), there are exactly twelve exceptional ones. Some of the exceptional amalgams are embedded into sporadic simple groups M22, M23, Co2, J4 and BM . For a locally projective amalgam A , the minimal degree m = m ( A )o f 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