LOCALLY FINITE CLASSICAL TITS CHAMBER SYSTEMS WITH TRANSITIVE GROUP OF AUTOMORPHISMS IN CHARACTERISTIC 3
Dedicated to Professor J. Tits on the occasion of his sixtieth birthday
ABSTRACT. The classification of locally finite classical Tits chamber systems cg of finite rank admitting a transitive group G of automorphisms, such that the stabilizer in G of some chamber is finite, is now complete by work of several authors. In the following, the case, that on a rank 2 residue of if, some exceptional flag-transitive subgroup of Aut(U~(3)) or Aut(PSp4(3)) is induced by G, is treated.
In recent years a great deal of effort has been spent trying to classify groups acting transitively on locally finite classical Tits chamber systems such that the stabilizer of a chamber is finite. A classification (apart from the case of spherical diagrams, i.e. the case of finite buildings using , or the A 7chamber system of type C3) will in almost all cases be only local. Namely, for any such group and chamber system, one gets a (for non-spherical diagrams possibly infinite) number of groups and chamber systems 'of the same type' by lifting the group to the universal 2-cover (which is an infinite building, if some condition on spherical rank-3 residues is fulfilled, by [-12]) and projecting down to some finite or infinite chamber system. Thus, in general, a classification will consist only of the determination of all such types, to be more precise: of the determination of all possible diagrams and the set of isomorphism types of the (finite) stabilizers of rank-2 residues in the corresponding groups (a classification of the chamber systems up to parabolic isomorphism). Sometimes, the isomorphism types of the stabilizers of rank-2 residues also determine the embeddings of the stabilizers of chambers (resp. rank-1 residues) into the stabilizers of rank-1 residues (resp. rank-2 residues). Then also the universal 2-covers are uniquely determined. This classification is now complete, due to work of a number of authors. The last open case is treated in this paper.
Connected locally finite classical Tits chamber systems with connected diagram have a natural characteristic p, which is the characteristics of all its nontrivial rank-2 residues. If this characteristic is large, i.e. at least 5, or if the rank-1 residues have at least five chambers in the characteristic 2 or 3 case,