Folding Down Classical Tits Chamber Systems

Abstract

Let Ĝ be some classical p-adic group, then the existence of a discrete subgroup of G acting chamber transitively on the affine building Δ of Ĝ is a rare occurency, if the rank of the building is at least three. This is part of a theorem by Kantor, Liebler and Tits. In the cases where such a subgroup exists, one has always p = 2 or 3 and constructions of the exceptional groups can be found in [4], [5], [6], [7], [8] and [12]. Many (if not all) of the constructions — the starting point being [4] — made use of some “diagram automorphism” acting on the vertices contained in a chamber of Δ. These automorphisms of the resulting groups can, however, also be used to fold down the groups — to get subgroups acting on buildings of smaller rank “contained in Δ”; again this idea was first shown to be successful in ([4], section 6). In the following we apply this method to some rank-3-cases; thereby we get groups acting on rank-2-buildings of affine type, i.e. on trees. As a result, we obtain small-dimensional faithful representations for the amalgamated sum of the maximal parabolics of some rank-2-Chevalley groups and for the two biggest groups in Goldschmidt’s list of all primitive amalgams of index (3.3)([3]).