Abstract. If a group Γ acts simply transitively on the vertices of a thick building of type Ãn, Γ must have a presentation of a simple type. In the case n = 1, when Δ is a tree, the possible groups Γ are well understood. Recently, the case n = 2 was studied. We now consider the case n ≥ 3, and are lead to combinatorial objects Τ which we call Ãn-triangle presentations. Associated to any Ãn-triangle presentation Τ there is a group ΓΤ. We show that the Cayley graph of any group Γt is the 1-skeleton of a building ΔΤ of type Ãn. For n = 3 and n = 4, and for any prime power q, we exhibit an Ãn-triangle presentations Τ, and an embedding of ΓΤ into PGL(n + 1, Fq(X)). In these cases, the building ΔΤ is isomorphic to the building associated to SL(n + 1, Fq((X))).

Introduction.

It was shown in that if Δ is an affine building with connected diagram, and if there is a group Γ of automorphisms of Δ acting transitively on the set νΔ of vertices of Δ, then the diagram of Δ must be Ãn for some n ≥ 1. Now let Δ be a thick building of type Ãn. Let Γ be a group of automorphisms of Δ which acts simply transitively on νΔ.