Groups acting simply transitively on the vertices of a building of type Ã2 , I

If a group Γ acts simply transitively on the vertices of an affine building Δ with connected diagram, then Δ must be of type Ã_{n-1) for some n ≥ 2, and Γ must have a presentation of a simple type. The case n=2, when Δ is a tree, has been studied in detail. We consider the case n=3, motivated particularly by the case when Δ is the building of G=PGL(3,K), K a local field, and when Γ ≤ G. We exibit such a group Γ when K=F_q((X)), q any prime power. Our study leads to combinatorial objects which we call triangle presentations. These triangle presentations give rise to some new buildings of type Ã2