Groups acting simply transitively on the vertices of a building of type \uc32 , I

If a group \u393 acts simply transitively on the vertices of an affine building \u394 with connected diagram, then \u394 must be of type \uc3_{n-1) for some n 65 2, and \u393 must have a presentation of a simple type. The case n=2, when \u394 is a tree, has been studied in detail. We consider the case n=3, motivated particularly by the case when \u394 is the building of G=PGL(3,K), K a local field, and when \u393 64 G. We exibit such a group \u393 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 \uc3