Groups acting simply transitively on the vertices of a building of type� 2, II: The casesq=2 andq=3

Abstract

If (P, L) is a projective plane and ℐ is a ‘triangle presentation compatible with a point-line correspondence λ:P →L’, then ℐ gives rise to a group Γℐ and a thick building Δℐ of typeA 2 on the vertices of which Γℐ acts simply transitively. We find all triangle presentations (up to natural equivalence) compatible with some point-line correspondence λ:P →L, when (P, L) is the projective plane of orderq=2 orq=3. For some, but not all, of these ℐ, Δℐ is isomorphic to the building associated withG=PGL(3,K) whereK is a local field with discrete valuation and residual field of orderq. We identify the ℐ for which this is the case, and in these cases, find embeddings of Γℐ intoG. We also describe the arithmetic nature of these groups.