If (P,L) is a projective plane and T is a triangle presentation compatible with a point-line correspondence λ:P --> L', then T give rise to a group Γ_T and a thick building Δ_T of type Ã2 on the vertices of which Γ_T acts simply transitively. We find all triangle presentations compatible with some point-line correspondence λ:P --> L, when (P,L) is the projective plane of order q=2 or q=3. For some, but not all, of these T, Δ_T is isomorphic to the building associated to G=PGL(3,K) where K is a local field with discrete valuation and residual field of order q. We identify the T for which this is the case, and in these cases, find embeddings of Γ_T into G. We also describe the arithmetic nature of these groups.
Groups acting simply transitively on the vertices of a building of type Ã2 , II
MANTERO, ANNA MARIA;ZAPPA, ANNA
1993-01-01
Abstract
If (P,L) is a projective plane and T is a triangle presentation compatible with a point-line correspondence λ:P --> L', then T give rise to a group Γ_T and a thick building Δ_T of type Ã2 on the vertices of which Γ_T acts simply transitively. We find all triangle presentations compatible with some point-line correspondence λ:P --> L, when (P,L) is the projective plane of order q=2 or q=3. For some, but not all, of these T, Δ_T is isomorphic to the building associated to G=PGL(3,K) where K is a local field with discrete valuation and residual field of order q. We identify the T for which this is the case, and in these cases, find embeddings of Γ_T into G. We also describe the arithmetic nature of these groups.
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