Characterizations by Automorphism Groups of some Rank 3 Buildings – I. Some Properties of Half Strongly-Transitive Triangle Buildings

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In a sequence of papers, we will show that the existence of a (half) strongly-transitive automorphism group acting on a locally finite triangle building Δ forces Δ to be one of the examples arising from PSL3(K) for a locally finite local skewfield K Furthermore, we introduce some Moufang-like conditions in affine buildings of rank 3, and characterize those examples arising from algebraic, classical or mixed type groups over a local field. In particular, we characterize the p-adic-like affine rank 3 buildings by a certain p-adic Moufang condition, and show that such a condition has zero probability to survive in hyperbolic rank 3 buildings. This shows that a construction of hyperbolic buildings as analogues of p-adic affine buildings is very unlikely to exist.

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