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Characterizations by automorphism groups of some rank 3 buildings. I: Some properties of half strongly-transitive triangle buildings. (English) Zbl 0938.51007

In this series of four papers, the authors classify certain affine hyperbolic buildings of rank three in terms of their automorphism groups.
In Part I and II, they introduce the notion of a half strongly-transitive automorphism group: such a group acts transitively on all pairs of chambers \(C,D\) with distance \(\delta(C,D)=w\), for all \(w\in W\). For spherical buildings, this is equivalent with the notion of a strongly transitive group, since opposite chambers determine a unique apartment. Constructions of J. Tits [Invent. Math. 43, 283-295 (1977; Zbl 0399.20037)] and K. Tent [to appear in J. London Math. Soc.] show that it is impossible to classify buildings with strongly transitive automorphism groups.
In Part I & II, the authors prove the following remarkable result: A locally finite triangle building with a half strongly-transitive automorphism group is Moufang and arises from a locally finite skew field with a discrete valuation. The idea of the proof is to construct a sequence of Hjelmslev planes and to prove that these planes are Moufang.
A Corollary of this result (which is not mentioned in the paper) is that an infinite totally disconnected compact projective plane which admits a 4-gon transitive automorphism group is Moufang, provided that it admits a projective valuation (private communication by the second author).
In Part III, Van Steen discusses various Moufang-like conditions for affine buildings. The definition of an affine Moufang building given in M. Ronan’s book [‘Lectures on buildings’, Academic Press (1989; Zbl 0694.51001)] has the disadvantage thas it is not satisfied by affine buildings over the \(p\)-adics. The author proposes a ‘\(p\)-adic Moufang condition’ and a ‘root-Moufang condition’. She shows that a triangle building is a Bruhat-Tits building if and only if it satisfies the root-Moufang condition (the Bruhat-Tits buildings are the affine buildings associated to discrete valuations). If an affine building of rank three satisfies the \(p\)-adic Moufang condition, then it is a Bruhat-Tits building, and the Bruhat-Tits triangle buildings over the \(p\)-adics satisfy the \(p\)-adic Moufang condition.
In Part IV the authors consider the \(p\)-adic Moufang condition for hyperbolic buildings. The main result is that such a building is of type \(\Delta_{3-4-6}\) (i.e. its residues are triangles, quadrangles and hexagons), provided that the characteristic is different from 2 and 3. This result confirms the (open) conjecture that no \(p\)-adic hyperbolic Moufang buildings exist.

MSC:

51E24 Buildings and the geometry of diagrams
51C05 Ring geometry (Hjelmslev, Barbilian, etc.)
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