Van Maldeghem, H.; Van Steen, K. Characterizations by automorphism groups of some rank 3 buildings. I: Some properties of half strongly-transitive triangle buildings. (English) Zbl 0938.51007 Geom. Dedicata 73, No. 2, 119-142 (1998). 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. Reviewer: L.K.H.Kramer (Würzburg) Cited in 3 ReviewsCited in 4 Documents MSC: 51E24 Buildings and the geometry of diagrams 51C05 Ring geometry (Hjelmslev, Barbilian, etc.) Keywords:triangle buildings; affine buildings; Hjelmslev planes; valuations; \(p\)-adic Moufang condition; automorphism groups; hyperbolic buildings Citations:Zbl 0938.51008; Zbl 0938.51009; Zbl 0938.51010; Zbl 0399.20037; Zbl 0694.51001 × Cite Format Result Cite Review PDF Full Text: DOI