Tits, Jacques Spheres of radius 2 in triangle buildings. I. (English) Zbl 0744.51013 Finite geometries, buildings, and related topics, Pap. Conf., Pingree Park/CO (USA) 1988, 17-28 (1990). [For the entire collection see Zbl 0741.00065.]A sphere of radius two in a building of type \(\tilde A_ 2\) is the graph obtained by looking at all vertices at distance \(\leq 2\) of a certain fixed vertex. If the order \(q\) of the building (\(q+1\) is the number of chambers through a panel) is 2, then the author shows that there are only two isomorphism classes and both occur. This is used to show that the four buildings obtained by amalgamation of three Frobenius groups of order 21 are pairwise non-isomorphic and two of them are not classical. The paper ends with an estimate (lower bound) of the number of isomorphism classes of spheres of radius 2 that can appear in triangle buildings of order \(q\). This number increases very rapidly with \(q\).(Note that by work of the reviewer [Geom. Dedicata 24, 123-206 (1987; Zbl 0648.51016)], these spheres can be identified with Hjelmslev-planes of level 2, more generally, spheres of radius \(n\) are Hjelmslev-planes of level \(n\), from which readily follows that for \(q=2\) and \(n=2\), there are only two possibilities.). Reviewer: H.Van Maldeghem (Gent) Cited in 3 ReviewsCited in 8 Documents MSC: 51E24 Buildings and the geometry of diagrams Keywords:triangle buildings; sphere of radius two Citations:Zbl 0741.00065; Zbl 0648.51016 × Cite Format Result Cite Review PDF