Moufang polygons and irreducible spherical BN-pairs of rank 2. I
Let \(G\) be a group with an irreducible spherical BN-pair \((B,N)\) such that \(B=(B\cap N)U\) for a normal nilpotent subgroup \(U\) of \(B\). The authors show that \(G\) corresponds to a Moufang polygon \(\Gamma\) -- hence the action of \(G\) on \(\Gamma\) is known explicitly by the classification of \textit{J. Tits} and \textit{R. M. Weiss} [Moufang polygons, Springer (2002; Zbl 1010.20017)] -- under the assumption that \(\Gamma\) is not an octagon; this assumption on \(\Gamma\) is removed by \textit{K. Tent} [Adv. Math.181, No. 2, 308-320 (2004; Zbl 1044.20013)]. In view of the results of \textit{K. Tent} and \textit{H. Van Maldeghem} [Forum Math. 13, No. 6, 853-862 (2001; Zbl 0989.20026)], this means that one has now a classification of split BN-pairs of rank 2 which extends the well-known result of \textit{P. Fong} and \textit{G. M. Seitz} [Invent. Math. 21, 1-57 (1973; Zbl 0295.20048) and ibid. 24, 191-239 (1974; Zbl 0295.20049)] on BN-pairs in finite groups. The proof is achieved by elementary geometric arguments, in the style of \textit{R. Weiss} [Invent. Math. 51, 261-266 (1979; Zbl 0409.05033)].
scientific article; zbMATH DE number 1903517