On irreducible \((B,N)\)-pairs of rank 2 (Q2753168)

A \(BN\)-pair in a group \(G\) is the same as a strongly transitive action of the group on some building \(\Delta\). If \(\Delta\) is spherical, irreducible, and of rank at least 3, then \(\Delta\) is known, and in the finite case, \(G\) is also known. This is due to \textit{J. Tits} [Buildings of spherical type and finite \(BN\)-pairs, Lect. Notes Math. 386, Springer, Berlin (1974; Zbl 0295.20047)]. A \(BN\)-pair of rank 1 is the same as a doubly transitive group, so not much can be said about this case in general. In a famous paper, \textit{P. Fong} and \textit{G. M. Seitz} [Invent. Math. 21, 1-57 (1973; Zbl 0295.20048) and Invent. Math. 24, 191-239 (1974; Zbl 0295.20049)] classified all finite groups acting strongly transitively on buildings of rank 2. They assumed that the chamber stabilizer splits as \(B=UT\), with \(U\subseteq B\) normal, and they proved that the associated building is a finite Moufang polygon. Their proof relies heavily on finite group theory and on the result by \textit{W. Feit} and \textit{G. Higman} [J. Algebra 1, 114-131 (1964; Zbl 0126.05303)]. Since then, it remained an open problem if a similar result holds in the infinite case (where the result by Feit-Higman is not valid). It was clear that such a proof would require very different methods.NEWLINENEWLINENEWLINEIn the present paper, the authors prove the following results. Suppose that \(G\) acts strongly transitively on a generalized \(n\)-gon, and that \(B=TU\), with \(U\subseteq B\) normal. If \(U\) is nilpotent, or if \(G\) acts effectively and if \(Z(U)\neq 1\), then \(n=3,4,6,8,12\). Moreover, \(Z(U)\) consists of central elations if \(G\) acts effectively and if \(n\neq 4,6\). For \(n=3\) (projective planes), they obtain a complete classification assuming that \(U\) is nilpotent. Assuming that \(n=3,4,6\) and that the action is effective, they prove that the building is a Moufang polygon, provided that the action is highly transitive, i.e. that \(T\) acts transitively on the doubly-punctured panels of its fixed apartment.NEWLINENEWLINENEWLINEBuilding on this paper, the authors meanwhile managed to prove the following remarkable result: if \(G\) is as above, with \(U\) nilpotent, and if \(n\neq 8\), then \(\Delta\) is a Moufang polygon [\textit{K. Tent}, \textit{H. Van Maldeghem}, Moufang polygons and irreducible spherical \(BN\)-pairs of rank 2, to appear in Adv. Math.]. Thus, they proved the Fong-Seitz result in the infinite case, except for \(n=8\). (There is strong evidence that the case \(n=8\) is not really an exception.) Their result is a significant simplification even in the finite case; the original proof by Fong-Seitz is about 100 pages long.