Generalized polygons with split Levi factors (Q5929486)

Irreducible spherical BN-pairs (or Tits systems) of rank \(r\geq 2\) give rise to spherical buildings of rank \(r\) and if \(r\geq 3\), then such buildings were classified by Jacques Tits (but note that the corresponding spherical BN-pairs of rank \(r\geq 3\) are not classified and can be rather ``wild''). On the one hand, spherical buildings of rank \(r\geq 3\) automatically satisfy the so-called Moufang condition, that is a condition on the existence of certain groups of automorphisms (the ``root groups''). On the other hand there exist many spherical buildings of rank 2 not satisfying the Moufang condition and yet arising from an irreducible spherical BN-pair of rank 2. The paper under review finds a necessary condition on the Levi factors of a group with a BN-pair in order to conclude that the BN-pair corresponds to a Moufang building of rank 2 (the so-called generalized polygons). In geometric terms the main result of the author is as follows. Let \(\Gamma\) be a generalized polygon, let \(G\) be an automorphism group acting strongly transitively (i.e. transitive on the set of pairs \((C,\Sigma)\), where \(C\) is a chamber in an apartment \(\Sigma\)). Fix a chamber \(D=\{x_0,x_1\}\) and let \(L_i\), \(i=0,1\), be the group induced on the set of chambers through \(x_i\) by the stabilizer \(P_i\) (a parabolic subgroup) of \(x_i\) in \(G\). Let \(B_i\) be the stabilizer of \(D\) in \(L_i\), and let \(T_i\) be the stabilizer of one further chamber \(D_i\neq D\) through \(x_i\) in \(B_i\). If \(B_i\) is the semi-direct product of its commutator subgroup \([B,B]\) with \(T_i\), for all \(i\in\{0,1\}\), then \(\Gamma\) is a Moufang polygon and \(G\) contains its little projective group. The author applies her Main Result to the (general linear and Suzuki) groups \(\text{PGL}_2\) and GSz. She concludes with two variations on her Main Result. The first one replaces the condition ``for all \(i\in\{0,1\}\)'' by ``for some \(i\in\{0,1\}\)'' and the conclusion remains true if \(\Gamma\) is a generalized quadrangle and if both \(T_0\) and \(T_1\) are Abelian; the second says something about generalized \(n\)-gons with a group \(G\) of automorphisms acting transitively on the set of ordered ordinary \((n+1)\)-gons such that the \(T_i\) above are Abelian.