BN-pairs with projective or affine lines. (Q2781435)

Let \(\Gamma\) be a generalized \(n\)-gon and let \(G\) be an automorphism group of \(\Gamma\) which acts transitively on the ordered ordinary \(n\)-gons in \(\Gamma\) (this means that \(G\) contains a BN-pair of rank \(2\)). Assume that the groups induced by \(G\) on panels (point rows or line pencils) of \(\Gamma\) are permutation equivalent to \(\text{PSL}_2F\) or \(\text{AGL}_1F\) for some field \(F\) with at least \(4\) elements. The authors prove that \(\Gamma\) is a Moufang polygon and that \(G\) contains the little projective group of \(\Gamma\). (By the classification of Tits-Weiss [\textit{J. Tits} and \textit{R. M. Weiss}, Moufang polygons (2002; Zbl 1010.20017)], the possibilities for \(\Gamma\) and \(G\) are known explicitly.)NEWLINENEWLINE This result has applications in the context of the Cherlin-Zil'ber conjecture, which states that an infinite simple group \(G\) of finite Morley rank is an algebraic group over an algebraically closed field. Using results of \textit{E. Hrushovski} [Ann. Pure Appl. Logic 45, No. 2, 139-155 (1989; Zbl 0697.03023)], the authors determine explicitly all such groups \(G\) which have a definable BN-pair with panels of Morley rank \(1\), see also \textit{L. Kramer} and the authors [Isr. J. Math. 109, 189-224 (1999; Zbl 0933.20020)]. For these groups, the Cherlin-Zil'ber conjecture is confirmed.