On groups of finite Morley rank with a split \(BN\)-pair of rank 1. (Q555583)

This work relates to groups of finite Morley rank and the Cherlin-Zilber ``algebraicity'' conjecture that infinite simple such groups are algebraic. The approach involves Moufang sets. The paper focuses on the case where the group is assumed to have a split BN-pair of Tits rank 1, as bigger ranks are investigated by \textit{L. Kramer}, \textit{K. Tent}, \textit{H. Van Maldeghem} [Isr. J. Math. 109, 189-224 (1999; Zbl 0933.20020)] and by \textit{K. Tent} [Lond. Math. Soc. Lect. Note Ser. 291, 173-183 (2002; Zbl 1040.20026)]. A split BN-pair of Tits rank 1 is a triple \((B,N,U)\) where \((B,N)\) is a BN-pair, \(H=B\cap N\) has index 2 in \(N\), and \(B\) is the semi-direct product of \(U\) by \(H\). The author then characterizes \(\text{(P)SL}_2(F)\) among groups of finite Morley rank admitting a split BN-pair of Tits rank 1, under some extra group-theoretic assumptions on \(U\) and its conjugates, in particular the assumption that \(U\) is Abelian and has a \(p\)-element for some prime \(p>2\). The main result may thus be seen as a complement to \textit{T. De Medts} and \textit{K. Tent} [J. Group Theory 11, No. 5, 645-655 (2008; Zbl 1161.20027)], and to \textit{J. Wiscons} [J. Group Theory 13, No. 1, 71-82 (2010; Zbl 1194.20034)], where similar \(U\) was assumed to have an element of infinite order, of order \(2\), respectively.