Moufang sets from groups of mixed type (Q2498870)

All known Moufang sets, that is, rank 1 buildings satisfying the Moufang condition, are sharply 2-transitive permutation groups or they arise from classical groups, algebraic groups or mixed groups. In the paper under review the authors construct examples of Moufang sets from mixed Moufang quadrangles and hexagons and from exceptional Moufang quadrangles of type \(F_4\). They are obtained via semilinear involutions or Ree-type polarities. In the first case semilinear involutions that act isotropically on the point set but anisotropically on the line set of Moufang quadrangles related to mixed groups of type \(C_2\) are determined. The set of fixed points together with the centralizer of the involution gives then a Moufang set. These Moufang sets can also be found in certain Jordan division algebras. Similarly, in the second case, the authors show that there are no semilinear involutions of Moufang hexagons of mixed type that only fix points so that no new Moufang sets are obtained. The last case however is most interesting as it yields new examples. Based on their construction of some exceptional Moufang quadrangles of type \(F_4\), the authors [Can. J. Math. 51, 347--371 (1999; Zbl 0942.51002)] obtain Moufang sets that have the remarkable property that their root groups are 2-groups of nilpotency class 3, the only other known proper Moufang sets with root groups of nilpotency class \(>2\) are those constructed from Ree groups. The examples comprise all Moufang sets that can be obtained from a semilinear involution of a Moufang octagon and the authors conjecture that the converse is also true, that is, that each of the examples comes from a Moufang octagon in this way.