Construction of RGD-systems of type (4, 4, 4) over F_2

Abstract

We investigate the structure of RGD-systems over F_2. For this purpose we introduce the notion of commutator blueprints which prescribe the commutator relations between prenilpotent pairs of positive roots. To each RGD-system one can associate a commutator blueprint and such a commutator blueprint will be called integrable. We give necessary and sufficient conditions of an integrable commutator blueprint. Moreover, we construct uncountably many different integrable commutator blueprints of type (4, 4, 4). The existence of these integrable commutator blueprints disproves the general validity of the extension theorem for isometries of 2-spherical thick twin buildings. Additionally, we obtain the first example of a 2-spherical Kac-Moody group over a finite field which is not finitely presented. Furthermore, we construct the first example of a 2-spherical RGD-system with finite root groups which does not have property (FPRS).