The Curtis–Tits-presentation

AbstractLet B be an irreducible, spherical Moufang building of rank ⩾2, A an apartment of B and Φ the set of roots of A with corresponding root-subgroups Ar⩽Aut(B), r∈Φ. Let G=〈Ar|r∈Φ〉 be the group of ‘Lie-type B’. If Π is a fundamental system of Φ denote by Xr=〈Ar,A−r〉, r∈Π and Xrs=〈Xr,Xs〉 for r≠s∈Π. Then the following: Let Ĝ be the amalgamated product of the Xrs, amalgamated over the Xr and π:Ĝ→G the natural homomorphism. Then kerπ⩽Z(Ĝ), is known as the Curtis–Tits-presentation for G.In this paper the proof of a generalization of this theorem, which uses only local information, is given