Simple connectedness of the 3-local geometry of the Monster (Q1368036)

The Monster group \(M\) gives rise to a 3-local geometry \(\mathcal M\) which is determined in terms of a certain subgroup of \(M\) extending the simple orthogonal group \(\Omega_8^-(3)\). The main result of the paper is that \(\mathcal M\) is simply connected, which is equivalent to the fact that \(M\) is the universal completion of the amalgam of maximal parabolic subgroups corresponding to the flag-transitive action of \(M\) on \(\mathcal M\). In order to prove this theorem, the authors make use of the simple connectedness results for the 2-local geometry of \(M\) and for a subgeometry in \(\mathcal M\) which is the 3-local geometry of the Fischer group \(M(24)\). The result makes unnecessary the refereeing to the classification of finite simple groups in a recent \(Z_3\)-orbifold construction of the Moonshine module.