Simple Connectedness of the 3-Local Geometry of the Monster Original Research Article

We consider the 3-local geometry image of the Monster groupMintroduced as a locally dual polar space of the group Ω8−(3) and independently in the context of minimalp-local parabolic geometries for sporadic simple groups. More recently the geometry appeared implicitly within theZ3-orbifold construction of the Moon-shine moduleVmusic natural. In this paper we prove the simple connectedness of image. This result makes unnecessary the refereeing to the classification of finite simple groups in theZ3-orbifold construction ofVmusic naturaland realizes an important step in the classification of the flag-transitivec-extensions of the classical dual polar spaces. We make use of the simple connectedness results for the 2-local geometry ofMand for a subgeometry in image which is the 3-local geometry of the Fischer groupM(24).