Extended F-4-buildings and the Baby Monster

Let Theta be the Baby Monster graph which is the graph on the set of {3, 4} -transpositions in the Baby Monster group B in which two such transposition s are adjacent if their product is a central involution in B. Then Theta is locally the commuting graph of central (root) involutions in E-2(6)(2). Th e graph Theta contains a family of cliques of size 120, With respect to the incidence relation defined via inclusion these cliques and the nonempty in tersections of two or more of them form a geometry E(B) with diagram [GRAPHICS] for t = 4 and the action of B on E(B) is flag-transitive. We show that E(B) contains subgeometries E(E-2(6) (2)) and E(Fi(22)) With diagrams c. F-4(2) and c.F-4(1). The stabilizers in B of these subgeometries induce on them f lag-transitive actions of E-3(6)(2) : 2 and Fi22 : 2 respectively, The geom etries E(B), E(E-2(6)(2)) and E(Fi(22)) possess the following properties: ( a) any two elements of type 1 are incident to at most one common element of type 2 and (b) three elements of type 1 are pairwise incident to common el ements of type 2 if and only if they are incident to a common element of ty pe 5. The paper addresses the classification problem of c.F-4(t)-geometries satisfying (a) and (b). We construct three further examples for t = 2 with flag-transitive automorphism groups isomorphic to 3 E-2(2)(2) : 2, E-6(2) : 2 and 2(26). F-4(2) and one for t = 1 with flag-transitive automorphism g roup 3 Fi(22) : 2 We also study the graph of an arbitrary (non-necessary fl ag-transitive) c.F-4(t)-geometry satisfying (a) and (b) and obtain a comple te list of possibilities for the isomorphism type of subgraph induced by th e common neighbours of a pair of vertices at distance 2, Finally, we prove that E(B) is the only c.F-4(4)-geometry, satisfying (a) and (b).