Extended F4-buildings and the Baby Monster

Abstract.

Let Τ 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 transpositions are adjacent if their product is a central involution in B. Then Τ is locally the commuting graph of central (root) involutions in ² E ₆(2). The graph Τ contains a family of cliques of size 120. With respect to the incidence relation defined via inclusion these cliques and the non-empty intersections of two or more of them form a geometry ℰ(B) with diagram for t=4 and the action of B on ℰ(B) is flag-transitive. We show that ℰ(B) contains subgeometries ℰ(² E 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 flag-transitive actions of ² E ₆(2):2 and Fi ₂₂:2, respectively. The geometries ℰ(B), ℰ(² E 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 elements of type 2 if and only if they are incident to a common element of type 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⋅²E₂:2, E₆(2):2 and 2²⁶ .F₄(2) and one for t=1 with flag-transitive automorphism group 3⋅Fi ₂₂:2. We also study the graph of an arbitrary (non-necessary flag-transitive) c.F ₄(t)-geometry satisfying (a) and (b) and obtain a complete list of possibilities for the isomorphism type of subgraph induced by the 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).