Extended F4-buildings and the baby monster - Ivanov, A. A. (2011)
Let Θ 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 Θ is locally the commuting graph of central (root) involutions in 2E6(2). The graph Θ 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 ε(B) with diagram for t = 4 and the action of B on ε(B) is flag-transitive. We show that ε(B) contains subgeome¬tries ε(2E6(2)) and ε(Fi22) with diagrams c.F4(2) and c.F4(1). The stabilizers in B of these subgeometries induce on them flag-transitive actions of 2E6(2) : 2 and Fi22 : 2, respectively. The geometries ε(B), ε(2E6(2)) and ε(Fi22) 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.F4(t)-geometries satisfying (a) and (b). We construct three further examples for t = 2 with flag-transitive au¬tomorphism groups isomorphic to 3•2E2(2) : 2, E6(2) : 2 and 226.F4(2) and one for t = 1 with flag-transitive automorphism group 3 • Fi22 : 2. We also study the graph of an arbitrary (non-necessary flag-transitive) c.F4(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 ε(B) is the only c.F4(4)-geometry, satisfying (a) and (b).