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The Baby Monster group B acts naturally on a geometry E(B) with diagramc.F_4(t) for t=4 and the action of B on E(B) is flag-transitive. It possessesthe following properties:  (a) any two elements of type 1 are incident to at most one common element oftype 2, and  (b) three elements of type 1 are pairwise incident to common elements of type2 iff they are incident to a common element of type 5.  It is shown that E(B) is the only (non-necessary flag-transitive)c.F_4(t)-geometry, satisfying t=4, (a) and (b), thus obtaining the firstcharacterization of B in terms of an incidence geometry, similar in vein to oneknown for classical groups acting on buildings. Further, it is shown that E(B)contains subgeometries E(^2E_6(2)) and E(Fi22) with diagrams c.F_4(2) andc.F_4(1). The stabilizers of these subgeometries induce on them flag-transitiveactions of ^2E_6(2):2 and Fi22:2, respectively. Three further examples for t=2with flag-transitive automorphism groups are constructed. A complete list ofpossibilities for the isomorphism type of the subgraph induced by the commonneighbours of a pair of vertices at distance 2 in an arbitrary c.F_4(t)satisfying (a) and (b) is obtained.Comment: to appear in Inventiones Mathematica

Extended F4-buildings and the Baby Monster