Rank three geometry and positive curvature
  An axiomatic characterization of buildings of type $\CC_3$ due to Tits is used to prove that any cohomogeneity two polar action of type $\CC_3$ on a positively curved simply connected manifold is equivariantly diffeomorphic to a polar action on a rank one symmetric space. This includes two actions on the Cayley plane whose associated $\CC_3$ type geometry is not covered by a building. 
Fuquan Fang, Karsten Grove, and Gudlaugur Thorbergsson
math.DG math.GR math.GT