The sphericity of the phan geometries of type B_n and C_n and the phan-type theorem of type F_4

摘要

We adapt and refine the methods developed by Abramenko and Devillers-K?hl-Mühlherr in order to establish the sphericity of the Phan geometries of types B_n and C_n and their generalizations. As an application we determine the finiteness length of the unitary form of certain hyperbolic Kac-Moody groups. We also reproduce the finiteness length of the unitary form of the groups Sp_(2n)(F_(q2) [t, t~(-1)]). Another application is the first published proof of the Phan-type theorem of type F_4. Within the revision of the classification of the finite simple groups this concludes the revision of Phan's theorems and their extension to the nonsimply laced diagrams. We also reproduce the Phan-type theorems of types B_n and C_n.