The sphericity of the Phan geometries of type Bn and Cn and the Phan-type theorem of type F4

Details The sphericity of the Phan geometries of type Bn and Cn and the Phan-type theorem of type F4 The sphericity of the Phan geometries of type Bn and Cn and the Phan-type theorem of type F4

2009-01-09

arxiv.org/abs/0901.1156v2

Mathematics

Group Theory

Scientific paper

We adapt and refine methods developed by Abramenko and Devillers--K\"ohl--M\"uhlherr in order to establish the sphericity of the Phan geometries of type 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}(GF(q^2)[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 non-simply laced diagrams. We also reproduce the Phan-type theorems of types B_n and C_n.

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