The sphericity of the Phan geometries of type 𝐵_{𝑛} and 𝐶_{𝑛} and the Phan-type theorem of type 𝐹₄

Köhl (né Gramlich), Ralf; Witzel, Stefan

doi:10.1090/S0002-9947-2012-05694-7

The sphericity of the Phan geometries of type $B_n$ and $C_n$ and the Phan-type theorem of type $F_4$
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by Ralf Köhl (né Gramlich) and Stefan Witzel PDF

Trans. Amer. Math. Soc. 365 (2013), 1577-1602

Abstract:

We adapt and refine the methods developed by Abramenko and Devillers–Köhl–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 $\mathrm {Sp}_{2n}(\mathbb {F}_{q^2}[t,t^{-1}])$.

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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