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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

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


Authors: Ralf Köhl (né Gramlich) and Stefan Witzel
Journal: Trans. Amer. Math. Soc. 365 (2013), 1577-1602
MSC (2010): Primary 51E24; Secondary 20G30, 20E42, 51A50
DOI: https://doi.org/10.1090/S0002-9947-2012-05694-7
Published electronically: September 25, 2012
MathSciNet review: 3003275
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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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Additional Information

Ralf Köhl (né Gramlich)
Affiliation: Mathematisches Institut, Universität Gießen, Arndtstraße 2, 35392 Gießen, Germany
Email: ralf.koehl@math.uni-giessen.de

Stefan Witzel
Affiliation: Mathematisches Institut, Universität Münster, Einsteinstraße 62, 48149 Münster, Germany
Email: s.witzel@uni-muenster.de

Received by editor(s): January 5, 2009
Received by editor(s) in revised form: August 11, 2011
Published electronically: September 25, 2012
Article copyright: © Copyright 2012 Ralf Köhl and Stefan Witzel