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Transactions of the American Mathematical Society
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
Published electronically: September 25, 2012
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05694-7
PII: S 0002-9947(2012)05694-7
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 K\"{o}hl and Stefan Witzel