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Finiteness properties of Chevalley groups over the ring of (Laurent) polynomials over a finite field [Elektronische Ressource] / von Stefan Witzel

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Finiteness Properties of Chevalley Groupsover the Ring of (Laurent) Polynomialsover a Finite FieldVom Fachbereich Mathematikder Technischen Universit at Darmstadtzur Erlangung des Grades einesDoktors der Naturwissenschaften(Dr. rer. nat)genehmigteDissertationvonDipl.-Math. Stefan Witzelaus Darmstadt1. Referent: PD dr. Ralf Gramlich2.t: Prof. Dr. Kai-Uwe Bux3. Referent: Prof. Dr. Michael JoswigTag der Einreichung: 9. Dezember 2010Tag der mundlic hen Prufung: 2. Februar 2011Darmstadt 2011D 17Fur meine ElternAbstractA group G is of type F if there is a K(G; 1) complex that has nite n-skeleton.nThe property F is equivalent to being nitely generated and the property F is1 2equivalent to being nitely presented. The niteness length of G is the maximaln for which G is of type F if it exists and is in nite otherwise. A rich sourcenof groups with nite niteness length consists of S-arithmetic groups in positivecharacteristic, that is, groups of the form G(O ) where G is an algebraic groupSde ned over a global function eld k andO is the ring ofS-integers for a nite setSS of places of k.In this thesis we determine the niteness length of the groups G(O ) where GSis anF -isotropic, connected, noncommutative, almost simple F -group andO isq q S1 1one ofF [t],F [t ], andF [t;t ]. That is,k =F (t) andS contains one or both ofq q q qthe places s and s corresponding to the polynomial p(t) = t respectively to the0 1point at in nity.

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Published 01 January 2011
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Language English