Higher Finiteness Properties of Reductive Arithmetic Groups in Positive Characteristic: the Rank Theorem

View/ Open
Date
2011-05-8MFO Scientific Program
Research in Pairs 2009Series
Oberwolfach Preprints;2011,05Author
Bux, Kai-Uwe
Köhl, Ralf
Witzel, Stefan
Metadata
Show full item recordOWP-2011-05
Abstract
We show that the finiteness length of an $S$-arithmetic subgroup $\Gamma$ in a noncommutative isotropic absolutely almost simple group $\mathcal{G}$ over a global function field is one less than the sum of the local ranks of $\mathcal{G}$ taken over the places in $S$. This determines the finiteness properties for arithmetic subgroups in isotropic reductive groups, confirming the conjectured finiteness properties for this class of groups. Our main tool is Behr-Harder reduction theory which we recast in terms of the metric structure of euclidean buildings.