Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem

Details Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem

2011-02-02

arxiv.org/abs/1102.0428v1

Mathematics

Group Theory

Scientific paper

We show that the finiteness length of an $S$-arithmetic subgroup $\Gamma$ in a noncommutative isotropic absolutely almost simple group $G$ over a global function field is one less than the sum of the local ranks of $G$ 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.

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