On the geometry of global function fields, the Riemann-Roch theorem, and finiteness properties of S-arithmetic groups

Details On the geometry of global function fields, the Riemann-Roch theorem, and finiteness properties of S-arithmetic groups On the geometry of global function fields, the Riemann-Roch theorem, and finiteness properties of S-arithmetic groups

2010-08-21

arxiv.org/abs/1008.3664v2

Mathematics

Group Theory

Scientific paper

Harder's reduction theory provides filtrations of euclidean buildings that allow one to deduce cohomological and homological properties of S-arithmetic groups over global function fields. In this survey I will sketch the main points of Harder's reduction theory starting from Weil's geometry of numbers and the Riemann-Roch theorem, describe a filtration that is particularly useful for deriving finiteness properties of S-arithmetic groups, and state the rank conjecture and its partial verifications that do not restrict the cardinality of the underlying field of constants. As a motivation for further research I also state a much more general conjecture on isoperimetric properties of S-arithmetic groups over global fields (number fields or function fields).

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