Cohomology of buildings and finiteness properties of An-groups

Borel and Serre calculated the cohomology of the building associated to a reductive group and used the result to deduce that torsion-free S-arithmetic groups are duality groups. By replacing their group-theoretic arguments with proofs relying only upon the geometry of buildings, we show that Borel and Serre's approach can be modied to calculate the cohomology of any locally nite ane building. As an application we show that any nitely presented e An-group is a virtual duality group. A number of other niteness conditions for e An-groups are also established.