Cohomology of buildings and finiteness properties of $\widetilde{A}_n$-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 modified to calculate the cohomology of any locally finite affine building. As an application we show that any finitely presented $\widetilde{A}_n$-group is a virtual duality group. A number of other finiteness conditions for $\widetilde{A}_n$-groups are also established.