The Completed Cohomology of Arithmetic Groups

The cohomology of arithmetic groups (with real coefficients) is usually understood in terms of automorphic forms. Such methods, however, fail (at least naively) to capture information about torsion classes in integral cohomology. We discuss a formalism -- completed cohomology -- which provides a way of studying the cohomology in a p-power congruence tower. We discuss (in particular) two advantages of the formalism: it allows for methods from non-commutative Iwasawa theory to be applied, and it is highly suggestive of a conjecture which predicts the precise growth of the mod-p cohomology in any arithmetic group up a p-power congruence tower, as well as more subtle conjectures regarding torsion in the cohomology of Shimura varieties.