Cohomology of buildings and of their automorphism groups

Abstract

Buildings were created by Tits. They serve as a vehicle for understanding real semi-simple groups and their p-adic analogs. These are undoubtedly the two most interesting cases, but existing constructions provide many more interesting examples, some of which are remarkably symmetric. While classical (spherical or Euclidean) buildings correspond to spherical or Euclidean reflection groups, we are mainly concerned with buildings related to other (infinite) reflection groups. The class of examples which this paper addresses is that of Kac–Moody buildings, as described by Tits [Ti]. It is a very large family, and it is precisely the great symmetry of these buildings that we are using. Our initial interest in the cohomology of buildings was that we suspected that buildings provided a rich source of Kazhdan groups. Recall that a locally compact group G is a Kazhdan group (or has Property (T)) if the trivial representation of G is isolated in the space of all irreducible unitary representations of G. Equivalently, G is a Kazhdan group if H1 ct(G, ρ) = 0: i.e., its first continuous cohomology group with coefficients in any unitary representation vanishes. In this paper we use the latter definition (for their equivalence, cf. [HV]). It turns out that not only can we give an almost definitive statement about when the automorphism group of a Kac–Moody