Thin buildings
  Let X be a building of uniform thickness q+1. L^2-Betti numbers of X are reinterpreted as von-Neumann dimensions of weighted L^2-cohomology of the underlying Coxeter group. The dimension is measured with the help of the Hecke algebra. The weight depends on the thickness q. The weighted cohomology makes sense for all real positive values of q, and is computed for small q. If the Davis complex of the Coxeter group is a manifold, a version of Poincare duality allows to deduce that the L^2-cohomology of a building with large thickness is concentrated in the top dimension. 
Jan Dymara