Thin buildings

Abstract

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 Poincaré duality allows to deduce that the $L^2$–cohomology of a building with large thickness is concentrated in the top dimension.