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.Comment: This is the version published by Geometry & Topology on 24 May 200

Cohomology of Coxeter groups with group ring coefficients: II

For any Coxeter group W, we define a filtration of H^*(W;ZW) by W-submodules and then compute the associated graded terms. More generally, if U is a CW complex on which W acts as a reflection group we compute the associated graded terms for H_*(U) and, in the case where the action is proper and cocompact, for H^*_c(U).Comment: This is the version published by Algebraic & Geometric Topology on 15 September 200

Weighted L2L^2-cohomology of Coxeter groups

Given a Coxeter system $(W,S)$ and a positive real multiparameter \bq, we study the "weighted $L^2$-cohomology groups," of a certain simplicial complex $\Sigma$ associated to $(W,S)$. These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to $(W,S)$ and the multiparameter $q$. They have a "von Neumann dimension" with respect to the associated "Hecke - von Neumann algebra," $N_q$. The dimension of the $i^th$ cohomology group is denoted $b^i_q(\Sigma)$. It is a nonnegative real number which varies continuously with $q$. When $q$ is integral, the $b^i_q(\Sigma)$ are the usual $L^2$-Betti numbers of buildings of type $(W,S)$ and thickness $q$. For a certain range of $q$, we calculate these cohomology groups as modules over $N_q$ and obtain explicit formulas for the $b^i_q(\Sigma)$. The range of $q$ for which our calculations are valid depends on the region of convergence of the growth series of $W$. Within this range, we also prove a Decomposition Theorem for $N_q$, analogous to a theorem of L. Solomon on the decomposition of the group algebra of a finite Coxeter group.Comment: minor change

Weighted L²-cohomology of Coxeter groups

Given a Coxeter system (W, S) and a positive real multiparameter q, we study the “weighted L²-cohomology groups,” of a certain simplicial complex Σ associated to (W, S). These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to (W, S) and the multiparameter q. They have a “von Neumann dimension” with respect to the associated “Hecke- von Neumann algebra, ” Nq. The dimension of the ith cohomology group is denoted bi q (Σ). It is a nonnegative real number which varies continuously with q. When q is integral, the bi q(Σ) are the usual L²-Betti numbers of buildings of type (W, S) and thickness q. For a certain range of q, we calculate these cohomology groups as modules over Nq and obtain explicit formulas for the bi q(Σ). The range of q for which our calculations are valid depends on the region of convergence of the growth series of W. Within this range, we also prove a Decomposition Theorem for Nq, analogous to a theorem of L. Solomon on the decomposition of the group algebra of a finite Coxeter group