The Fattened Davis Complex and the Weighted L^2-(Co)Homology of Coxeter Groups

Abstract

Associated to a Coxeter system $(W,S)$ there is a contractible simplicial complex $\Sigma$ called the Davis complex on which $W$ acts properly and cocompactly by reflections. Given a positive real multiparameter $\Q$ indexed by $S$, one can define the weighted $L^2$--(co)homology groups of $\Sigma$ and associate to them a nonnegative real number called the weighted $L^2$--Betti number. Unfortunately, not much is known about the behavior of these groups when $\Q$ lies outside a certain restricted range, and weighted $L^2$--Betti numbers have proven difficult to compute. We propose a program to compute the weighted $L^2$--(co)homology of $\Sigma$ by introducing a thickened version of this complex which we call the fattened Davis complex. A salient feature of this complex is that our construction produces a homology manifold with boundary possessing $\Sigma$ as a $W$--equivariant retract. This allows us to use many standard algebraic topology tools such as Poincar\'{e} duality for computing the $L^2$--(co)homology of $\Sigma$, and we successfully perform computations for many examples of Coxeter groups. Within the spectrum of weighted $L^2$--(co)homology there is a conjecture of interest called the Weighted Singer Conjecture. The conjecture claims that if $\Sigma$ is an $n$--manifold (equivalently, the nerve of the corresponding Coxeter group is an $(n-1)$--sphere), then the weighted $L^2$--(co)homology groups of $\Sigma$ vanish above dimension $\frac{n}{2}$ whenever $\Q\leq\mathbf{1}$. We present a proof of the conjecture in dimension three that encompasses all but nine Coxeter groups. Then, under some restrictions on the nerve of the Coxeter group, we obtain partial results whenever $n=4$ (in particular, the conjecture holds for $n=4$ if the nerve of the corresponding Coxeter group is a flag complex). We also prove a version of this conjecture in dimensions three and four whenever $\Sigma$ is a manifold with (nonempty) boundary, and then extend our results in dimension four to prove a general version of the conjecture for the case where the nerve of the Coxeter group is assumed to be a flag triangulation of a $3$--manifold.