Weighted Singer Conjecture

Let $(W,S)$ be a Coxeter system with nerve $L$, and let $\boldsymbol{q}$ be a vector of weights with coordinatewise order $\boldsymbol{q}\leq\mathbf{1}$. Suppose that $L$ is a triangulation of $S^{n-1}$, so that the Davis complex $\Sigma(W,S)$ is a contractible $n$-manifold.

Weighted Singer Conjecture. The weighted $L^2$-homology vanishes above the middle dimension:

$$L_{\boldsymbol{q}}^2H_i(\Sigma(W,S))=0\quad\text{for }i>\frac{n}{2}\text{ and }\boldsymbol{q}\leq\mathbf{1}.$$

This strengthens Singer's conjecture, which predicts concentration of the $L^2$-cohomology of a closed, aspherical $n$-manifold in the middle dimension. By weighted Poincaré duality, the assertion is equivalent to vanishing for $\boldsymbol{q}\geq\mathbf{1}$ and $i<n/2$. The general conjecture remains open.