Coxeter Groups, Ruins, and Weighted $L^2$-cohomology

Given a Coxeter system $(W,S)$ and a multiparameter $\mathbf{q}$ of real numbers indexed by $S$, one can define the weighted $L^2$-cohomology groups and associate to them a nonnegative real number called the weighted $L^2$-Betti number. We show that for ranges of $\mathbf{q}$ depending on certain subgroups of $W$, the weighted $L^2$-cohomology groups of $W$ are concentrated in low dimensions. We then prove new vanishing results for the weighted $L^2$-cohomology of certain low-dimensional Coxeter groups. Our arguments rely on computing the $L^2$-cohomology of certain complexes called ruins, as well as the resolution of the Strong Atiyah Conjecture for hyperbolic Coxeter groups. We conclude by extending to the weighted setting the computations of Davis and Okun for the case where the nerve of a right-angled Coxeter group is the barycentric subdivision of a PL-cellulation of an $(n-1)$-manifold with $n=6,8$.