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

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