  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 $\mathbf{q}$,
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. Not
much is known about the behavior of these groups when $\mathbf{q}$ lies outside
a certain restricted range, and weighted $L^2$-Betti numbers have proven
difficult to compute. In this article we propose a program to compute the
weighted $L^2$-(co)homology of $\Sigma$ by considering a thickened version of
this complex. The program proves especially successful provided that the
weighted $L^2$-(co)homology of certain infinite special subgroups of $W$
vanishes in low dimensions. We then use our complex to perform computations for
many examples of Coxeter groups, in most cases providing explicit formulas for
the weighted $L^2$-Betti numbers.
