This article consists of two parts. First, we propose a program to compute the weighted
–(co)homology
of the Davis complex by considering a thickened version of this complex.
The program proves especially successful provided that the weighted
–(co)homology
of certain infinite special subgroups of the corresponding Coxeter group vanishes
in low dimensions. We then use our complex to perform computations for
many examples of Coxeter groups. Second, we prove the weighted Singer
conjecture for Coxeter groups in dimension three under the assumption that
the nerve of the Coxeter group is not dual to a hyperbolic simplex, and
in dimension four under the assumption that the nerve is a flag complex.
We then prove a general version of the conjecture in dimension four where
the nerve of the Coxeter group is assumed to be a flag triangulation of a
–manifold.
Keywords
weighted L^2 cohomology, fattened Davis complex, Coxeter
groups, Singer conjecture
