Statements and Dilemmas Regarding the $\ell^2$-homology of Coxeter groups

Abstract: We generalize the methods in previous work to provide a program for proving Singer's Conjecture for Coxeter systems. Specifically, we consider even Coxeter systems with nerves that are flag triangulations of $\BS^{n-1}$, $n=2k$. We prove that Singer's Conjecture in dimensions $n-2$ and $n-1$, along with the vanishing of the $\ltwo$-homology of certain subspaces called "two-letter" ruins above dimension $k+1$, imply Singer's Conjecture in dimension $n$. This is, so far, an incomplete program. The author intends this paper to serve as a reference for those inquiring about Singer's Conjecture and about even Coxeter systems.