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

TL;DR

This paper advances the program for proving Singer's Conjecture for even Coxeter systems with specific nerve structures, linking lower-dimensional cases and subspace homologies to higher-dimensional conjectures.

Contribution

It generalizes previous methods to establish a framework connecting lower-dimensional Singer's Conjecture cases and subspace homologies for even Coxeter systems.

Findings

Proves implications of Singer's Conjecture in dimensions n-2 and n-1.

Links vanishing of certain subspace homologies to higher-dimensional conjecture.

Provides a reference framework for researchers studying Singer's Conjecture.

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.